The Four-Vectors (4-Vectors) and Lorentz Invariants of Special
Relativistic (SR) theory are fundamental entities that accurately,
precisely, and beautifully describe the physical properties of the world
around us. While it is known that SR is not the "deepest" theory, it is
valid for the majority of the known universe. It is believed to apply to
all forms of interaction, including that of fundamental particles and
quantum effects, with the only exception being that of large-scale
gravitational phenomena, where spacetime itself is significantly curved,
for which General Relativity (GR) is required. The SR 4-vector notation is
one of the most powerful tools in understanding the physics of the
universe, as it simplifies a great many of the physical relations.
This remains a work in progress.
Please, send comments/corrections to
John
A vector is a mathematical object which has both magnitude and direction.
It is a powerful tool for describing physical phenomena. A common 3-vector
is the velocity vector (vx, vy, vz),
which tells you in what direction and how fast something is moving.
One might use the (x, y, z) coordinates to write down the velocity vector
of some object in the laboratory. That would be an example of a
rectilinear coordinate system. Another person might use a coordinate
system that is rotated wrt. the first observer, with components (y', x',
z). The same vector might also be described by the (r, θ, φ) spherical
coordinate system. Within a given coordinate system, each component is
typically orthogonal to each other component. While these different
coordinate systems will usually have different numbers in the vector
3-tuple, they nevertheless describe the same vector and the same physics.
Hence, the vector can be considered the "primary" element, which is then
described by any number of different coordinate systems, which simply
represent one point-of-view of the given vector.
The extension of 3-vectors to that of 4-vectors is a simple idea. Let's
imagine some <event> in spacetime. The location of the <event> in the
Newtonian world would be it's 3-position (x,y,z), and the time (t) at
which it occurs. In the Newtonian world these are totally separate ideas.
SR unites them into a single object. The location of the <event> in the SR
world would be it's 4-position (ct,x,y,z). All that we have done is to
insert the time (t) into the vector as another component. The factor of
(c) is put with it to make the dimensional units work out right. (
[m/s]*[s] = [m]). So, each component now has overall units of [m] for this
4-vector. This rather simple idea, combined with the postulates of SR,
lead to some amazing results and elegant simplifications of physical
concepts...
There are two postulates which lead to all of SR-Special Relativity:
(1) The laws of physics are the same for all inertial reference frames.
This means the form of the physical laws should not change for different
inertial observers. This can be also restated as "All inertial observers
measure the same interval magnitude between two <event>'s". I say it this way
because all of experimental physics ultimately boils down to taking a
measurement. Yet another way to say it is that the result of any Poincaré Transformation (includes Lorentz Transformations) leaves the Invariant Interval unchanged.
(2) The speed of light (c) in vacuum is the same for all inertial
reference frames. This is the result of millions of independent
measurements, all confirming the same observation. This differentiates SR from classical Galilean invariance, which also obeys the first postulate.
4-vectors are tensorial entities which display Poincaré Invariance,
meaning they leave invariant the differential squared interval (ds)2
= (cdt)2-dx2-dy2-dz2. A
consequence of this invariant measurement is that any physical equation
which is written in Poincaré Invariant form is automatically valid for
any inertial reference frame, regardless of how coordinate systems are
arranged. Transformations which leave these vectors unchanged include
fixed translations through space and/or time, rotations through space, and
boosts (coordinate systems moving with constant velocity) through
spacetime. Since 4-vectors are tensors, and Poincaré Invariant, they can
be used to describe and explain the physical properties that are observed
in nature. Although the vector components may change from one reference
frame to another, the 4-vector itself is an invariant, meaning that it
gives valid physical information for all inertial observers. Likewise, the
scalar products of Lorentz Invariant 4-vectors are themselves invariant
quantities, known as Lorentz Scalars. Lorentz Invariance is a
subset of the more general Poincaré Invariance.
The reason that I really like 4-vectors and their notation is that they beautifully and
elegantly display the relations between lots of different physical
properties. They also devolve very nicely into the limiting/approximate
Newtonian cases of {|v|<<c} by letting {γ →1 and dγ/dt →0}. SR
tells us that several different physical properties are actually dual
aspects of the same thing, with the only real difference being one's point
of view, or reference frame. Examples include: (Time , Space), (Energy ,
Momentum), (Power , Force), (Frequency , WaveNumber), (ChargeDensity ,
CurrentDensity), (EM-ScalarPotential , EM-VectorPotential), (Time Differential,
Spatial Gradient), etc. Also, things are even more related than that. The
4-Momentum is just a constant times 4-Velocity. The 4-WaveVector is just a
constant times 4-Momentum. In addition, the very important
conservation/continuity equations seem to just fall out of the notation.
The universe apparently has some simple laws which can be easy to write
down by using a little math and a super notation.
QM = Quantum Mechanics SR = Special Relativity
SM = Statistical Mechanics GR = General Relativity
length/time |
[m] meter <*> [s] second |
Count of the quantity of separation or distance; Location of <event>'s in spacetime |
mass |
[kg] kilogram |
Count of the quantity of matter; (the "stuff" at an <event>) |
EMcharge |
[C] Coulomb |
Count of the quantity of electric charge; the Coulomb is more
fundamental than the Ampere |
temperature |
[ºK] Kelvin |
Count of the quantity of heat (statistical) |
Velocity
vgroup or v or u: v
= cβ = c2/vp = group velocity = <event> velocity= [0..c], {u is historically used in SR
notation}
vphase or vp: vp = c/β = c2/v = phase velocity = celerity
= [c..Infinity]
Minkowski Flat (Pseudo-Euclidian) Spacetime Metric:
ημν = ημν = SR gμν = Diag[+1,-1,-1,-1]
Dimensionless SR Factors:
β = (v/c) = (vgroup/c) = (c/vphase)=
[0..1]: Relativistic Beta factor, the fraction of the speed of light c
β = (u/c): Vector form of Beta factor, u is the velocity
γ[u] = dt/dτ: Lorentz Gamma Scaling Factor (Relativistic Gamma factor)
γ = (1 / √[1-(v/c)2] ) = (1 / √[1-(u·u/c2)]
): Lorentz Gamma Scaling Factor (~1 for v<<c), (>>1 for v~c)
γ = (1 / √[1-β2] ) = (1 / √[1-β·β] ): Lorentz Gamma
Scaling Factor (~1 for β<<1), (>>1 for β~1)
γ = (1 / √[1-β2] ) = 1/√[(1+|β|)/(1-|β|)] : Useful for Doppler Shift Eqns
φ = Ln[γ(1+ β)] ~ Atanh[β]: BoostParameter/Rapidity (which remains
additive in SR, unlike v)
eφ = γ(1+β) = √[(1+β)/(1-β)]
β = Tanh[φ], γ = Cosh[φ], γβ = Sinh[φ], φ = Rapidity (which remains
strictly additive in SR, unlike v)
D = 1 / [γ(1 - β Cos[θ] )] = 1 / [γ(1 - β·n )]: Relativistic
Doppler Factor (sometimes called a relativistic beaming factor)
D+ = γ(1 + β Cos[θ] ): Forward jet Doppler shift
D- = γ(1 - β Cos[θ] ): Counter-jet Doppler shift
Temporal Factors:
τ = t / γ : Proper Time = Rest Time (time as measured in a frame at rest)
dτ = dt / γ : Differential of Proper Time
d/dτ = γ d/dt = U·∂ : Differential wrt Proper Time
Useful SR Formulas:
V·V = Vo·Vo : Invariant interval is often easier to
calculate in rest frame coordinates
√[1+x] ~ (1+x/2) for x ~ 0 : Math relation often used to simplify
Relativistic eqns. to Newtonian eqns.
1/√[1+x] ~ (1-x/2) for x ~ 0 : Math relation sometimes used to simplify
Relativistic eqns. to Newtonian eqns.
δuv = Delta function = (1 if u = v, 0 if u ≠ v)
γ = (1 / √[1-(u·u/c2)]) = c/√[c2-v2]
= c/√[c2-u·u]
γ2 = c2/(c2-v2) = c2/(c2-u·u)
= 1/(1- β2)
c2/γ2 = (c2-v2)
v γ = c √[γ2-1]
β γ = √[γ2-1]
(1-β2)γ2 = 1
(1-β2) = 1/γ2
β2γ2 = γ2-1
β2γ2 +1 = γ2
β2γ = (γ-1/γ)
c2 dγ = γ3 v dv
d(γ v) = c2 dγ / v = γ3 dv
dγ = γ3 v dv / c2 = γ3 β dβ
dβ = dv / c
dγ/dv = γ3 v / c2
d(γ-1)/dv = - γ v / c2
γ' = dγ/dt = (γ3 v dv/dt)/c2 = (γ3 u·a)/c2
= (u·ar)/c2
γ'' = dγ'/dt = d2γ/dt2 = (γ3/c2)*[(3γ2/c2)(u·a)2
+ (u'·a) + (u·a')]
u2 = u2
u·u' = uu' = ua
(|u x a|)2 + (u·a)2 = u2a2
sin2 + cos2
= 1
(∇·∇)[1/r]
= Δ[1/r] = -4πδ3(r)
(∇·∇)[1/|r-r'|]
= Δ[1/|r-r'|] = -4πδ3(r-r') Green's function for Poisson's Eqn
**NOTE**
All results below use the time-positive SR Minkowski Metric ημν =
Diag[+1,-1,-1,-1].
If you wish to do GR, with other metrics gμν, then some results
below may need GR modification, such as the GR √[-g] for whichever metric
you are using...
You have been warned.
There are several different SR notations available that are,
mathematically speaking, equivalent.
However, some are easier to employ than others. I have used that one which
seems the most practical and least error-prone.
If you mix notations, you will get errors! Always check notation
conventions in SR & 4-Vector references, they are all relative ;-)
Minkowski SR Metric (time 0-component positive), for which {in Cartesian coordinates} ημν
= ημν = SR gμν = SR gμν = Diag[+1,-1] =
Diag[+1,-1,-1,-1]
Signature[ημν] = -2
Generic 4-Vector Definition:
A = Au
= (a0,a) = (a0,a1,a2,a3) time (a0) in the 0th coord. (
some alternate notations use time as a4 )
Specific coordinate system representations:
A = Au
= (a0,a) = (a0,a1,a2,a3) => (at,ax,ay,az) {for rectangular/Cartesian coords}
A = Au
= (a0,a) = (a0,a1,a2,a3) => (at,ar,aθ,az) {for cylindrical coords}
A = Au
= (a0,a) = (a0,a1,a2,a3) => (at,ar,aθ,aφ) {for spherical coords}
Note that the superscripted variables are not exponents, they are upper tensor indices
Intervals:
TimeLike/Temporal (+ interval) = 0 coordinate ( some alternate notations
use time as - interval and space as the + interval)
LightLike/Null (0 interval)
SpaceLike/Spatial (- interval) = 1,2,3 coordinates
Temporal Components: Future(+), Now(0), Past(-)
4-Vector Name: always references the "Spatial" 3-vector component
(basically trying to extend the Newtonian 3-vector to SR 4-vector)
4-Vector Magnitude: usually references the "temporal" scalar
component (because many vectors in the rest frame only have a temporal
component)
4-Vector Tensor Indices: I use the convention of [Greek symbols dim{0..3} = time+space], [Latin symbols dim{1..3} = only space]
4-Vector Symbols: A = Aμ = (a0,a)
= (a0,ai) = (a0,a1+a2+a3) = (a0,a1,a2,a3),
where the raised index indicates dimension, not exponent
4-Vector Definition: A = Aμ , always references the upper tensor index unless otherwise noted
4-Vector c-Factor: almost always applied to "Temporal" scalar component, as
necessary to give consistent dimensional units for all vector components
(a0,a1,a2,a3) <==>
(ct,x,y,z) = (ct,x)
*Note* c-Factor can be on the top, as ( ct , x , y , z ) = [m], or on
bottom, as ( E/c , px , py , pz ) = [kg m
s-1]. It all depends on the particular 4-Vector and its components.
*Note* P = (E/c,p) = (mc,p);
the 4-Momentum is a good case showing top or bottom, with E = mc2
4-Vector Computer HTML Representation:
SR 4-vector = {BOLD UPPERCASE} = A
time scalar component = {regular lowercase} = a0
space 3-vector component = {bold lowercase} = a = ai = (a1,a2,a3)
Contraction & Dilation Relativistic Component: v --> vo
in a rest-frame, typically v = γ vo (dilation) or v = (1/γ) vo
(contraction)
eg.
t = γ to (time dilation) - pertains to temporal separation
between two <event>'s
L = (1/γ) Lo (length contraction) - pertains to the spatial
separation between two parallel world lines
Generally, time-like quantities get dilated, space-like quantities get
contracted by motion
Also, I typically denote "at-rest" invariant quantities with a "naught",
or "o", i.e.:
Lo (invariant rest length = proper length), relativistic length
L = (1/γ) Lo
Vo (invariant rest volume), relativistic volume V = (1/γ) Vo
mo (invariant rest mass), relativistic mass m = γ mo
Eo (invariant rest energy), relativistic energy E = γ Eo
ωo (invariant rest ang-frequency), relativistic ang-frequency ω
= γ ωo
ρo (invariant rest charge-density), relativistic charge-density
ρ = γ ρo
no (invariant rest number-density), relativistic number-density
n = γ no
to (invariant rest time = proper time), relativistic time
t = γ to = γ τ
etc.
This avoids the confusion of some texts which use just "m" as invariant
mass, or just "ρ" as invariant charge-density.
It also helps to avoid confusion such as:
If the mass m of an object increases with velocity, wouldn't it
have be a black hole in some reference frames (near c), since the mass
increases with velocity.
Answer - no. The rest mass mo does not
change. The relativistic mass is simply an "apparent" mass, how the
object is velocity-related to an observer, not how much "stuff" is in
it...
The apparent increase is fully due to the gamma factor( γ ), which is
simply an indication of the amount of relative motion.
Imaginary unit: ( i ) used only for QM phenomena, not for SR frame
transformations or metric. To follow up on a quote from MTW " ict was put to
the sword ".
This allows all the purely SR stuff to use only real numbers.
Imaginary/complex stuff apparently only enters the scene via QM.
( some alternate notations use the imaginary unit ( i ) in the
components/frame transformations/metric )
So, in summary, this notation allows:
easy recovery of Newtonian cases by allowing (γ→1, dγ→0) when
(|v|<<c)
easy separation of SR vs Newtonian concepts, with the Newtonian 3-vector (a)
extending naturally into the SR 4-vector (A)
easy separation of SR vs QM concepts, no ict's -- ( i ) only enters into
QM concepts, such as Photon Polarization, Quantum Probability Current,
etc.
easy separation of relativistic quantities vs. invariant quantities, E = γ
Eo
reduction in number of minus signs (-), eg. U·U = c2, P·P
= (moc)2: the square magnitudes of velocity,
momentum, wavevector, and other velocity-based vectors are positive
The main assumption of SR, or GR for that matter, is that the structure of spacetime is described by a metric gμν. A metric tells how the spacetime is put together, or how distances are measured within the spacetime. These distances are known as intervals. In GR, the metric may take a number of different values, depending on various circumstances which determine its curvature. We are interested in the flat/pseudo-Euclidean spacetime of SR, also known as the Minkowski Metric, for which gμν => ημν = ημν = Diag[+1,-1,-1,-1].
