John's Online SR 4-Vector & Tensor Mathematics Calculator


SR 4-tensors, enjoy! A powerful, online, Calculator, written in Javascript.
It employs tensor math, and will have hundreds of tensors and tensor functions in proper time. :)
Currently has over 100 SR Tensors and calculates the Invariant (Lorentz Scalar Self-Product) of all 4-Vectors and the Invariant (Trace & Determinant & InnerProduct) of all 4-Tensors.
It lists, but doesn't really calculate, the EigenValues for type (1,1)-Tensors. I am working on Tensor Addition and Tensor Products, Exterior Products, Index Raising and Index Lowering.
It calculates the symmetry of 2-index Tensors.
See also, my Online RPN Complex-capable Scientific Calculator

** Note ** This is still a work in progress... If you notice any errors, please bring them to my attention, I will fix them. Thanks!
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Welcome to Relativistic Quantum Reality: Virtual worlds of imaginary particles: The dreams stuff is made of: Life, the eternal ghost in the machine...
This site is dedicated to the quest for knowledge and wisdom, through science, mathematics, philosophy, invention, and technology. 
May the treasures buried herein spark the fires of imagination like the twinkling jewels of celestial light crowning the midnight sky...

Quantum Mechanics is derivable from Special Relativity
See SRQM - QM from SR - Simple RoadMap (.html)
See SRQM - QM from SR - Simple RoadMap (.pdf)
See SRQM - Online SR 4-Vector & Tensor Calculator

***Learn, Discover, Explore***


Site last modified: 2019-Feb-03


Tensor Matrix

Tensor Information


(0,0)-Tensors
4-Scalars
Lorentz Invariant Scalars

(1,0)-Tensors
4-Vectors

(0,1)-Tensors
4-CoVectors
One-Forms

(2,0)-Tensors
4-Tensors

(1,1)-Tensors
4-MixedTensors, inc. Lorentz Transformations
( Tr[]:Type | Det[]:Proper = +1,Improper = -1 | IP[]:ST_Dim = 4 )
(0,2)-Tensors
4-LoweredTensors

Universal:




εoμo = 1/c2


SR Particle Dependent:








SR Fluid Dependent:






SR Situation|Interaction Dependent:






SR Conservation Laws:






Physical:



























Mathematical:















Projection Tensors:
Minkowski Diagram
(V)ertical + (H)orzintal = η
Temporal + Spatial = TimeSpace






Lorentz Transforms:


Proper (Matter):






Improper (Matter):





Proper (AntiMatter):






Improper (AntiMatter):






Projection Tensors:








Projection Tensors:


Calculated:






















DecimalDigits:





(0,0)-Tensors
4-Scalars
Lorentz Invariant Scalars

(1,0)-Tensors
4-Vectors

(0,1)-Tensors
4-CoVectors
One-Forms

(2,0)-Tensors
4-Tensors

(1,1)-Tensors
4-MixedTensors, inc. Lorentz Transformations
( Tr[]:Type | Det[]:Proper = +1,Improper = -1 | IP[]:ST_Dim = 4 )
(0,2)-Tensors
4-LoweredTensors

Tensor Matrix 2
xxx

Tensor Information 2


xxx




I always choose to have the 4-Vector (or 3-vector) refer to the upper index tensor of the same name.
4-Vector A = Aμ = (a0,a1,a2,a3) = (a0,a) = (a0,ai)
3-vector a = ai = (a1,a2,a3)
In addition, I like the convention of having the (c) factor in the temporal part for correct dimensional units. {eg. 4-Position R = (ct,r) overall units = [length]}
This also allows the SR 4-Vector name to match the classical 3-vector name, which is useful when considering Newtonian limiting-cases.

I use:
UPPER CASE BOLD, Greek index range = {0..3}, for 4-Vectors (A = Aμ), and sometimes an under-line symbol ( A ) when writing-by-hand.
lower case bold, Latin index range = {1..3}, for 3-vectors (a = ai), and sometimes an over-arrow (vector) symbol (a⃑ ) when writing-by-hand.