"Flat" SpaceTime |
t |
x |
y |
z |
t |
1 |
0 |
0 |
0 |
x |
0 |
-1 |
0 |
0 |
y |
0 |
0 |
-1 |
0 |
z |
0 |
0 |
0 |
-1 |
ημν = SR gμν = SR gμν =
DiagonalMatrix[1,-1,-1,-1]: Minkowski Spacetime Metric-the "flat"
spacetime of SR {in Cartesion coordinates}
A = Aμ = (a0,ai) = (a0,a1,a2,a3) => (at,ax,ay,az)
:
Typical
SR 4-vector (using all upper indices)
Aμ = (a0,ai) = (a0,a1,a2,a3) => (at,ax,ay,az) :
Typical SR 4-covector (using all lower indices)
We can always get the alternate form by applying the Minkowski Metric Tensor: Aμ
= ημνAν and Aμ = ημν Aν
Basically, this has the effect of putting a
minus sign on the space component(s)
A = Aμ = (a0,ai) = (a0,a1,a2,a3) = (a0, a1, a2, a3) = (a0,a)
:Typical
SR 4-vector (all upper indices)
Aμ = (a0,ai) = (a0,a1,a2,a3) = (a0,-a1,-a2,-a3) = (a0,-a):
Typical SR 4-covector (converted using Metric Tensor)
It is occasionally convenient to choose a particular basis to simplify component calculations
Typical bases are rectangular, cylindrical, spherical
A = Aμ = (a0,ai) = (a0,a1,a2,a3) = (a0,a) = (a0,a1,a2,a3) = (a0,a) => (at,ax,ay,az)
:Typical SR 4-vector (choosing the rectangular basis)
Aμ = (a0,ai) = (a0,a1,a2,a3) = (a0,-a) = (a0,-a1,-a2,-a3) = (a0,-a) => (at,ax,ay,az) = (at,-ax,-ay,-az)
:
Typical SR 4-covector (choosing the rectangular basis)
B = Bμ = (b0,b1,b2,b3) = (b0,b) = (bt,bx,by,bz)
:
Another
typical SR 4-vector
A·B = ημν Aμ Bν = Aν Bν
= Aμ Bμ = +a0b0-a1b1-a2b2-a3b3
= (+a0b0-a·b): The Scalar Product or Invariant Product relation,
used to make SR invariants
c(A + B) = (cA + cB) scalar multiplication
A·A = A2 = (+a02 - a12
- a22 - a32) = (+a02
- a·a) magnitude squared, which can be { - , 0 , + }
A = |A| = √|A2| >= 0 absolute magnitude or
length, which can be { 0 , + }
A·B = B·A commutative, with the exception of the (∂)
operator, since it only acts to the right
A·(B + C) = A·B + A·C distributive
d(A·B) = d(A)·B + A·d(B)
differentiation
B = d(A)/dθ, where θ is a Lorentz Scalar Invariant
Aproj = (A·B)/(B·B) B =
Projection of A along B
A|| = (A·B)/(B·B) B =
Component of A parallel to B
A⊥ = A - A||
A⊥ =A - (A·B)/(B·B) B =
Component of A perpendicular to B
Let A = (a0,a) be a general 4-Vector and T = U/c = γ(c, u)/c = γ(1, β) =
(γ, γβ) be the unit-temporal 4-Vector
Then...
(A·T) = (a0,a)·γ(1, β) = γ(a0-a·β) = 1(a0o-a·0) = a0o
which is a Lorentz Invariant way to get (a0o), the rest temporal component of A
Let A = (a0,a) be a general 4-Vector and S = γβn(β·n, n) be the unit-spatial 4-Vector
Then...
-(A·S) = -(a0,a)·γβn(β·n, n) = -γβn(a0β·n, a·n) = -1(a0a·0 - a·n) = a·n
which is a Lorentz Invariant way to get (a·n), the rest spatial component of A along the n direction
If A is a time-like vector, then you can also do the following:
(A·T)2 - (A·A) = (a0o)2 - (a0a0-a·a) = (a0o)2 - (a0)2 + (a·a) = (a·a)
Sqrt[(A·T)2 - (A·A)] = Sqrt[(a·a)] = |a|
which is the Lorentz Invariant way to get the magnitude |a|, the spatial component magnitude of a time-like A
Also:
(A·A) = (a0)2 - (a·a) = invariant = (a0o)2 ,where (a0o) is the Lorentz Scalar Invariant "temporal rest value" for those vectors that can be at rest, and just the invariant for others
(a0)2 = (a0o)2 + (a·a)
Consider the f
ollowing identity:
γ = 1/Sqrt[1-β2] : γ2 = 1 + γ2β2
Multiply by the square of a Lorentz Scalar Invariant (a0o):
γ2(a0o)2 = (a0o)2 + γ2(a0o)2β2
Compare Terms:
(a0)2 = (a0o)2 + (a·a)
γ2(a0o)2 = (a0o)2 + γ2(a0o)2β2
Notice the following correspondences:
a0 = γa0o, The temporal component is the Gamma Factor times the rest value
a = γa0oβ = a0β, The spatial component is temporal component times t
he Beta Factor
Try it with the 4-Momentum P = (E/c,p)
(E/c) = γ(Eo/c) or E = γEo
p = γ(Eo/c)β = γ(Eo/c)(v/c) = γ(Eov/c2) = γ(mov) = γmov
p = (E/c)β = (mc)β = (mv) = γmov
If β=1 then |p| = E/c or E = |p|c, which is correct for photons
It also shows that as β → 1: γ → Infinity, (a0o) → 0, a0 = (γa0o) → some finite value
Special Relativity is interesting in that it is one area of physics where {Infinity*Zero = Finite Value} for certain variables.
eg. E = γEo
For photons, the rest energy Eo = 0, the gamma factor γ = Infinity, but the overall energy of photon E = finite value for a given obs
erver.
One can do this with any SR rest value variable. Always pair γ={1..Infinity} with ao={large..0} to get a finite value a = γao={something finite}
These correspondences can also be generated by letting A = LorentzScalar (a0o) * TemporalUnit 4-Vector T
A = (a0o)T = (a0o)γ(1,β) = γ(a0o)(1,β) = (γa0o)(1,β) = (a0)(1,β) = (a0,a0β) = (a0,a)
If β→0 , then A --> (a0,a0β) → (a0o,0), which has (A·A) = (a0o)2 as expected
If β→1 , then A --> (a0,a0n) → a0(1,n), which has (A·A) = (a0)2 (12 - n·n) = 0 as expected
==========
if A·A = const
then
dA1dA2dA3 / A0
dA0dA2dA3 / A1
dA0dA1dA3 / A2
dA0dA1dA2 / A3
are all scalar invariants
from Jacobian derivation
============
if Aμ dXμ = invariant for any dXμ, then Aμ
is a 4-vector
ημν Λμα Λνβ = ηαβ This is basically the reason why the Scalar Product relation gives invariants
ημν (A'μ)(B'ν) = ημν (ΛμαAα)(ΛνβBβ) = ημν (ΛμαΛνβ)(Aα)(Bβ) = ηαβ (Aα)(Bβ) = ηαβ (AαBβ)
Thus, A'·B' = A·B, where the primed 4-vectors are just Lorentz Transformed versions of the unprimed ones
Λαμ Λμβ = dαβ
A'μ = Λμν Aν: Lorentz
Transform (Transformation tensor which gives relations between alternate
boosted inertial reference frames)
Λμ'ν = (∂Xμ'/∂Xν)
Λμν = (for x-boost)
γ |
-(vx/c)γ |
0 |
0 |
-(vx/c)γ |
γ |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
or
γ |
-βxγ |
0 |
0 |
-βxγ |
γ |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
General Lorentz Transformation
Λμν = (for n-boost)
γ |
-βxγ |
-βyγ |
-βzγ |
-βxγ |
1+(γ-1)(βx/β)2 |
( γ-1)(βxβy)/(β)2 |
( γ-1)(βxβz)/(β)2 |
-βyγ |
( γ-1)(βyβx)/(β)2 |
1+( γ-1)(βy/β)2 |
( γ-1)(βyβz)/(β)2 |
-βzγ |
( γ-1)(βzβx)/(β)2 |
( γ-1)(βzβy)/(β)2 |
1+( γ-1)(βz/β)2 |
Λuv = (for x-boost, y & z unchanged)
Cosh[φ] |
-Sinh[φ] |
0 |
0 |
-Sinh[φ] |
Cosh[φ] |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
Rz = (for x-y rotation about z-axis, t & z unchanged)
1 |
0 |
0 |
0 |
0 |
Cos[φ] |
-Sin[φ] |
0 |
0 |
Sin[φ] |
Cos[φ] |
0 |
0 |
0 |
0 |
1 |
A few 4-vectors are known to have complex components. The Polarization
4-vector and ProbabilityCurrent 4-vector are a couple of these.
It will be assumed that all physical 4-vectors may potentially be complex,
although, as far as I know, these only come into play via QM...
i = √[-1] :Imaginary Unit
e0: Unit vector in the temporal direction
(typically not used since the temporal unit is always considered a scalar)
e1, e2, e3
:Unit Vectors in the spatial x, y, z directions (used instead of i,
j, k so that there is no confusion with the imaginary unit
i)
Note that for the following 4-vectors, the superscript is the tensor
index, not exponentiation.
A = (a0c + a1c e1+
a2c e2+ a3c
e3): Complex 4-vector has complex components, 1
along time and 3 along space
Scalar[A] = a0c: Just the time component
Vector[A] = a1c e1
+ a2c e2 + a3c
e3: Just the spatial components
A = Scalar[A] + Vector[A]
A = ( (a0r + a0i
) + (a1r + a1i
) e1 + (a2r
+ a2i ) e2 +
(a3r + a3i ) e3
): Complex 4-vector has real + imaginary
components, 1 each along time and 3 each along space
Re[A] = ( (a0r
) + (a1r ) e1
+ (a2r ) e2
+ (a3r ) e3
): Only the real components
Im[A] = ( (a0i
) + (a1i ) e1
+ (a2i ) e2
+ (a3i ) e3
): Only the imaginary components
A = Re[A] + i Im[A]
A = (a0r + i a0i,ar
+ i ai) : Standard 4-vector
A* = (a0r - i a0i,ar
- i ai): Complex conjugate 4-vector, just changes the
sign of the imaginary component
A = (a0r + i a0i,ar
+ i ai) : A* = (a0r
- i a0i,ar - i ai)
B = (b0r + i b0i,br
+ i bi) : B* = (b0r
- i b0i,br - i bi)
A·B = [( a0r b0r
- ar·br ) - ( a0i
b0i - ai·bi )] + i [( a0r
b0i - ar·bi ) +
( a0i b0r - ai·br
)] : General scalar product
A·A = [( a0r2
- ar·ar ) - ( a0i2
- ai·ai )]
+ 2i [( a0r a0i
- ar·ai )]
= |A|2 : Scalar product of 4-vector with itself gives
the magnitude squared
A·A* = [( a0r2
+ a0i2 ) - ( ar·ar
+ ai·ai )]
= Re[A·A*]: Scalar product of 4-vector with its complex
conjugate is Real, thus Im[A·A*] = 0
∂·B = [( ∂/c∂tr b0r
+ ∇r·br ) - ( ∂/c∂ti b0i
+ ∇i·bi )]
+ i [( ∂/c∂tr b0i
+ ∇r·bi ) + ( ∂/c∂ti b0r
+ ∇i·br )]
= [( ∂/c∂tr b0r
+ ∇r·br ) - ( ∂/c∂ti b0i
+ ∇i·bi )]
= Re[∂·B]
The 4-Divergence of a Complex 4-Vector is Real, assuming that:
The real gradient acts only on real spaces & the imaginary gradient
acts only on imaginary spaces, thus Im[∂·B] = 0
I believe this is due to the physical functions being complex analytic
functions.
i = √[-1] :Imaginary Unit
π = 3.14159265358979... :Circular Const
c = Speed of Light Const = 1/√[εoμo] ~ 2.99729x108
[m/s]
h = Planck's Constant - relates particle to wave - Action constant
ћ = (h/2π) = Planck's Reduced Const , aka. Dirac's Const - same idea as
transforming between cycles and radians for angles
In essence, the reduced Planck constant is a conversion factor between
phase (in radians) and action (in joule-seconds)
kB = Boltzmann's Const ~ 1.3806504(24)×10−23 [J/ ºK]
relates temperature to energy
mo = Rest Mass Const (varies with particle type)
q = Electric Charge Const (varies with particle type)
Note:
I do not set various fundamental physical constants
to dimensionless unity, (i.e. c = h = G = kB = 1).
While doing so may make the mathematics/geometry a bit easier, it
ultimately obscures the physics.
While pure 4-Vectors may be Math, SR 4-Vectors is Physics. I prefer to
keep the dimensional units.
Also, it is much easier to set them to unity in a final formula than to
figure out where they go later if you need them.