Note that the ( i ) in the index is not the imaginary ( i ).  Try to use other (non-i) index letters when imaginary (i) is elsewhere in a tensor equation to avoid confusion.
Likewise, try to use different index letters than the other variables in an equation.
Many, many other texts do not use different letters.  It can be very confusing and ambiguous.

I also adopt Wolfgang Rindler's convention of using:
 lower case bold (e) for the Electric field 3-vector e = ei → (ex,ey,ez)
 lower case bold (b) for the Magnetic field 3-vector b = bi → (bx,by,bz)
 *Note* (e) and (b) are not 4-Vectors, nor the spatial parts of 4-Vectors, but combined are components of the 2-index Faraday EM Tensor Fμν
 You can get close though...
4-EM Force FEM = γq[ (u·e)/c, (e) + (u⨯b) ]
In a rest frame: FEMo = q(0,e), so (e) is the spatial part of  (FEMo/q), but only in the rest frame.
e = ei = cFi0 and b = bk = -(1/2)εijkFij

Following the same idea, I am also implementing:
 lower case bold (a) for the Electromagnetic 3-vector VectorPotential (spatial component)
 while retaining the historical (φ) for the Electromagnetic ScalarPotential (temporal component)
 A = (φ/c,a) → (φ/c,ax,ay,az)

Also, I use the "at-rest" notation "naught" (o) to differentiate from the 0-index component of 4-Vector.
Thus
v0 is the 0th-index component of Vμ = (v0,v1,v2,v3)
v0 is the 0th-index component of Vμ = (v0,v1,v2,v3)
vo is the "at-rest" Lorentz Scalar, which usually relates two separate 4-Vectors, eg. Vμ = voTμ
v0o is the "at-rest" 0th-index component and Lorentz scalar for V·T = v0o
Based on this, I prefer restmass = mo instead of m0.


All SR 4-Vectors have the following properties:
==============================================

A = Aμ = (a0,ai) = (a0,a) = (a0,a1,a2,a3) → (at,ax,ay,az): A typical 4-Vector (contravariant = upper index)

       Aμ = (a0,ai) = (a0,-a) = (a0,a1,a2,a3) → (at,ax,ay,az): A typical 4-Covector (covariant = lower index)
                          = (a0,-a) = (a0,-a1,-a2,-a3) → (at,-ax,-ay,-az):

with Aμ = ημνAν and Aμ = ημνAν: Tensor index lowering and raising with the Minkowski Metric ημν

One can also use A = Aμ = (A0, Ai) = (A0,A1,A2,A3) to demonstrate 4-Vector mathematics,
but once you start doing physics, most of the component values (physical variable names) are lowercase letters,
An exception is that the Energies are usually uppercase E = Energy, U = PotentialEnergy, H = Hamiltonian.
The energy-densities are usually lowercase: ue = ε = e = ρe = EnergyDensity, u = PotentialEnergyDensity, etc.
Therefore, I prefer A = Aμ = (a0,ai) = (a0,a) = (a0,a1,a2,a3), which also avoids the weirdness of (A0,a).
However, there are a few cases where the Uppercase letters are helpful.
4-Velocity U = Uμ = (U0,Ui) = γ(c,u) = (γc,γu)
The relativistic spatial component of 4-Velocity is ( Ui = γui = γu ), which is actually equal to the Lorentz gamma factor (γ) * spatial Newtonia 3-velocity ( u = ui ).
This type of thing also occurs in the 4-Acceleration and the 4-Force, and a few others.