4-Vector Name |
4-Vector Components |
Units (mksC) - Description |
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4-Displacement |
ΔR = (cΔt, Δr) |
[m], Δt = Temporal Displacement, Δr
or Δx = Spatial Displacement, (Finite Differences) |
4-Differential |
dR = (cdt, dr) |
[m], dt = Temporal Differential, dr or dx = Spatial Differential, (Infinitesimals) |
4-Gradient |
The tensor gradient (one-form) is technically defined as |
[m-1], ∂ is the partial
derivative, ∇ => (∂/∂x i + ∂/∂y j +
∂/∂z k) is the gradient operator |
4-MomentumGradient |
∂P = ∂/∂Pμ = (c∂E, -∇p) |
[kg-1 m-1 s], ∂E is the
momentum-space partial derivative, ∇p = momentum-space gradient |
4-WaveGradient |
∂K = ∂/∂Kμ = (c∂ω, -∇k) |
[m], ∂ω is the wave-space partial derivative, ∇k = wave-space gradient (1/ћ)∂K[P·U] |
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4-Position |
R = Rμ = (ct, r), eg. radial coords |
[m], t = Time (temporal), r or x
= 3-Position (spatial) |
4-Velocity |
U = dR/dτ |
[m s-1], γ = relativistic
factor, ur = Relativistic
3-Velocity, u = dr/dt = Newtonian 3-Velocity |
4-Acceleration |
A = dU/dτ |
[m s-2], ar
= Relativistic 3-Acceleration, a = du/dt =
Newtonian 3-Acceleration ar = (γur)' = γ' ur + γ ur' = γ' u + γ a = (γ3/c2)(u·a) u + γ a a = du/dt = u' γ' = dγ/dt = (γ3/c2)(u·a) = (u·ar)/c2 Interesting note: The temporal component has units of frequency, before the c factor, and is given by γ(dγ/dt)=γ(γ') γ(c γ')=γ(u·ar)/c γ'=(u·ar)/c2 4-Spin also has a temporal component in this form, given by u·s/c I now wonder if all 4-vectors which are tangent to the worldline possess this "cyclic" feature... |
4-Jerk |
J = dA/dτ |
[m s-3], jr
= Relativistic 3-Jerk, j = da/dt = Newtonian
3-Jerk |
4-Snap |
S = dJ/dτ |
[m s-4], sr
= Relativistic 3-Snap, s = dj/dt = Newtonian
3-Snap |
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4-Momentum |
P = moU = (Eo/c2)U
= ћK |
[kg m s-1], E = Energy, pr
= Relativistic 3-Momentum, p = mdr/dt = Newtonian
3-Momentum |
4-MomentumDensity |
G = (u/c, g) = (pmc, g)
= po_m γ(c, u) |
[kg m-2 s-1], u
= EnergyDen = ne, pm = MassDen = u/c2 |
4-Force |
F = dP/dτ |
[kg m s-2], dE/dt = Power, fr
= Relativistic 3-Force, f = Newtonian 3- Force |
4-Force Density |
Fd = F/Vo or F/δVo Fd = γ(du/cdt, fdr) = dG/dτ?? |
[kg m-2 s-2], 4-Force
divided by rest volume element |
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4-WaveVector |
K = (ωo/c2)U = (mo/ћ)U = (1/ћ)P |
[rad m-1], ω = AngularFrequency
[rad/s], k = WaveNumber or WaveVector [rad/m] |
4-Frequency |
Ν = (ν, c/λ n) |
[cyc s-1], ν = ω/2π, λ = 2π/k |
4-CycWaveVector |
Kcyc = (ν/c,1/λ n) = (1/cT,1/λ n) |
[cyc m-1], ν = CyclicalFrequency
[cyc/s], λ = WaveLength [m/cyc] |
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4-NumberFlux |
N = (cn, nf) = no
γ(c, u) = n(c, u) |
[(#) m-2 s-1], no
= RestNumberDensity [#/m3], n = γno =
NumberDensity [#/m3] |
4-VolumetricFlux |
V = VoU?? |
[(m3) m-2 s-1], Vo = RestVolume |
4-ElectricCurrentDensity |
J = (cρ, j) = ρo γ(c, u) = ρ(c,
u) |
[(C) m-2 s-1], ρo
= RestElecChargeDensity [C/m3], ρ = γρo =
ElecChargeDensity |
4-MagneticCurrentDensity |
Jmag = (cρmag, jmag)
= ρo_mag γ(c, u) |
[(MagCharge) m-2 s-1],
ρo_mag = RestMagChargeDensity, ρmag = γρo_mag
= MagChargeDensity |
4-ChemicalFlux |
|
[(mol) m-2 s-1], |
4-MassFlux |
G = (u/c, g) = (cρm, g)
= ρo_m γ(c, u) = ρo_mU |
[(kg) m-2 s-1], u
= EnergyDen = ne, pm = MassDen = u/c2 |
4-PoyntingVector |
S = (cu, s) = uo γ(c, u)
= uoU = c2G |
[(J) m-2 s-1], u
= EnergyDen = ne, s = EnergyFlux = PoyntingVector = uu
= c2g = Ne |
4-EntropyFlux |
S = (cs, sf) |
[(J ºK-1) m-2 s-1], so = RestEntropyDensity = qo/T, sf
= EntropyFlux |
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4-InverseTempFlux |
β = βo U = (1/kBTo)
U |
Considered on Thermodynamic
principles |
4-MomentumTemperature |
PT = P/kB = (pt0/c,
pT) = (T/c, pT)
= ((E/kB)/c, p/kB) |
[ºK m-1 s], |
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4-Potential Flux?? |
V = (cq, qf) = q γ(c, u)
= qU?? |
Potential Flow for Velocity?? In fluid dynamics, a potential flow is described by means of a velocity potential , φ being a function of space and time. The flow velocity v is a vector field equal to the negative gradient, ∇, of the velocity potential φ: Incompressible flowIn case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity v has zero divergence: ∇·v = 0with the dot denoting the inner product. As a result, the velocity potential φ has to satisfy Laplace's equation ∇·∇ φ = 0where Δ = ∇·∇ is the Laplace operator. In this case the flow can be determined completely from its kinematics: the assumptions of irrotationality and zero divergence of the flow. Dynamics only have to be applied afterwards, if one is interested in computing pressures: for instance for flow around airfoils through the use of Bernoulli's principle. The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow: Steady flowMain article: Steady flow Incompressible flowMain article: Incompressible flow Irrotational flowMain article: Irrotational
flow VorticityMain article: Vorticity
The velocity potentialMain article: Potential flow An lamellar vector field is a synonym for an irrotational vector field.[1] The adjective "lamellar" derives from the noun "lamella", which means a thin layer. In Latin, lamella is the diminutive of lamina (but do not confuse with laminar flow). The lamellae to which "lamellar flow" refers are the surfaces of constant potential. An irrotational vector field which is also solenoidal is called a Laplacian vector field. The fundamental theorem of vector calculus states that any vector
field can be expressed as the sum of a conservative vector field
and a solenoidal field. The fundamental theorem of vector calculus states that any vector
field can be expressed as the sum of a conservative vector field
and a solenoidal field. The condition of zero divergence is
satisfied whenever a vector field v has only a vector
potential component, because the definition of the vector
potential A as: automatically results in the identity (as can be shown, for
example, using Cartesian coordinates): The converse also holds: for any solenoidal v there exists a vector potential A such that v = ∇ x A. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.) In vector calculus, a Laplacian vector field is a vector
field which is both irrotational and incompressible. If the field
is denoted as v, then it is described by the following
differential equations: Since the curl of v is zero, it follows that v
can be expressed as the gradient of a scalar potential (see
irrotational field) φ : Then, since the divergence of v is also zero, it follows from equation (1) that ∇·∇ φ = 0 which is equivalent to Therefore, the potential of a Laplacian field satisfies Laplace's
equation. In case of an incompressible flow the velocity potential satisfies the Laplace's equation. However, potential flows have also been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows. Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, and groundwater flow. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable. A velocity potential is used in fluid dynamics, when a
fluid occupies a simply-connected region and is irrotational. In
such a case, where u denotes the flow velocity of the fluid. As a
result, u can be represented as the gradient of a scalar
function Φ: Φ is known as a velocity potential for u. A velocity potential is not unique. If a is a constant then Φ + a is also a velocity potential for u. Conversely, if Ψ is a velocity potential for u then Ψ = Φ + b for some constant b. In other words, velocity potentials are unique up to a constant. Unlike a stream function, a velocity potential can exist in three-dimensional flow. |
4-HeatFlux |
Q = qU |
[(J) m-2 s-1] = [(W)
m-2], |
4-DarcyFlux |
Q = (cq, qf) = q γ(c, u)
= qU = ( c (βφ)P , qf) |
[(m3) m-2 s-1] = [m s-1], |
4-ElectricChargeFlux |
Q = (cq, qf) = q γ(c, u) = qU = ( c ρ, j) |
[(C) m-2 s-1], |
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4-SpinMomentum |
W = (w0,w) = (u·w/c,w) |
[spin-momentum], |
4-Spin |
S = (s0,s) = (u·s/c,s) |
[ J s], = [spin] Spin =
IntrinsicAngMomentum, u·s/c = component such that U·S
= 0 ===== |
4-SpinDensity | ||
4-Rotation |
Omega |
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4-Polarization |
Ε = (ε0, ε) = (ε·u/c,ε)
for a massive particle |
[1], ε = PolarizationVector **This
4-vector has complex components in QM** |
4-SpinPolarization |
In the rest frame, where K = (m,0), choose a unit
3-vector n as the quantization axis. |
I am suspecting that the s(s+1) value can be derived from the Laplacian acting on a pure radial function. From mathworld.wolfram.com, |
4-PauliMatrix |
σμ = (σ0 ,σ1 ,σ2 ,σ3) = (σ0 ,σ) = (I ,σ) Σ = Σμν = Diag[σ0,-σ]?? where Σ·Σ = -2σ0?? |
The components of this 4-vector are
actually the Pauli Spin Matrices |
4-DiracGamma | Γ = Γμ = (γ0,γ) | According to some books, not strictly a 4-vector, the Dirac Gamma
Matrices are actually matrices to represent intrinsic spin.
However, I have seen index raising/lowering Γμ = ημνΓμ Γμ Γν = ημν + (1/2)σμν σμν = [Γμ,Γν] Not sure about a Lorentz Boost |
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4-VectorPotential |
A = (φ/c, a) |
[kg m <chargetype>-1 s-1], for arbitrary field |
4-Potential ***Break with standard notation*** better to use the 4-VectorPotential | Φ = (φ,c a) | Φ = (φ,c a) where Φ = cA ***this is bad notation based on our 4-vector naming convention*** the c-factor should be in the temporal component the 4-vector name should reference the spatial component I simply include it here because it can sometimes be found in the literature |
4-VectorPotentialGrav 4-VectorPotentialGEM ** Note ** This is an approximation only For more accurate results the full GR theory must be used This is just to illustrate a formal analogy between EM and Gravitational formula's in a semi-classical limit. |
Agrav = (Φgrav/c, agrav) |
[kg m <chargetype>-1 s-1], for arbitrary field |
4-VectorPotentialMomentum |
QEM = (EEM/c, pEM) |
[kg m s-1], |
4-Potential |
ΦEM = (ΦEM,c aEM) |
[kg m2 C-1 s-2],
ΦEM = ScalarPotenialEM ,aEM
= VectorPotenialEM |
4-MomentumEM |
PT = (ET/c + qΦEM/c,
pT + qaEM) |
[kg m s-1], **Momentum including
effects of potentials** H = γmoc2 + q φEM |
4-GradientEM |
DEM = (∂/c∂t + iq/ћ ΦEM/c, -∇
+ iq/ћ aEM) |
[m-1], **Gradient including
effects of EM potentials** |
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***Position space & Momentum Space Differentials*** |
4-Differential |
dX = (cdt, dx) |
[m], dt = Temporal Differential, dx = Spatial Differential |
4-Volume Element (Flux?) |
dV = (dv0,dv) |
[m3], A vector-valued volume
element is just a 4-vector that is perpendicular to all spatial
vectors in the volume element, and has a magnitude that's
proportional to the volume. |
4-Momentum Differential |
dP = (dE/c, dp) |
[kg m s-1], dE = Temporal Momentum Differential, dp = Spatial Momentum Differential |
4-MomentumSpace |
dVp = (dvp0,dvp) |
[kg3 m3 s-3],
A
vector-valued MomentumSpace volume element is just a 4-vector
that is perpendicular to all spatial vectors in the
MomentumSpace volume element, and has a magnitude that's
proportional to the MomentumSpace volume. |
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***Special 4-Vectors*** |
4-Zero |
Zero = (0,0) = (0,0,0,0) |
[*], All components are 0 in all reference
frames, the only vector with this property |
4-Null |
Null = (a,a) = (a,an) = a(1,n) |
[*], Any 4-vector for which the temporal
component magnitude equals the spatial component magnitude |
4-Unit Temporal |
T = U/c = γ(c, u)/c = γ(1, β) =
(γ, γβ) |
[1] = dimensionless, The Unit Temporal
4-Vector |
4-Unit Null |
N = (1,n) |
[1] = dimensionless, A Unit Null 4-Vector |
4-Unit Spatial |
S = γ[βn] (n·β,n) = (γ[βn]n·β,γ[βn]
n) |
[1] = dimensionless, A Unit Spatial
4-Vector |
4-Basis Vectors |
Bt = (1,0,0,0) |
A tetrad of 4 mutually orthogonal,
unit-length, linearly-independent, basis vectors |
4-Basis Vectors |
Bn1 = √[1/2] (1,0,0,1) |
A tetrad of complex, linearly-independent,
null basis vectors |
4-Basis Vectors |
Bn1 = (1,1,0,0) |
A tetrad of real, linearly-independent,
null basis vectors |
4-ProbabilityCurrentDensity |
Jprob = (cρprob, jprob) = (iћ/2mo)(ψ*<-∂->ψ) |
[# m-2 s-1],
4-Probability Current Density is proportional to the 4-Momentum |
<Event> R |
Mass mo = ρo_mVo |
WaveAngFreq ωo |
ElecCharge q = ρoVo |
MassDensity ρo_m |
ChargeDensity ρo |
NumberDensity no |
<event> |
particle |
wave |
elec. charge |
mass |
charge |
number |
pos: R = (ct, r) |
mo at R |
ωo at R |
q at R |
ρo_m at R |
ρo at R |
no at R |
vel: U = dR/dτ |
P = moU = (Eo/c2)U |
K = (ωo/c2)U = (1/ћ)P |
Jq = qU |
G = ρo_mU = (uo/c2)U |
J = ρoU |
N = noU |
accel: A = dU/dτ |
F = dP/dτ |
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Fd = dG/dτ |
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jerk: J = dA/dτ |
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snap: S = dJ/dτ |
<Event>(SR) |
<Event>Movement |
MassEnergy |
Particle-WaveDuality |
QuantumMechanics(QM) |
SpaceTimeVariations |
R = (ct, r) |
dR/dτ = U = γ(c,u) |
U = P/mo |
P = ћ K |
*** K = i ∂ *** |
∂ = (∂/c∂t,-∇) |
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or K = (ωo/c2)U |
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d/dτ[R] = (i ћ / mo) ∂ <Event> motion ~
spacetime structure --- depends on i ћ / mo
So, the following assumptions within SR-Special Relativity lead to
QM-Quantum Mechanics:
R = (ct,r) |
Location of an <event> (i.e. a particle) within spacetime |
U = dR/dτ |
Velocity of the <event> is the derivative of <event> position wrt. Proper Time |
P = moU |
Momentum is just the Rest Mass of the particle/<event> times its velocity |
K = (1 / ћ )P |
A particle's wave vector is just the momentum divided by Dirac's constant, but uncertain by a phase factor |
∂ = -i K |
The change in spacetime corresponds to (-i) times the wave vector, whatever that means... |
R·R = (Δ s)2 = (ct)2-r·r = (ct)2-|r|2
: dR·dR = (ds)2 = (c dt)2-dr·dr
= (c dt)2-|dr|2 : Invariant Interval
U·U = c2
P·P = (moc)2
K·K = (moc / ћ)2 = (ωo/c)2
∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂2/c2∂t2-∇·∇
= -(moc / ћ)2 : Klein-Gordon Relativistic Wave Eqn.
Each relation may seem simple, but there is a lot of complexity generated
by each level.
*see QM from SR
(Quantum Mechanics derived from Special Relativity)*
This can be further explored:
∂·∂ + (moc / ћ)2 = 0
(∂·∂ + (moc/ћ)2 ) Ψ = 0, |
Ψ is a scalar Klein-Gordon eqn for massive spin-0 field |
(∂·∂ + (moc/ћ)2 ) A = 0 |
A is a 4-vector Proca eqn for massive spin-1 field |
(∂·∂) Ψ = 0 |
Ψ is a scalar Free-wave eqn for massless (mo = 0) spin-0 field |
(∂·∂) A = 0 |
A is a 4-vector Maxwell eqn for massless (mo = 0) spin-1 field, no current sources |
Momentum/Gradient Relations(Correspondences)
P = i ћ ∂ = -∂(Sact) |
∂ = (∂/c∂t,-∇) |
AEM = (0,0) *special case* |
PEM = P+qAEM = i ћ DEM |
DEM = ∂+iq/ћ AEM |
AEM = (VEM/c,aEM) |
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Relations involving the 4-Position or 4-Displacment: |
R·R = (Δs)2 =
(ct)2-r·r = (ct)2-|r|2
|
Spacetime position of an <event> wrt. an origin <event> |
dR·dR = (ds)2 |
Differential interval magnitude - the fundamental invariant
differential form |
ΔR·ΔR = (Δs)2 = (c Δt)2-Δr·Δr = (c Δt)2-|Δr|2 |
Spacetime displacement interval magnitude - used to derive SR |
∂·R = 4 |
The divergence of open spacetime is equal to the number of independent dimensions (t,x,y,z) |
K·R = -Φ = (ωt-k·r) |
Phase of a SR wave; Ψ = a E e -iK·R Photon Wave Equation (Solution to Maxwell Equation) |
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R·U = (ct,r)·γ(c,u) = γ(c2t - r·u) = cγ(ct - r·u/c) |
Part of expression used in Liénard-Wiechert potential |
ΔX·U =
(cΔt,Δx)·γ(c,u) = γ(c2Δt - Δx·u) = (c2Δto) = (c2Δτ) Let T = U/c = γ(c, u)/c = γ(1, β) = (γ, γβ) be the unit-temporal 4-Vector Then... (ΔX·T) = (cΔt,Δx)·γ(1, β) = γ(cΔt-Δx·β) = (cΔto) = (cΔτ) |
ΔX·U = 0 is an interesting 4-vector condition/definition for
simultaneity (The displacement of an external <event> is normal to either <event>'s worldline) ΔX is a displacement vector from <Event> A to <Event> B U is an observer's 4-Velocity wrt. one of the <event>'s The standard definition of simultaneity is when Δt = 0. This gives ΔX·U = -γ(Δx·u). But since we can always choose an observer rest frame we get ΔX·U = 0, which is thus the Lorentz Invariant condition/definition for simultaneity. However, since Δt is one of the components of a 4-vector, it is only true for certain classes of observers. Let's examine the cases when ΔX·U = γ(c2Δt - Δx·u) = 0 then (c2Δt - Δx·u) = 0, since γ is always >=1 then (c2Δt = Δx·u) For simultaneity, Δt = 0, therefore Δx·u = 0 if Δx = 0, then <Event> A and <Event> B are co-local, ie. at the same spatial point if u = 0, then the observer is at rest wrt. the <Event>'s A and B if Δx·u = 0, but Δx > 0 and u > 0, then the observer's motion must be perpendicular to a spatial line from <Event> A to <Event> B |
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Relations involving the 4-Velocity: |
U·U = c2 |
The magnitude of 4-Velocity is always c2 |
U1·U2
= γ[u1]γ[u2](c2-u1·u2)
= γ[urel]c2 |
Relative Gamma Factor |
A1·U1 = 0, where A is dU/dτ |
The 4-Acceleration of a given particle is always normal to its own worldline |
P1·U2 = γ[u2](E-p1·u2)
= Erel |
Relative Energy |
K1·U2 = γ[u2](ω-k1·u2)
= ωrel |
Relative Ang. Frequency |
F·U = (moA+(dmo/dτ)U)·U
= c2(dmo/dτ) = γc2(dmo/dt) |
Power Law |
U·∂ = γ(∂/∂t + u·∇) = γ d/dt = d/dτ |
Relativistic Convective (Time) Derivative, Intrinsic Derivative |
∂·U = 0 (always??) |
The General Continuity Equation, one might say the conservation
of <event> flux. |
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Relations involving the 4-Acceleration: |
A·A = -a2 = -γ4[a2 + (γ/c)2(u·a)2] |
Magnitude squared of acceleration |
U·A = 0, where A is dU/dτ |
The 4-Acceleration of a given particle is always normal to its
own worldline |
A·S | Part of the proportionality factor of a 4-Spin Vector
Fermi-Walker Transported in time Would also apply to any constant spatial 3-vector that is "attached" to a particle Since U·S = 0 then d/dτ [U·S] = 0 = d/dτ[U]·S + U·d/dτ[S] = A·S + U· d/dτ[S] U· d/dτ[S] = - A·S if we assume d/dτ[S] = (k)*U then U·d/dτ[S] = kU·U = kc2 = -A·S k = -A·S/c2 then d/dτ[S] = (-A·S/c2)U, which is Fermi-Walker Transport of the 4-Spin and leads to Thomas Precession. Fermi-Walker Transport is the way of transporting a purely spatial vector along the worldline of the particle in such a way that it is as "rotationless" as possible, given that it must remain orthogonal to the worldline. This choice also implies that d/dτ [S·S] = 0, since d/dτ [S·S] ~ [U·S] = 0, which means that the magnitude of the 4-Spin is constant |
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Others: |
P·P = (moc)2 = (Eo/c)2 |
Square Magnitude of the 4-Momentum |
P1·P2 = γ[u1]γ[u2]mo1
mo2(c2-u1·u2) |
Relativistic Billiards... |
Γ·P= (moc) ( Σ·P )·( Σ·P ) = (Ps)·(Ps) = (moc)2 ( Σ1·P1 )·( Σ2·P2 ) = (Ps1)·(Ps2) = γ[ur12](mo1)(mo2)c2 ?? |
Momentum representation of Dirac Equation, the 4-DiracGamma Γ with
4-Momentum P (Γ·P)Ψ= (moc)Ψ (ΓμPμ)Ψ= (moc)Ψ or, in operator form iћ(Γμ∂μ)Ψ= (moc)Ψ (γ0p0 - γ·p )Ψ= (moc)Ψ One can get the equivalent result using the Pauli Spin Matrix Tensor Σ as well 4-Momentum (inc. spin) Ps = Σ·P |
N·N = (noc)2 |
Square Magnitude of the 4-NumberCurrentDensity |
J·J = (poc)2 = (qnoc)2 |
Square Magnitude of the 4-ElectricCurrentDensity |
K·K = (moc / ћ)2 = (ωo/c)2 |
Square Magnitude of the 4-WaveVector |
∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂2/c2∂t2-∇·∇ = -(moc / ћ)2 |
Klein-Gordon Relativistic Wave Eqn. |
∂·J = ∂ρ/∂t +∇·j = 0 |
Continuity Equation - Conservation of Electric Charge |
E·K = 0 |
The Polarization of a photon is orthogonal to direction of wave
motion (E·K = 0) (cancellation of "scalar" polarization) |
AEM·AEM = (VEM/c,aEM)·(VEM/c,aEM) = (VEM/c)2-aEM·aEM = ???? |
Square Magnitude of the Electromagnetic field |
PT·R = -S = (-∂S/c∂t,∇[S])·R | Action S |
There is an important distinction between an invariant
quantity and a conserved quantity.
An invariant quantity has the same value wrt. all inertial systems, but
may possibly change upon physical interaction for a system of multiple particles
(e.g. a fission/fusion reaction
"redistributes" the rest masses).
A conserved quantity maintains the same value both before and after an
interaction, although the component values may appear different in
different frames.
In 4-vector notation:
An invariant quantity is a Lorentz Scalar, the dot product of two
4-Vectors, A·B = invariant = same value for all inertial
observers.
A conserved quantity is a component of a 4-Vector that has 4-Divergence =
0, ∂·V = 0.
Relativistic Invariant Quantities (Lorentz Scalars~A·B),
although perhaps there might be some that are simply just scalars...
also known as Relativistic Covariance = Relativistic
Invariance = Lorentz Invariance
Lorentz Scalars = World Scalars = Invariant Scalars = Lorentz Invariants
Any quantity involving counting of particles or states | Entropy, Number of particles in a volume, Number of microstates/macrostate, etc. | ||
c = √[U·U] |
Speed of Light: c (in vacuum) E ~ cp |
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h = √[P·P/L·L] = P·L /
L·L |
Planck's const: h E ~ hν |
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kB = √[P·P/PT·PT] = P·PT / PT·PT?? |
Boltzmann's const: kB, E ~ kBT |
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γrel = V·U / U·U = V·U / V·V |
Relative Relativistic Gamma Factor, between 4-velocities U and V |
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Δs = √[ΔR·ΔR] = √[c2Δt2 - Δx2 - Δy2 - Δz2] |
Displacement |
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Δσ = √[-ΔR·ΔR] = √[-(c2Δt2 - Δx2 - Δy2 - Δz2)] |
Proper Distance, for SpaceLike Intervals |
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ds = √[dR·dR] = √[c2dt2
- dx2 - dy2 - dz2] |
Differential Length of World Line
Element, the Spacetime Interval ds |
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dτ = √[dR·dR/U·U] |
Differential Proper Time, aka. the Eigentime differential |
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d/dτ = U·∂ = γ(∂/∂t + u·∇) = γ d/dt |
Derivative wrt Proper Time d/dτ |
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Δ = ∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) |
D'Alembertian/wave operator |
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d4x |
Invariant 4-volume |
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d4p = dP·dVp |
Spacetime momentum-space differential "4-volume" element |
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d4k = dK·dVk |
Spacetime wavevec-space differential "4-volume" element?? |
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d3xd3p = dVphase = dμ[t] = dV·dVp |
Invariant phase volume |
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δ4(x-y) |
4-D Dirac Delta Function |
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G(x|x') |
Green's Function where
∂·∂ G(x|x') = δ4(x-x') |
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εo |
The Electric Constant εo | ||
μo = |(∂·∂)A| / | J | |
The Magnetic Constant μo | ||
f(t,x,p) |
One Particle Distribution Function |
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N[t] = f(t,x,p)
dμ[t] = ∫N·dV |
Number of particles in a volume element | ||
d3p/(2E) |
Invariant phase-space element |
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d3p/(2E) = p2dpdΩ/(2E) | Spacetime
phase-space differential 3-volume element, Cartesian vs. spherical
basis dΩ =sin(θ) dθ dφ = Solid Angle Element in direction of Ω |
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(2E)δ3(p-p'), (2ω)δ3(k-k') | Spacetime phase-space
differential 3-volume element, Dirac form need to get exact units, etc. corrected δ4(p-p') => Lorentz Invariant δ( E - Eo )δ3(p-p') δ( E - Eo )δ3(p-p') / δ( F[E] ) Divide by another Lorentz invariant δ( E - Eo )δ3(p-p') / δ( E - Eo )/|2E| δ( E - Eo ) |2E| δ3(p-p') / δ( E - Eo ) |2E| δ3(p-p') |
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d3p/E E d3x d3xd3p Iν/ν3 |
d3x = c dt
dA⊥ : The photons contained in a cylinder of base area
A traveling a distance dx = c dt d3p = (h/c)3 ν2 dν d[cos(θ)] dφ : Spherical coords Iν d ν d[ cos( θ)] dφ dA⊥ dt = Energy carried by photons in range (ν, ν+dν) 2 ν2/c2 dν d[ cos( θ)] d φ dA⊥ dt = 2 d3pd3x / h3 (c2 Iν )/( 2 ν2 )= Energy per mode n = (c2 Iν )/( 2 h ν3 ) = Number of photons per quantum state, itself an invariant thus Iν /ν3 is invariant |
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u[ε,Ω]/ε3 | Specific spectral energy density over dimensionless energy cubed | ||
Iε[Ω]/ε3 | Intensity over dimensionless energy cubed | ||
j[ε,Ω]/ε2 | Emissivity over dimensionless energy squared | ||
Iν/ν3 | Spectral
Intensity / ν3 Lorentz Invariant related to Relativistic Beaming or Doppler Beaming |
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mo = √[P·P/U·U] = P·U/U·U |
RestMass of a Particle mo ( 0 for photons, + for massive ) |
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ρmo = mono
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ProperMassDensity ρmo of
a continuum in the co-moving frame of no |
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q = √[J·J/N·N] = J·N/N·N |
RestElectricCharge of a Particle q |
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3-vector |
Spin so |
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magnetic moment |
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Eo = P·U = moc2 |
RestEnergy of a Particle ( 0 for photons, + for massive ) |
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ωo = K·U = moc2/ћ |
RestAngFrequency of a Particle ( 0 for photons, + for massive ) |
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ΦT = - KT·R |
*** Phase of a wave ***, |
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S = Saction = - PT·R
= ∫[dt L;ti,tf] |
Action Variable S of Action
Integral |
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γL = - (P + qA)·U |
Relativistic Lagrangian L * gamma
(γ) |
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L |
Lagrangian Density |
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L1 H1 |
Extended
Lagrangian/Hamiltonian formalism L1 = (+/-?) -γL L1 + H1 = PT·U H1 seems to end up being identically 0 in all frames, H1 = H - E = 0, so I guess it is a Lorentz scalar as well I'm not totally sure of the sign for L1, it seems to differ in the various papers I have read on this In any case, we have the following: L + H = pT·u {non-covariant, but true relativistically/Newtonian, for the conventional Lagrangian/Hamiltonian} L1 + H1 = PT·U {covariant generally true, for the extended Lagrangian/Hamiltonian} L = T - V, only in Newtonian |
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H / γ = Pt·U | Relativistic
Hamiltonian H / gamma (γ) Start with Lagrangian L[qi,qi'], a function of coords qi and their time derivatives qi' Conjugate momenta pi = ∂L/∂qi' Then Hamiltonian H = Σ [piqi';i] - L Then, Eqns. of Motion p' = -∂H/∂q q' = ∂H/∂p ex. Lagrangian of a free particle L = -moc2/γ pi = ∂L/∂qi' = γmoui (or p = γmou) H = Σ [piqi';i] - L = p·u - L = γmou·u + moc2/γ = γmoc2 where γ2 = c2/(c2-v2) = c2/(c2-u·u) p' = -∂H/∂q = 0, since H = γmoc2 has no explicit dependence on q q' = ∂H/∂p = u So, p' ~ a = 0 ie. no acceleration q' = u as we expect Hamiltonian for a free particle: H = γmoc2 = E ; H/γ = moc2 = Eo ------- ex. Lagrangian of a charged particle in EM field
L = -(P + QEM)·U/γ L = -(P·U + QEM·U)/γ L = -P·U/γ - QEM·U/γ L = -moU·U/γ - qAEM·U/γ L = -moc2/γ - qAEM·U/γ L = -moc2/γ - q(ΦEM/c, aEM)·γ(c, u)/γ L = -moc2/γ - q(ΦEM/c, aEM)·(c, u) L = -moc2/γ - q(ΦEM - aEM·u) L = -moc2/γ - qΦEM + qaEM·u L = -moc2/γ - qΦoEM/γ L = -(moc2 + qΦoEM)/γ pcanonical = ptotal = ∂L/∂q' = (γmou) + (qaEM) = (pdynamical) + (qaEM) = (pkinetic) + (ppotential) Hence, γmou = ptotal - qaEM Equation of motion: (leading to negative gradient of potential) dp/dt = ∂L/∂x = - q(∂ΦEM/∂x - ∂aEM/∂x·u)
H = pT·u - L H = γmou·u + qaEM·u - L H = γmou·u + qaEM·u + moc2/γ + q(ΦEM - aEM·u) H = γmou·u + moc2/γ + q(ΦEM) H = p·u + moc2/γ + q(ΦEM) H = γmoc2 + qΦEM H = E + V = (rest+kinetic) + (potential) H = moc2 + (γ-1)moc2 + qΦEM H = (rest) + (kinetic) + (potential) also, since E=√[p·p c2 + mo2c4] H = √[pT2c2 + mo2c4] + qΦEM H = √[(pkinetic - qaEM)2 c2 + mo2c4] + qΦEM q' = ∂H/∂p = (p - qaEM) / √[(p - qaEM)2 /c2 + mo2] p' = -∂H/∂q = q(∇ aEM) ·u - q∇ΦEM this leads to the Lorentz force (here E and B are the classical electric and magnetic fields, not 4-vectors): pT' = f = q(E + v x B) representing the rate at which the EM field adds relativistic momentum to a charged particle dp/dτ = γq(E + v x B) The non-relativistic Lagrangian L is an approximation of the relativistic one: L = -(moc2 + qΦoEM)/γ -L = (moc2 + qΦoEM)/γ = √[1-(v/c)2](moc2 + qΦoEM) ~ (moc2 + qΦoEM) - (1/2)(moc2v2/c2 + qΦoEMv2/c2) ~ (moc2 + qΦoEM) - (1/2)(mov2 + 0) L ~ (1/2)(mov2) - (moc2 + qΦoEM) L ~ (Kinetic) - (Rest+Potential) = T - V (for v << c) The large constant coming from the restmass is simply ignored in classical mechanics. The gamma factor in the Lagrangian corresponds to the time dilation of an object moving at v. In QM words: the number of phase changes (ticks) over the trajectory of the particle (the t' axis) is less by a factor gamma. The non-relativistic Hamiltonian H is an approximation of the relativistic one: H = γ(moc2 + qΦoEM) H = (1/√[1-(v/c)2])(moc2 + qΦoEM) ~ [1+(v/c)2/2])(moc2 + qΦoEM) = (moc2 + qΦoEM)+(1/2)(moc2v2/c2 + qΦoEMv2/c2) ~ (moc2 + qΦoEM) -+(1/2)(mov2 + 0) H ~ (1/2)(mov2) + (moc2 + qΦoEM) H ~ (Kinetic) + (Rest+Potential) = T + V (for v << c) In QM words: the number of phase changes (ticks) over the t axis is higher by a factor gamma. Thus, L ~ T-V and H ~ T+V only in the non-relativistic limit (v<<c) |
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T = (1/2)moU·U= (1/2)P·U ?? |
Relativistic Kinetic Energy Term T |
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ψ[R] , ψ*[R] |
Scalar Quantum Wave Function |
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no = √[N·N/U·U] = N·U/U·U = n/γ |
Particle RestNumberDensity (for stat mech) |
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so = √[S·S/U·U] = S·U/U·U |
RestEntropyDensity (for stat mech) |
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Ωo = Ω |
Ω = # of microstates = (N!) / (n0!n1!n2!...) |
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No = N |
(Stable) Particle Number: N = nV =
(n/γ)(γ V) = noVo = No |
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Po = P |
Pressure of system (eg. of a tensorial perfect fluid): P = Po |
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So = S = kB ln Ω |
Entropy: S = sV = (s/γ)(γ V) = soVo = So , |
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To = γ T |
RestTemperature (according to Einstein/Planck def.) |
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q = γ Q |
RestHeat |
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Vo = γ V |
RestVolume |
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dS = kB d(ln Ω) = δQ / T |
Change in Entropy |
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Π (pα,xα)
= |
Invariant equilibrium distribution
function for relativistic gas |
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Fuv Fuv
= 2(B2 - E2/c2) |
EM invariant |
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Gcd Fcd
= εabcdFabFcd
=(2/c)(B·E) |
EM invariant |
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P =
(2q2/3c3) γ4(aperp2+
γ2a2||) P = (μoq2a2γ6)/(6πc) in parallel Generally P = μoq2(A·A)/(6πc) |
Radiated Power P, total power is Lorentz invariant for processes with symmetry in the rest frame | ||
Iv/v3 | Spectral Intensity / v3 | ||
Helicity | A massless particle moves with the speed of light, so a real observer (who must always travel at less than the speed of light) cannot be in any reference frame where the particle appears to reverse its relative direction, meaning that all real observers see the same chirality. Because of this, the direction of spin of massless particles is not affected by a Lorentz boost (change of viewpoint) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: the helicity is a relativistic invariant |
Conserved Quantities (components of V, such that the 4-Divergence ∂·V = 0 )
∂·J = ∂p/∂t +∇·j = 0 |
Conservation of 4-CurrentDensity (EM
charge): p & j |
∂·N = ∂/∂t(γ no)+∇·(γ
nou) |
Conservation of 4-NumberFlux (Particle
NumberDen, NumFlux): n & nf
|
∂·P = (1/c2)∂E/∂t +∇·p
= 0 |
Conservation of 4-Momentum (Energy~Mass,
Momentum): E & p |
∂·K = ∂/c∂t(w/c)+∇·k |
Conservation of 4-WaveVec (AngFreq,
WaveNum): w & k |
∂·AEM = (1/c2)∂VEM/∂t +∇·aEM = 0 |
Conservation of 4-VectPotentialEM
(applies in the Lorenz Gauge): VEM & aEM
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∂·U = ∂/∂t(γ[u])+∇·(γ[u] u) |
Conservation of 4-Velocity: (Flux-Gauss'
Law)??: γ & γ u |
ημν = ημν =
Diag[1,-1,-1,-1] |
Minkowski Metric (flat spacetime) |
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1 if a=b, |
Kronecker Delta |
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= +1 if {abcd} is an even permutation of
{0123} |
Levi-Civita symbol |
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Fuv =
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EM Field Tensor |
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Gcd = (1/2)εabcdFab =
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Dual EM Field Tensor |
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[Generally {including shear forces}]
[For a Perfect Fluid {no viscosity}] Tμν = (ρeo+p)UμUν/c2 - pημν : where ημν = Diag[1,-1,-1,-1] Tμν =
(both of these are Lorentz Scalars) [For Dust {no particle interaction}] Tμν = (ρeo)UμUν/c2 = (ρmo)UμUν Tμν =
and p = po = 0 for dust {particles do not interact} |
Energy-Momentum Stress Tensor The stress-energy tensor of a relativistic
fluid can be written in the form Here
The heat flux vector and viscous shear tensor are transverse to the world lines, in the sense that
This means that they are effectively three-dimensional quantities, and since the viscous stress tensor is symmetric and traceless, they have respectively 3 and 5 linearly independent components. Together with the density and pressure, this makes a total of 10 linearly independent components, which is the number of linearly independent components in a four-dimensional symmetric rank two tensor. |
4-Vector(s) | Type | Relativistic Law | Newtonian Limit Low Velocity (v<<c) or Low Energy (E<<moc2) Basically, β --> 0, γ --> 1 |
R = (ct,r) | 4-Position | (ct,r) is single 4-vector entity t and r related by Lorentz transform |
t independent from r t is independent scalar, r is independent 3-vector |
ΔR = (cΔt,Δr) | 4-Displacement | Relative Simultaneity Δt' = γ(Δt - β·Δr/c) |
Absolute Simultaneity Δt' = Δt |
U = dR/dτ | 4-Velocity | Relativistic Composition of Velocities urel = =[u1+u2]/(1+β1·β2) =[u1+u2]/(1+u1·u2/c2) Imposes Universal Speed Limit of c |
Additive Velocities u12 = u1 + u2 Unlimited Speed |
A = dT/dτ | 4-Acceleration | Relativistic Larmor Formula Power radiated by moving charge P = = ( q2/ 6πεoc3)(A·A) = (μoq2/6πc)(A·A) = (μoq2/6πc) γ6/ (a2 - (|u x a|)2/c2) |
Newtonian Larmor Formula Power radiated by a non-relativistic moving charge P = (μoq2/6πc)(a2) |
P = moU | 4-Momentum | Einstein Energy-Mass Relation E = γ moc2 = Sqrt[ mo2c4 + p·p c2 ] |
Total Energy = Rest Energy + Kinetic Energy E = moc2 + (p2/2mo) |
∂·P | Divergence of 4-Momentum | Local? Conservation of 4-Momentum | Conservation of Energy, Conservation of Momentum |
P1·P2 | Particle Interaction | Conservation of 4-Momentum | Conservation of Energy, Conservation of Momentum, sometimes Conservation of Kinetic Energy |
K =(ω/c,k) =(1/ћ)P = (mo/ћ)U = (ωo/c2)U |
4-WaveVector and 4-Velocity |
Relativistic Doppler Effect, inc. Transverse Doppler Effect ao_obs = = ao_emit / γ(1 - (n·v/c)) = ao_emit / γ(1 - (n·β)) = ao_emit √[1+|β|]√[1-|β|] / (1 - (n·β)) Relativistic Aberration Effect cos(ø_obs) = [cos(ø_emit)-β]/[1-βcos(ø_emit)] Relativistic Wave Speed, all elementary particles, matter or photonic λf = c/β = vphase |
Regular Doppler Effect ao_obs = ao_emit √[1+|β|]√[1-|β|] Newtonian Aberration = None cos(ø_obs)= cos(ø_emit) Newtonian Wave Speed, only photonic particles (a rare case when the lightspeed case is chosen for Newtonian description) λf = c |
P and K | 4-Momentum and 4-WaveVector |
Compton Scattering (λ'-λ) = (h/moc)(1-cos[ø]) (moc2)(1/E'-1/E) = (1-cos[ø]) Ratio of photon energy after/before collision P[E,ø] = 1/[1+(E/moc2)(1-cos[ø])] see also Klein-Nishina formula |
Thompson Scattering Ratio of photon energy after/before collision: E<<moc2 P[E,ø] --> 1 |
∂ = -iK | 4-Gradient | D'Alembertian & Klein-Gordon Equation ∂t2/c2 = ∇·∇-(moc/ћ)2 |
Schroedinger Equation (i ћ)( ∂t ) = - (ћ)2(∇)2/2mo |
∂·J | Divergence of 4-Current | Conservation of 4-EM_CurrentDensity ∂·J = ∂/c∂t(cp)+∇·j = ∂p/∂t +∇·j = 0 |
Conservation of 4-EM_CurrentDensity ∂·J = ∂/c∂t(cp)+∇·j = ∂p/∂t +∇·j = 0 |
Jprob | Probability CurrentDensity | Conservation of ProbabilityCurrentDensity ρ = (iћ/2moc2)(ψ* ∂t[ψ]-∂t[ψ*] ψ) j = (-iћ/2mo)(ψ* ∇[ψ]-∇[ψ*] ψ) ∂·Jprob = ∂ρ/∂t +∇·j = 0 ρ = γ(ψ*ψ) for time separable wave functions Relativistically, this is conservation of the number of worldlines thru a given spacetime <event> |
Conservation of Probability ∂·Jprob = ∂ρ/∂t +∇·j = 0 ρ = (ψ*ψ) for time separable wave functions Typically set so that the sum over all quantum states in space = 1 At low energies/velocities, this appears as the conservation of probability of a given wavefunction at a given spacetime <event> - In other words, the probability interpretation of a wavefunction is just a Newtonian approximation to the more correctly stated conservation of relativistic worldlines. This is why the problem of positive definite probabilities and probabilities >1 vanishes once you consider anti-particles and conservation of charged currents. |
AEM = (ΦEM/c, aEM) | 4-VectorPotential | 4-VectorPotenial of a moving point charge (Lienard-Wiechert potential) AEM = (q/4πεoc) U / [R·U]ret [..]ret implies (R·R = 0, the definition of a light signal) ΦEM = (γΦo) = (γq/4πεor) aEM = (γΦo/c2)u = (γqμo/4πr)u |
Scalar Potential and Vector Potential of a stationary point charge ΦEM = (q/4πεor) aEM = 0 Scalar Potential and Vector Potential of a slowly moving point charge (|v|<<c implies γ-->1) ΦEM = (Φo) = (q/4πεor) aEM = (Φo/c2)u = (qμo/4π r)u |
QEM = (EEM/c, pEM) = q AEM = q (ΦEM/c, aEM) |
4-VectorPotentialMomentum |
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|
PEM = (E/c + qΦEM/c, p +
qaEM) = γ mo(c,u) PEM = Π = P + qAEM = moU + qAEM =(H/c,pEM) = (γmoc+q ΦEM/c,γmou+q aEM) |
4-MomentumEM 4-CanonicalMomentum 4-TotalMomentum |
Minimal Coupling ============= Total 4-Momentum = Particle 4-Momentum + Potential(Field) 4-Momentum |
|
D = ∂ + iq/ћ AEM | Minimal Coupling Prescription |
KG equation, with minimal coupling to an EM
potential D·D = = -(moc/ћ)2 (∂ + iq/ћ AEM)·(∂ + iq/ћ AEM) + (moc/ћ)2 = 0 |
Schroedinger Equation (with external potential) (i ћ)( ∂t ) = V[x] - (ћ)2(∇)2/2mo |
thus, for photonic frequency Doppler shifts,
if {n toward and β toward obs}, then νobs = νemit
√[(1+|β|)/(1-|β|)] -->Doppler BlueShift
if {n toward and β 90° to obs}, then νobs = νemit
/ γ -->the transverse Doppler effect
if {n toward and β away from obs}, then νobs = νemit
√[(1-|β|)/(1+|β|)] -->Doppler RedShift
U = γ(c, u), P = (E/c,p), d(P) =
(dE/c,dp)
U·d(P) = γ(c dE/c-u·dp) = γ(dE-u·dp)
= γ(T dS - P dV + μ dN) = (To dSo - Po
dVo + μo dNo) = 0 ??
U·d(P) = γ(dE-u·dp) = (TodSo - PodVo + μodNo) = const = ? 0 ? |
U·P = γ(E-u·p) = (To So - Po Vo + μo No) = moc2 ? for a spatially homogeneous system: relativistic Gibbs-Duhem eqn. |
Invariants |
P = Pressure = Po |
N = ParticleNum = No |
S = Entropy = So |
Variables |
V = Volume = (1/γ)Volo |
μ = ChemPoten = (1/γ)μo |
T = Temperature = (1/γ)Tempo |
V*P (particle superstructure = Vol*Press)
μ*N (particle structure = ChemPoten*ParticleNum)
T*S (particle substructure = Temp*Entropy)
Time t = γ to
Length L = Lo/γ
Heat Q = q/γ
dq = TodSo
InertialMassDen(of radiation field) q = P/vV = γ q
Total Particle Number N = No is an invariant, because the
NumberDensity n varies as n = γ no, but this is balanced by
Volume V = Vo/γ
NumberDenstiy n = γ no where NumberFlux 4-Vector N =
(cn,nf) = no γ(c, u) = noU,no
= No/(Δ_xo*Δ_yo*Δ_zo)
N = n * V = (γ no)*(Vo/γ) = no* Vo
= No
N·N = (noc)2
Total Entropy S = So is an invariant, because the
EntropyDensity s varies as s = γ so, but this is balanced by
Volume V = Vo/γ
EntropyDensity s = γ so where EntropyFlux 4-Vector S =
(cs,sf) = so γ(c, u) = soU,so
= So/(Δ_xo*Δ_yo*Δ_zo)
S = s * V = (γ so)*(Vo/γ) = so* Vo
= So
S·S = (soc)2
Action S = S(ct,x,y,z)
dS/dτ = 0
dS/dτ = U·∂(S) = γ(∂S/∂t + u·∇(S)) = 0
see Menzel pg.172
√[1+x] ~ (1+x/2) for x<<1 This mathematical formula is used
to derive the Newtonian limit of the various relativistic entities
γ = (1 / √[1-(v/c)2] )
γ --> 1 for v<<c
All of the formulas below can also be generated from the 4-Velocity
Relation and multiplying by the appropriate Lorentz scalar:
U·U = γ[u]2(c2-u·u) = c2
γ2(1-β·β) = 1
γ2 = 1 + γ2β2
γ = ±√[1 + γ2β2]
We choose the positive root since γ is always positive
γ = √[1 + γ2β2]
γ ~ [1 + γ2β2/2] for (γ2β2
<< 1)
4-Momentum
P = (E/c, p)
P·P = (Eo/c)2 = (moc)2
E2 = Eo2 + p·p c2
E = Eo√[ 1 + p·p c2 / Eo2]
E ~ Eo( 1 + p·p c2 / 2 Eo2
+ ...) for | p·p c2 | << | Eo2
| discarding higher order terms...
E ~ ( Eo + p·p c2 / 2 Eo ) for |
p·p c2 | << | Eo2 |
E ~ ( Eo + p·p / 2 mo ) for | p·p c2
| << | Eo2 | where Eo = moc2
E ~ ( Eo + |p|2 / 2 mo ) for | p
c | << | Eo |
Total Energy = Rest Energy + Newtonian Momentum term
alternately:
γ ~ [1 + γ2β2/2]
γEo ~ Eo[1 + γ2β2/2]
E ~ [Eo + γ2Eoβ2/2]
E ~ [Eo + γ2moc2β2/2]
E ~ [Eo + γ2mo2c2β2/2mo]
E ~ [Eo + γ2mo2v2/2mo]
E ~ [Eo + p2/2mo]
4-WaveVector
K = (ω/c, k)
K·K = (ωo/c)2 = (Eo/ћc)2
= (moc / ћ)2
ω2 = ωo2 + k·k c2
ω = ωo √[ 1 + k·k c2 / ωo2
]
ω ~ ωo( 1 + k·k c2 / 2 ωo2
+ ...) for | k·k c2 | << | ωo2
| and choosing the positive root and discarding higher order terms...
ω ~ ( ωo + k·k c2 / 2 ωo )
for | k·k c2 | << | ωo2
|
ω ~ ( ωo + ћ k·k / 2 mo ) for | k·k
c2 | << | ωo2 | where ωo
= moc2 / ћ
ω ~ ( ωo + ћ |k|2 / 2 mo ) for | k
c | << | ωo |
Total Angular Frequency = Rest Angular Frequency + Newtonian Wave Number
term
4-Gradient (Wave equation)
∂ = ∂/∂xμ = (∂/c∂t, -∇) =
(∂t/c,
-∇)
∂·∂ = (∂to/c)2 = (- i moc / ћ )2:
Klein-Gordon Relativistic Wave eqn.
∂t2 = ∂to2 + ∇·∇
c2
∂t = ±∂to√[ 1 + ∇·∇ c2
/ ∂to2]
∂t ~ ∂to( 1 + ∇·∇ c2 /
2 ∂to2 + ...) for | ∇·∇ c2
| << | ∂to2 | and choosing the positive root
and discarding higher order terms...
∂t ~ ( ∂to + ∇·∇ c2 /
2 ∂to ) for | ∇·∇ c2 | << |
∂to2 |
∂t ~ ( ∂to - ћ ∇·∇ / i 2 mo
) for | ∇·∇ c2 | << | ∂to2
| where ∂to = - i moc2 / ћ
∂t ~ ( ∂to - ћ |∇|2 / i 2
mo ) for | ∇ c | << | ∂to |
or, in more standard form
i ћ ∂t ~ ( i ћ ∂to - ћ2 |∇|2
/ 2 mo ) for | ∇ c | << | ∂to |
where i ћ ∂to = Eo, the rest energy of the potential
V
i ћ ∂t ~ ( V(x,t) - ћ2 |∇|2
/ 2 mo ) for | ∇ c | << | ∂to |
Time dependent Schroedinger equation is just the Newtonian approximation
of the Klein-Gordon Relativistic Wave eqn.
4-ProbabilityCurrentDensity (change in form of Probability Density)
J = (cρ, j) = (iћ/2mo)(ψ*∂[ψ]-∂[ψ*]ψ)
taking the temporal component, the relativistic probability density
ρ = (iћ/2moc2)(ψ* ∂t[ψ]-∂t[ψ*]
ψ)
assume wave solution in following general form:
ψ = A f [k] e(-iωt) and ψ* = A* f [k]* e(+iωt)
then
∂t[ψ] = (-iω)A f [k] e(-iωt) = (-iω)ψ and ∂t[ψ*]
= (+iω)A* f [k]* e(+iωt) = (+iω)ψ*
then
ρ = (iћ/2moc2)(ψ* ∂t[ψ]-∂t[ψ*]
ψ)
ρ = (iћ/2moc2)((-iω)ψ*ψ-(+iω)ψ*ψ)
ρ = (iћ/2moc2)((-2iω)ψ*ψ)
ρ = (ћω/moc2)(ψ*ψ)
now use the Newtonian form of ω from above
ρ ~ [ћ( ωo + ћ |k|2 / 2 mo )/moc2](ψ*ψ)
ρ ~ [(ћωo/moc2) + (ћћ
|k|2 / 2 momoc2)](ψ*ψ)
ρ ~ [(ћωo/moc2) + (ћωoћωo
|k|2c2 / 2 ωoωomoc2moc2
)](ψ*ψ)
ρ ~ [(1) + ( |k|2c2 / 2 ωo2)](ψ*ψ),
but
| k c | << | ωo |
ρ ~ [(1) + (~0)](ψ*ψ) because 2nd term is very small in
non-relativistic limit
ρ ~ (ψ*ψ)
The standard probability density (ψ*ψ) is the Newtonian
approximation of the temporal component of the 4-ProbabilityCurrent
Alternately, use ω = γωo
ρ = (ћω/moc2)(ψ*ψ)
ρ = (ћγωo/moc2)(ψ*ψ)
ρ = (γ)(ψ*ψ)
ρ ~ (ψ*ψ) where γ->1 in the Newtonian limit
∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂2/c2∂t2-∇·∇
= -(moc / ћ)2: Klein-Gordon Relativistic Wave eqn.
DEM = (∂/c∂t + iq/ћ VEM/c,
-∇ + iq/ћ aEM) = ∂ + (iq/ћ)AEM
DEM·DEM = -(moc
/ ћ)2: Klein-Gordon Relativistic Wave eqn. in electromagnetic
potentials
(∂ + (iq/ћ)AEM)·(∂ + (iq/ћ)AEM)
= -(moc / ћ)2: Klein-Gordon Relativistic Wave eqn.
w/ electromagnetic potentials
(∂·∂) + (iq/ћ)(∂·AEM + AEM·∂)
+ (iq/ћ)2(AEM·AEM)
= -(moc / ћ)2: Klein-Gordon Relativistic Wave eqn.
w/ electromagnetic potentials
if (∂·AEM + AEM·∂)
= 0
then (∂·∂) + (iq/ћ)2(AEM·AEM)
= -(moc / ћ)2
(∂/c∂t,-∇)·(∂/c∂t,-∇) + (iq/ћ)2((VEM/c,
aEM)·(VEM/c,
aEM)) = -(moc / ћ)2
(∂2/c2∂t2-∇·∇) + (iq/ћ)2(
(VEM/c)2-(aEM·aEM) ) = -(moc
/ ћ)2
(∂2/c2∂t2+(iq/cћ)2(VEM2)-(∇·∇+(iq/ћ)2(aEM·aEM)
) = -(moc / ћ)2
The Klein-Gordon equation is more general than the Schrödinger equation,
but simplifies to the Schrödinger equation in the (φ/c)<<1 limit.
∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂2/c2∂t2-∇·∇
= -(moc / ћ)2: Klein-Gordon Relativistic Wave eqn.
∂2/c2∂t2 = ∇·∇-(moc
/ ћ)2
∂2/c2∂t2 = (imoc / ћ)2+∇·∇
(i ћ)2∂2/c2∂t2 = (i ћ)2(imoc
/ ћ)2+(i ћ)2∇·∇
(i ћ)2∂2/c2∂t2 = (moc)2+(i
ћ)2∇·∇
(i ћ)2∂2/∂t2 = (moc2)2*[1
+
(i ћ/moc)2∇·∇]
(i ћ)∂/∂t = ± (moc2)*Sqrt[1 + (i ћ/moc)2∇·∇]
(i ћ)∂/∂t ~ ± (moc2)*[1 + (1/2)*(i ћ/moc)2∇·∇
+ ...] for ( ћ)2*∇·∇<<(moc)2
,generally a very good approx. for non-relativistic systems
(i ћ)∂/∂t ~ ± [(moc2) + (i2 ћ2/2mo)∇·∇
+ ...]
choosing the positive root and discarding higher order terms...
(i ћ)∂/∂t ~ (moc2) - ( ћ2/2mo)|∇|2
(i ћ)∂/∂t ~ - ( ћ2/2mo)|∇|2
becomes
the time dependent Schrödinger eqn. for a free particle
Also, extensions into EM fields (or other types of relativistic
potentials) can be made using D = ∂ + iq/ћ AEM
where AEM is the EM vector potential and q is the EM
charge,
and allowing D·D = -(moc/ћ)2 to be the more
correct EM quantum wave equation.
D·D = -(moc/ћ)2
(∂ + iq/ћ AEM)·(∂ + iq/ћ AEM)
+ (moc/ћ)2 = 0
let A'EM = iq/ћ AEM
let M = moc/ћ
then (∂ + A'EM)·(∂ + A'EM)
+ (M)2 = 0
∂·∂ + ∂·A'EM + 2 A'EM·∂ + A'EM·A'EM
+ (M)2 = 0
now the trick is that factor of 2, it comes about by keeping track of
tensor notation...
a weakness of strick 4-vector notation
let the 4-Vector potential be a conservative field, then ∂·AEM
=
0
(∂·∂) + 2(A'EM·∂) + (A'EM·A'EM)
+
(M)2 = 0
expanding to temporal/spatial components...
( ∂t2/c2-∇·∇ ) + 2(φ'/c
∂t/c - a'·∇ ) + ( φ'2/c2- a'·a'
) + (M)2 = 0
gathering like components
( ∂t2/c2 + 2φ'/c ∂t/c
+ φ'2/c2 ) - (∇·∇ + 2
a'·∇ + a'·a' ) + (M)2 = 0
( ∂t2 + 2φ'∂t + φ'2
) - c2(∇·∇ + 2 a'·∇
+ a'·a'
) + c2(M)2 = 0
( ∂t + φ' )2 - c2(∇ + a'
)2 + c2(M)2 = 0
multiply everything by (i ћ)2
(i ћ)2( ∂t + φ' )2 - c2(i ћ)2(∇
+ a' )2 + c2(i
ћ)2(M)2 = 0
put into suggestive form
(i ћ)2( ∂t + φ' )2 = - c2(i ћ)2(M)2
+ c2(i ћ)2(∇ + a'
)2
(i ћ)2( ∂t + φ' )2 = i2c2(i
ћ)2(M)2 + c2(i ћ)2(∇
+ a' )2
(i ћ)2( ∂t + φ' )2 = i2c2(i
ћ)2(M)2 [1 + c2(i ћ)2(∇
+ a' )2/ i2c2(i
ћ)2(M)2 ]
(i ћ)2( ∂t + φ' )2 = i2c2(i
ћ)2(M)2 [1 + (∇ + a'
)2/ i2(M)2 ]
take Sqrt of both sides
(i ћ)( ∂t + φ' ) = ic(i ћ)(M) Sqrt[1 + (∇ + a' )2/ i2(M)2
]
use Newtonian approx Sqrt[1+x] ~ ±[1+x/2] for x<<1
(i ћ)( ∂t + φ' ) ~ ic(i ћ)(M) ±[1 + (∇ + a' )2/2 i2(M)2
]
(i ћ)( ∂t + φ' ) ~ ±[ic(i ћ)(M) + ic(i ћ)(M)(∇ +
a' )2/2 i2(M)2
]
(i ћ)( ∂t + φ' ) ~ ±[c(i2 ћ)(M) + c( ћ)(∇
+ a' )2/2(M) ]
remember M = moc/ћ
(i ћ)( ∂t + φ' ) ~ ±[c(i2 ћ)(moc/ћ)
+ c( ћ)(∇ + a' )2/2(moc/ћ)
]
(i ћ)( ∂t + φ' ) ~ ±[c(i2)(moc) +
(ћ)2(∇ + a' )2/2(mo)
]
(i ћ)( ∂t + φ' ) ~ ±[-(moc2) + (ћ)2(∇
+ a' )2/(2mo) ]
remember A'EM = iq/ћ AEM
(i ћ)( ∂t + iq/ћφ ) ~ ±[-(moc2) +
(ћ)2(∇ + iq/ћa )2/2mo
]
(i ћ)( ∂t ) + (i ћ)(iq/ћ)(φ) ~ ±[-(moc2)
+ (ћ)2(∇ + iq/ћa )2/2mo
]
(i ћ)( ∂t ) + (i2)(qφ ) ~ ±[-(moc2)
+ (ћ)2(∇ + iq/ћa )2/2mo
]
(i ћ)( ∂t ) -(qφ ) ~ ±[-(moc2) +
(ћ)2(∇ + iq/ћa )2/2mo
]
(i ћ)( ∂t ) ~ (qφ )±[-(moc2)
+ (ћ)2(∇ + iq/ћa )2/2mo
]
take the negative root
(i ћ)( ∂t ) ~ (qφ ) + [(moc2)
- (ћ)2(∇ + iq/ћa )2/2mo
]
Here is the general Newtonian result
(i ћ)( ∂t ) ~ (qφ ) + (moc2)
- (ћ)2(∇ + iq/ћa )2/2mo
or
(i ћ)( ∂t ) ~ (qφ ) + (moc2)
+ [( ћ / i )∇ + qa ]2/2mo
call (qφ ) + (moc2) = V[x]
(i ћ)( ∂t ) ~ V[x] - (ћ)2(∇
+ iq/ћa )2/2mo
typically the vector potential a is zero in most non-relativistic settings
(i ћ)( ∂t ) ~ V[x] - (ћ)2(∇)2/2mo
And there you have it, the Schrodinger Equation with a potential
The assumptions for non-relativistic equation were:
Conservative field AEM, then ∂·AEM
=
0
(∇ + a' )2/ i2(M)2
= (∇ + a' )2/ i2(moc/ћ)2
= (ћ)2(∇ + a' )2/
i2(moc)2 is near zero
i.e. (ћ)2(∇ + a'
)2 << (moc)2, a good approximation
for low-energy systems
Arbitrarily chose vector potential a=0
Or keep it around for a near-Pauli equation (we would just have to track
spins, not included in this derivation)
(K = mo/ћU = ωo/c2 U)
gives (c2/vphase n = u) Both the
wave vector and particle velocity point in the same direction; along the
worldline. The product of the phase velocity and the particle velocity
always equals c2. ( vphase * u = c2 ). In
the case of photons, the phase velocity = particle velocity = c. In the
case of matter particles, the phase velocity vphase = c2/u
> c and particle velocity u<c. What does this mean? Suppose that you
have a collection of particles traveling at identical velocities that all
flash at the same time. The vphase is the speed at which the
flash moves in other reference frames, and can be considered the speed of
propagation of simultaneity. For particles which are at rest, the vphase
is infinite, which makes sense since they all appear to flash
simultaneously. vphase (the phase velocity) is sometimes
known as the celerity.
(∂·∂)AEM = μo J+∂(∂·AEM)
Inhomogeneous Maxwell Equation
(∂·∂)AEM = μo J
Homogeneous Maxwell/Lorentz Equation (if ∂·AEM
= 0 Lorenz Gauge)
∂·J = ∂ρ/∂t +∇·j = 0 Conservation of
EMcurrent
Psi = a E e -iK·R Photon
Wave Equation (Solution to Maxwell Equation)
E·K = 0 The Polarization of a photon is orthogonal to the
WaveVector of that photon
V·Uobs = γv(c,v)·γ[uobs]
uobs = γvγ[uobs](c2-v·uobs)
V·Uobs[uobs = 0]/c2
= γv (RestFrame Invariant expression for relative gamma
factor)
P·Uobs = E/c γ[uobs]c-p·γ[uobs]
uobs = γ[uobs](E-p·uobs)
P·Uobs[uobs = 0]
= E (RestFrame Invariant expression for energy)
K·Uobs = w/c γ[uobs]c-k·γ[uobs]
uobs = γ[uobs](w-k·uobs)
K·Uobs[uobs = 0]
= w (RestFrame Invariant expression for angular frequency)
R·Uobs = ct γ[uobs]c-r·γ[uobs]
uobs = γ[uobs](c2t-p·uobs)
R·Uobs[uobs = 0]/c2
= t (RestFrame Invariant expression for time)
J·Uobs = cp γ[uobs]c-j·γ[uobs]
uobs = γ[uobs](pc2-j·uobs)
J·Uobs[uobs = 0]/c2
= ρ (RestFrame Invariant expression for ElecChargeDensity)
Fuv = ∂uAv-∂vAu
Electromagnetic Field Tensor (F0i = -Ei,Fij
= eijkBk)
L = -1/4 Fuv Fuv -
Ju Au : Lagrangian Density for EM field
L = -moc2/γ -V: Relativistic Lagrangian function of
a Particle in a Conservative Potential
VEM = q U·AEM/γ: Potential of
EM field
LEM = -moc2/γ - q U·AEM/γ
= - (P·P/mo + qU·AEM)/γ
= - (moU·U + qU·AEM)/γ
d/dτ = U·∂ = γ d/dt
U·∂/γ = ∂/∂t + u·∇ = d/dt : Convective Derivative
Larmor formula can be written in Lorentz invariant form
P = -( q2/ 6πεoc3)(A·A)
=
-(μoq2)/(6πc)(A·A) Guassian units?
= ( q2/ 6πεoc3) γ6/ (u'2
- (u x u')2/c2)
= ( q2/ 6πεoc3) γ6/ (β'2
- (β x β')2)
= (2q2/ 3c(1-β'2)3) γ6/ ( β'2
- ( β x β')2) SI Units?
alternate Larmor formula:
P = (-2/3)(q2/ mo2c3)(F·F)
SI
units?
P = -( q2/ 6πεomo2c3)(F·F)
Guassian
units?
Relativistic Power radiated by moving charge by Abraham-Lorentz-Dirac
force
P = (μoq2a2γ6)/(6πc)
====
Liénard-Wiechert potentials - potential due to a moving charge
Aμ(x) = (q/c4πεo) Uμ / ( Rν Uν
) where Rν is a null vector (Rν Rν = 0)
AEM = (q/c4πεo) U / (R·U)
where (R·R = 0, the definition of a light signal)
= (q/c4πεo) U / ( cγ ( |r|-r·u/c )
)
= (q/c24π εo)(c,u)/( |r|-r·u/c )
and therefore
φEM = (q / 4 π εo ) 1/[ r - r·u/c]ret
aEM = (μo q / 4 π) [u]/[ r - r·u/c]ret
where terms in square brackets [] indicate retarded quantities
(R·U) = (ct,r)·γ(c,u) = γ(c2t - r·u)
= cγ(ct - r·u/c)
tret = t - |x-x'|/c: (retarded time)
ru = r - r u/c = the virtual
present radius vector; i.e., the radius vector directed
from the position the charge would occupy at time t' if it had continued
with uniform velocity from its retarded position to the field point.
=====
F = - grad V(x): Particle moving in conservative
force field
mc2 + V(x) = E = const: Relativistic energy conservation in
conservative force fields
T = mc2-moc2 = (γ[u]-1) moc2
= (γ-1) moc2 Relativistic Kinetic Energy:
F·dX/dt = dT/dt: Also holds in Relativistic Mechanics
F·U = (moA+(dmo/dτ)U)·U
= c2(dmo/dτ) = γc2(dmo/dt)
Relativistic Perfect Fluids, where dissipative effects (viscosity,
heat conduction, etc.) are neglected.
Particle 4-Flow N is a conservative quantity whose balance eqn. is ∂·N
= 0
N = (cn, nf) = no
γ(c, u) = n(c, u) = noU
∂·N = ∂n/∂t +∇·(nu) = 0, where n=no
γ
∂·N = ∂no γ/∂t +∇·(no γu) = 0
In non-relativistic limit this becomes ∂no/∂t
+∇·( nou) = 0
Tαβ = ((ne+p)/c2)Uα Uβ - p ηαβ
∂βTαβ = 0,
Consevation of Energy-Momentum Tensor
An lamellar vector field is a synonym for an irrotational vector field.[1] The adjective "lamellar" derives from the noun "lamella", which means a thin layer. In Latin, lamella is the diminutive of lamina (but do not confuse with laminar flow). The lamellae to which "lamellar flow" refers are the surfaces of constant potential.
An irrotational vector field which is also solenoidal is called a Laplacian vector field.
The fundamental theorem of vector calculus states that any vector field
can be expressed as the sum of a conservative vector field and a
solenoidal field.
In vector calculus a solenoidal vector field (also known as an incompressible
vector field) is a vector field v with divergence zero:
∇·v = 0
The fundamental theorem of vector calculus states that any vector field
can be expressed as the sum of a conservative vector field and a
solenoidal field. The condition of zero divergence is satisfied whenever a
vector field v has only a vector potential component, because the
definition of the vector potential A as:
v = ∇ x A
automatically results in the identity (as can be shown, for example,
using Cartesian coordinates):
∇·v = ∇·(∇ x A) = 0
The converse also holds: for any solenoidal v there exists a vector potential A such that v = ∇ x A. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)
In vector calculus, a Laplacian vector field is a vector field
which is both irrotational and incompressible. If the field is denoted as
v, then it is described by the following differential equations:
∇ x v = 0
∇·v = 0
Since the curl of v is zero, it follows that v can be
expressed as the gradient of a scalar potential (see irrotational field) φ:
v = ∇ φ (1)
Then, since the divergence of v is also zero, it follows from equation (1) that
∇·∇ φ = 0
which is equivalent to
∇2 φ = 0
Therefore, the potential of a Laplacian field satisfies Laplace's
equation.
In fluid dynamics, a potential flow is a velocity field which is
described as the gradient of a scalar function: the velocity potential. As
a result, a potential flow is characterized by an irrotational velocity
field, which is a valid approximation for several applications. The
irrotationality of a potential flow is due to the curl of a gradient
always being equal to zero (since the curl of a gradient is equivalent to
take the cross product of two parallel vectors, which is zero).
In case of an incompressible flow the velocity potential satisfies the Laplace's equation. However, potential flows have also been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.
Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, and groundwater flow.
For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable.
A velocity potential is used in fluid dynamics, when a fluid
occupies a simply-connected region and is irrotational. In such a case,
∇ x u = 0
where u denotes the flow velocity of the fluid. As a result, u
can be represented as the gradient of a scalar function Φ:
u = ∇ Φ
Φ is known as a velocity potential for u.
A velocity potential is not unique. If a is a constant then Φ + a is also a velocity potential for u. Conversely, if Ψ is a velocity potential for u then Ψ = Φ + b for some constant b. In other words, velocity potentials are unique up to a constant.
Unlike a stream function, a velocity potential can exist in three-dimensional flow.
see Cosmological Physics
Relativistic Euler Equations:
dv/dt = - 1/[γ2(ρ + p/c2)](∇ p + p'v/c2):
Conservation of Momentum
d/dt[γ2(ρ + p/c2)] = p'/c2 - γ2(ρ
+ p/c2)∇·v: Conservation of Energy
where p' = ∂ p/∂ t
∂·J = 0 where J = noU (J is the Number
Flux here)
Relativistic Enthalpy w = (ρ + p/c2)
d/dt[γw/n] = p'/γnc2
Thus, in steady flow, γ * (enthalpy/particle) = const.
In non-relativistic limit these reduce to
dv/dt = - 1/[ρ](∇ p): Conservation of Momentum
d/dt[(ρ)] = - (ρ)∇·v: Conservation of Mass
p = Pressure
ΔE = - p ΔV
E = ρ c2 V
ΔV / V = - Δρo/ρo
Relativistic Bernoulli's eqn.
γ w / ρo = const
L = -(PT·U)/γ | H = γ(PT·U) | H + L = pT·u = γ(PT·U) - (PT·U)/γ |
L = -(PT·U)/γ L = -((P + Q)·U)/γ L = -(P·U + Q·U)/γ L = -P·U/γ - Q·U/γ L = -moU·U/γ - qA·U/γ L = -moc2/γ - qA·U/γ L = -moc2/γ - q(φ/c, a)·γ(c, u)/γ L = -moc2/γ - q(φ/c, a)·(c,u) L = -moc2/γ - q(φ - a·u) L = -moc2/γ - qφ + qa·u L = -moc2/γ - qφo/γ L = -(moc2 + qφo)/γ |
H = γ(PT·U) H = γ((P + Q)·U) H = γ(P·U + Q·U) H = γP·U + γQ·U H = γmoU·U + γqA·U H = γmoc2 + qγφo H = γmoc2 + qφ H = ( γβ2 + 1/γ )moc2 + qφ H = ( γmoβ2c2 + moc2/γ) + qφ H = ( γmou2 + moc2/γ) + qφ H = p·u + moc2/γ + qφ H = E + qφ H = ± c√[mo2c2+p2] + qφ H = ± c√[mo2c2+(pT-qa)2] + qφ |
H + L = γ(PT·U) - (PT·U)/γ H + L = (γ - 1/γ)(PT·U) H + L = ( γβ2 )(PT·U) H + L = ( γβ2 )((P + Q)·U) H + L = ( γβ2 )(P·U + Q·U) H + L = ( γβ2 )(moc2 + qφo) H + L = (γmoβ2c2 + qγφoβ2) H + L = (γmou·uc2/c2 + qφoγu·u/c2) H + L = (γmou·u + qa·u) H + L = (p·u + qa·u) H + L = pT·u |
Next, let's look at Quantum Commutation Relations...
Non-zero Commutation Relation between position and momentum:
[Xu,Pv] = - i ћ ηuv
this gives
[ x , px ] = [ y , py ] = [ z , pz ] = (i
ћ)
[ ct , E/c ] = [ t , E ] = (- i ћ) :assuming that one can treat the time
as an operator...
both of these yield the familiar uncertainty relations:
Generalized Uncertainty relation: (Δ A) * (Δ B) > = (1/2) |< i[A,B]
>| see Sudbury pg. 59 for a great derivation
(Δ x * Δ px > = ћ / 2) and (Δ t * Δ E > = ћ / 2)
or more generally
(Δ Ru * Δ Pv > = ћ δuv / 2)
or
(Δ Ru * Δ Kv > = δuv / 2)
(Δ x * Δ kx > = 1/2) and (Δ t * Δ w > = 1/2)
[ Ru , Rv ] = Ru Rv - Rv
Ru = 0 : All position coordinates commute
[ Pu , Pv ] = Pu Pv - Pv
Pu = 0 : All momentum coordinates commute
While I'm at it, a small comment about the quantum uncertainty relation. A
great many books state that the quantum uncertainty relations mean that a
"particle" cannot simultaneously have precise properties of position and
momentum. I disagree with that interpretation. The uncertainty relations,
the mathematical structure of the argument, say nothing about
"simultaneous" measurements. They do say something about "sequential"
measurements. A measurement of one variable places the system in a state
such that if the next measurement is that of a non-commuting variable of
the first, then the uncertainty must be of a minimum>0 amount. Also,
note that the uncertainty relations are not necessarily about the size of
h. Nor are they about the factor of ( i ) in the commutation relation. It
would appear that they are about the metric gμν itself, which
has a non-zero result for sequential, non-commuting measurements.
Also, a comment on the EPR results. Based on SR, one cannot say that the
measurement of one particle immediately "collapses" the physical state of
the other. Since the two entangled particles can be setup such that they
are space-like separated at the <event>'s of their respective measurement,
there exist coordinate frames in which the measurement of the 1st particle
occurs before that of the 2nd, exactly at the same time as the 2nd, and
after that of the 2nd. Thus, how is the first particle to "know" that it
must collapse the wavefunction of the 2nd, or that it must itself be
collapsed by the 2nd? The answer is of course that the one measurement does not affect the space-like separated other measurement.
--------
need to derive:
(Δ phix * Δ Lx > = ћ / 2)
where phix is angle about x, and Lx is angular
momentum about x
|
|
c
\ future /
\ | /
\ |
/ -- space-like
interval(-)
\|/now
/|\
/
| \
elsewhere
/ | \
/ past \
|
-c
(0,0) Zero-Null Vector
(+a,0) Future Pointing Pure TimeLike
(-a,0) Past Pointing Pure TimeLike
(0,b) Pure SpaceLike
(a,b) |a|>|b| TimeLike
(a,b) |a| =|b| Photonic-LightLike
(a,b) |a|<|b| SpaceLike
Any TimeLike 4-Vector (a,b) may be boosted into a Pure TimeLike
(ka,0) state
Any SpaceLike 4-Vector (a,b) may be boosted into a Pure SpaceLike
(0,kb) state
So far, Poincare Invariance appears to be an absolute conservation law of
all quantum field theories, as well as being a basis for Special
Relativity. A number of quantum field theories are based on the complex
(charged) scalar (Klein-Gordon) quantum field - which is mathematically
the simplest QFT that still contains a continuous global [U(1)] internal
symmetry. A real (Hermetian) scalar QFT is mathematically still simpler,
but the absence of "charge" renders it uninteresting for most purposes.
Poincare group (aka inhomogeneous Lorentz group) and its representations
The set of Lorentz transforms and spacetime translations (Λ,A) such that:
X'μ = Λμν Xν + Aμ
with conditions:
Det[Λ] = +1 (excludes discrete transforms of space inversion => proper)
Λ00 >= +1 (excluded discrete transforms of time
inversion => orthochronous, preserve direction of time)
Λμν (a Lorentz Transform - maps spacetime onto
itself and therefore preserves the inner product)
Λμν Λμλ = gνλ (the
Minkowski Metric)
Aμ = (Space-time Translation)
Unitary Operators representing these transforms:
U(A,1) = Exp[ i P·A ]
U(0,Λ) = Exp[ i Mμν Λμν ]
Poincare group has 10 generators (spacetime 4-generators)
Pμ (4 generators of space-time translation = Conservation of
4-Momentum)
Mμν (6 generators of Lorentz group = 3 orbital angular momenta
+ 3 Lorentz boosts)
[ Pμ, Pν ] = 0 (Energy/Momentum commutes with
itself)
[ Mμν, Pσ ] = - i ( Pμ gνσ - Pν
gμσ )
or
[ Mμν, Pσ ] = i ( gνσ Pμ - gμσ
Pν ) {one of these has a sign error I think}
[ Mμν, Mρσ ] = -i ( Mνσ gμρ -
Mμσ gνρ + Mρν gμσ - Mρμ
gνσ )
Then, define the spatial 3-generators:
"Spatial Rotation" generators Ji = -(1/2) εijk Mjk
(for i=1,2,3), are Hermetian, (Mjk)† = Mjk
"Lorentz Boost" generators Ki = Mi0 (for i=1,2,3),
are anti-Hermetian, (Mi0)† = - Mi0
[ Ji , Pk ] = i εikl Pl
[ Ji , P0 ] = 0 (Spin commutes Energy)
[ Ki , Pk ] = i P0 gik
[ Ki , P0 ] = - i Pi
[ Jm , Jn ] = i εmnk Jk
[ Jm , Kn ] = i εmnk Kk
[ Km , Kn ] = - i εmnk Jk
Covariance of physical laws under Poincare trans. imply that all
quantities defined in Minkowski space-time must belong to a representation
of the Poincare group. By def., the states that describe elementary
particles belong to irreducible representations of the Poincare group.
These representations can be classified by the eigenvalues of the Casimir
operators, which are the functions of the generators that commute with all
the generators. This property implies that the eigenvalues of the Casimir
operators remain invariant under group transforms.
Poincare Algebra ISO(1,3)
There are two Casimir operators of the Poincare group. They lead,
respectively, to mass and spin. Thus, mass and spin are inevitable
properties of particles in a universe where SR is valid.
(1) P2 = ημν Pμ Pν = Pμ
Pμ with corresponding eigenvalues P2 = m2
which measure the invariant mass of field configurations.
In the real world we observe only time-like or light-like four-momenta,
i.e. particles with positive or zero mass. Furthermore, the temporal
components are always positive.
With dimensional units this would be P2 = m2c2
(2) W2 = ημν Wμ Wν = Wμ
Wμ with corresponding eigenvalues W2 = ( w02
- w·w ) = - (w·w) = - (P02j2)
= - m2 s(s+1),
which measure the invariant spin of the particle, where there are (2s+1)
spin states
(or 2 polarization/helicity states for massless fields)
with Wμ as the Pauli-Lubanski (mixed) Spin-Momentum four vector
With dimensional units this would be W2 = - m2c2ћ2
s(s+1)
Note: Massless representation give P2 = m2 = 0 and W2
= - m2 s(s+1) = 0
For instance, for a photonic Pμ = E(1,0,0,1), one has Wμ
= M12 Pμ
so that M12 takes the possible eigenvalues ± s
Wσ = (1/2) εσμνρ Mμν Pρ
or
Wσ = - (1/2) εμνρσ Mμν Pρ
such that
[ Wσ , Pμ ] = 0
[ Mμν , Wσ ] = -i ( Wμ gνσ - Wν
gμσ )
[ Wλ , Wσ ] = i ελσαβ Wα Pβ
Further,
W = (w0,w) = (p·j , P0j
- p x k)
w0 = p·j
w = P0j - p x k
where
j = (M32,M13,M21) are the 3
components of angular momentum, where [J1,J2] = i J3
and cyclic permutations
k = (M01,M02,M03) are boosts in 3
Cartesian directions
Wigner's classification: (non-negative energy irreducible unitary
representations of the Poincare group)
The irreducible unitary representations of the Poincare' group are
classified according to the eigenvalues of P2 and W2
They fall into several classes:
1a) P2 = m2 > 0 and P0 > 0: Massive
particle
1b) P2 = m2 > 0 and P0 < 0: Massive
anti-particle??
2a) P2 = 0 and P0 > 0: Photonic
2b) P2 = 0 and P0 < 0: Photonic??
3) P2 = 0 and P0 = 0: P in the 4-Zero,
the vacuum
4) P2 = m2 < 0: Tachyonic
A complete set of commuting observables is composed of P2, the
3 components of p, W2, and one of the 4 components of Wμ
The eigenvalues of P2 (mass) and W2 (spin)
distinguish (possibly together with other quantum numbers) different
particles. This is the general result for finite-mass quantum fields that
are invariant under the Poincare transformation.
In the case of the scalar field, it is straightforward to identify the
particle content of its Hilbert space.
A 1-particle state |k> = at(k)|0> is
characterized by the eigenvalues
p0|k> = ћω(k)|k>, p|k>
= ћk|k>, W2|k> = 0
thus showing that the quanta of such a quantum field may be identified
with particles of definite energy-momentum and mass m, carrying a
vanishing spin (in the massive case) or helicity (in the massless
case). Relativistic QFT's are thus the natural framework in which to
describe all the relativistic quantum properties, including the processes
of their annihilation and creation in interactions, or relativistic
point-particles. It is the Poincare invariance properties, the
relativistic covariance of such systems, that also justifies, on account
of Noether's theorem, this physical interpretation.
One has to learn how to extend the above description to more general field
theories whose quanta are particles of nonvanishing spin or helicity. One
then has to consider collections of fields whose components also mix under
Lorentz transforms.
One may list the representations which are invariant under parity and
correspond to the lowest spin/helicity content possible.
(0,0) |
φ |
scalar field |
(1/2,0) (+) (0,1/2) |
ψ |
Dirac spinor |
(1/2,1/2) |
Aμ |
vector field |
(1,0) (+) (0,1) |
Fuv = ∂uAv-∂vAu |
EM field tensor |
X = xμσμ = |
( x0 + x3 |
x1 - i x2 ) |
|
|
|
|
|
|
|
|
|
|
|
( x1 + i x2 |
x0 - x3 ) |
then Det[X] = x2 = x·x = ημν xμ
xν
see Proceedings of the Third International Workshop on Contemporary
Problems in Physics, By Jan Govaerts, M. Norbert Hounkonnou, Alfred
Z. Msezane
see Conceptual Foundations of Modern Particle Physics, Robert Eugene
Marshak
see Fundamentals of Neutrino Physics and Astrophysics, Carlo Giunti
see Kinematical Theory of Spinning Particles, Martin Rivas
>Spin |
>Statistics |
>Relativistic Eqn. |
>Relativistic Eqn. |
>Non-Relativistic Eqn. Newtonian Limit √[1+x] ~ (1+x/2) for x<<1 v<<c |
>Field |
>Polarizations |
0 |
Boson: |
FreeWave |
Klein-Gordon (-Fock) |
Schrödinger ( iћ∂t-V+[ћ2∇2/2m])Ψ = 0 with minimal EM coupling (iћ∂t-qφ-[(p-qa)2/2m])Ψ = 0 (iћ∂t-qφ-[(iћ∇-qa)2/2m])Ψ = 0 (iћ∂t-qφ+[(ћ∇+iqa)2/2m])Ψ = 0 Note that qφ = V is approx. when a~0 |
Ψ |
2? |
1/2 |
Fermion |
Weyl (σ·∂) Ψ = 0 |
Dirac |
Pauli, (Schrödinger-Pauli) with minimal EM coupling (i ћ∂t - qφ -[(σ·(p-qa)2)/2m])Ψ = 0 |
Ψ |
2 |
1 |
Boson |
Maxwell |
Proca |
A |
2 (= 2 transverse) |
|
3/2 |
Fermion |
Gravitino? |
Rarita-Schwinger |
ψσ |
2 |
|
2 |
Boson |
Einstein |
? ? |
tensor = 2-tensor |
2 |
Duffin-Kemmer-Petiau Equation = Complex Proca Equation
Duffin-Kemmer Equation: ( βμ pμ - M ) ψ = 0 : for a
free spin-0 or spin-1 particle
Dim |
Type |
|
Hodge Dual |
0 |
scalar |
|
|
1 |
vector |
|
|
2 |
tensor |
|
|
3 |
pseudovector |
magnetic field, spin, torque, vorticity, angular momentum |
|
4 |
pseudoscalar |
magnetic charge, magnetic flux, helicity |
|
In Minkowski space (4-dimensions), the { 1 4 6 4 1} Hodge dual of an
n-rank (n<=2) tensor will be an (4-n) rank skew-symmetric pseudotensor
Hodge duals
*dt = dx ^ dy ^ dz
*dx = dt ^ dy ^ dz
*dy = - dt ^ dx ^ dz
*dz = dt ^ dx ^ dy
*(dt ^ dx) = -dy ^ dz
*(dt ^ dy) = dx ^ dz
*(dt ^ dz) = -dx ^ dy
*(dx ^ dy) = dt ^ dz
*(dx ^ dz) = -dt ^ dy
*(dy ^ dz) = dt ^ dx
Main article: Schrödinger equation
With a simple division, the Schrödinger equation for a single particle of mass m in the absence of any applied force field can be rewritten in the following way:
This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:
Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the Schrödinger equation, which in turn can be used to obtain the wavefunction at any time through an integral on the wavefunction at t=0:
Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the wavefunction satisfying Schrödinger's equation might have an origin other than diffusion.
Some examples of equivalent electrical and hydraulic equations:
type |
hydraulic |
electric |
thermal |
---|---|---|---|
quantity |
volume V [m3] |
charge q [C] |
heatQ [J] |
potential |
pressure p [Pa=J/m3] |
potential φ [V=J/C] |
temperature T [K=J/kB] |
flux |
current ΦV [m3/s] |
current I [A=C/s] |
heat transfer rate [J/s] |
flux density |
velocity v [m/s] |
j [C/(m2·s) = A/m²] |
heat flux [W/m2] |
linear model |
Poiseuille's law |
Ohm's law |
Fourier's law |
Classical Dynamics of Particles & Systems, 3rd Ed., Jerry B.
Marion & Stephen T. Thornton (Chap14)
Classical Electrodynamics, 2nd Ed., J.D. Jackson (Chap11,12)
Classical Mechanics, 2nd Ed., Herbert Goldstein (Chap7,12)
Electromagnetic Field, The, Albert Shadowitz (Chap13-15)
First Course in General Relativity, A, Bernard F. Schutz (Chap1-4)
Fundamental
Formulas of Physics, by Donald Howard Menzel (Chap6)
Introduction to Electrodynamics, 2nd Ed., David J. Griffiths
(Chap10)
Introduction to Modern Optics, 2nd Ed., Grant R. Fowles (var)
Introduction to Special Relativity, 2nd Ed., Wolfgang Rindler (All)
(**pg60-65,82-86**)
Lectures on Quantum Mechanics, Gordon Baym (Chap22,23)
Modern Elementary Particle Physics: The Fundamental Particles and
Forces?, Gordon Kane (Chap2+)
Path Integrals and Quantum Processes, Mark Swanson (var)
Quantum Electrodynamics, Richard P. Feynman (Lec7-rest)
Quantum Mechanics, Albert Messiah (Chap20)
Quantum Mechanics and the Particles of Nature: An Outline for
Mathematicians, Anthony Sudbery (Chap7)
Spacetime and Geometry: An Introduction to General Relativity, Sean
M. Carroll (var)
Statistical
Mechanics, by R. K. Pathria
(Chap6.5)
Theory of Spinors, The, E'lie Cartan (var)
Topics in Advanced Quantum Mechanics, Barry R. Holstein (Chap3,6,7)
Relativistic
Quantum
Fields, Mark Hindmarsh, Sussex, UK
Relativity
and
electromagnetism, Richard Fitzpatrick, Associate Professor
of Physics, The University of Texas at Austin
http://farside.ph.utexas.edu/teaching/em/lectures/node106.html
The Relativistic
Boltzmann Equation: Theory and Applications, Carlo
Cercignani, Gilberto Medeiros Kremer
Essential
Relativity: Special, General, and Cosmological, by
Wolfgang Rindler
Compendium
of
Theoretical Physics, by Armin Wachter, Henning
Hoeber
Relativistic
Quantum
Mechanics of Leptons and Fields, by Walter T. Grandy
This remains a work in progress.
Email me, especially if you notice errors (which I will fix ASAP) or have interesting comments.
Please, send comments to John Wilson
quantum
relativity
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