The main idea that makes a generic 4-Vector into an SR 4-Vector is that it must transform properly according to a Lorentz Transformation Λμ'ν (or sometimes written as Lμ'ν).
(Aμ' = Λμ'νAν) with Λμ'ν as the Lorentz Transformation tensor. { β = v/c = βx + βy + βz} and {γ = 1/√[1-β2]}
with both (Aμ' and Aν) both representing the same kind of SR 4-Vector (primed index after transform, unprimed index before transform).
We also have (Tr[Λμρ Λνσ] = Λμρ ημν Λνσ = ηρσ) as a general rule.
Likewise: {  (Λ-1)νμ = Λμν } and [(Λ-1)T]μν = Λνμ }



If you have a comment about this site, or find a bug, or want more features added, or just want to say 'hi'
Please, send comments to John Wilson
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Ambigram Quantum Relativity


See SRQM: QM from SR - The 4-Vector RoadMap (.html)
See SRQM: QM from SR - The 4-Vector RoadMap (.pdf)


SRQM 4-Vector : Four-Vector and Lorentz Scalar Diagram
SRQM 4-Vector : Four-Vector and Lorentz Scalar Diagram


SRQM + EM 4-Vector : Four-Vector and Lorentz Scalar Diagram
SRQM + EM 4-Vector : Four-Vector and Lorentz Scalar Diagram


SRQM + EM 4-Vector : Four-Vector and Lorentz Scalar Diagram With Tensor Invariants
SRQM + EM 4-Vector : Four-Vector and Lorentz Scalar Diagram With Tensor Invariants


SRQM 4-Vector : Four-Vector Stress-Energy & Projection Tensors Diagram
SRQM 4-Vector : Four-Vector Stress-Energy & Projection Tensors Diagram


SRQM 4-Vector : Four-Vector SR Quantum RoadMap
SRQM 4-Vector : Four-Vector SR Quantum RoadMap


SRQM + EM 4-Vector : Four-Vector SR Quantum RoadMap
SRQM + EM 4-Vector : Four-Vector SR Quantum RoadMap


SRQM 4-Vector : Four-Vector New Relativistic Quantum Paradigm
SRQM 4-Vector : Four-Vector New Relativistic Quantum Paradigm


SRQM + EM 4-Vector : Four-Vector New Relativistic Quantum Paradigm (with EM)
SRQM + EM 4-Vector : Four-Vector New Relativistic Quantum Paradigm (with EM)


SRQM 4-Vector : Four-Vector New Relativistic Quantum Paradigm - Venn Diagram
SRQM 4-Vector : Four-Vector New Relativistic Quantum Paradigm - Venn Diagram


SRQM 4-Vector : Four-Vector SpaceTime is 4D
SRQM 4-Vector : Four-Vector SpaceTime is 4D


SRQM 4-Vector : Four-Vector SpaceTime Orthogonality
SRQM 4-Vector : Four-Vector SpaceTime Orthogonality


SRQM 4-Vector : Four-Vector 4-Position, 4-Velocity, 4-Acceleration Diagram
SRQM 4-Vector : Four-Vector 4-Position, 4-Velocity, 4-Acceleration Diagram


SRQM 4-Vector : Four-Vector 4-Displacement, 4-Velocity, Relativity of Simultaneity Diagram
SRQM 4-Vector : Four-Vector 4-Displacement, 4-Velocity, Relativity of Simultaneity Diagram


SRQM 4-Vector : Four-Vector 4-Velocity, 4-Gradient, Time Dilation Diagram
SRQM 4-Vector : Four-Vector 4-Velocity, 4-Gradient, Time Dilation Diagram


SRQM 4-Vector : Four-Vector 4-Vector, 4-Velocity, 4-Momentum, E=mc^2 Diagram
SRQM 4-Vector : Four-Vector 4-Vector, 4-Velocity, 4-Momentum, E=mc^2 Diagram


SRQM 4-Vector : Four-Vector 4-Velocity, 4-WaveVector, Relativistic Doppler Effect Diagram
SRQM 4-Vector : Four-Vector 4-Velocity, 4-WaveVector, Relativistic Doppler Effect Diagram


SRQM 4-Vector : Four-Vector Wave-Particle Diagram
SRQM 4-Vector : Four-Vector Wave-Particle Diagram


SRQM 4-Vector : Four-Vector Compton Effect Diagram
SRQM 4-Vector : Four-Vector Compton Effect Diagram


SRQM 4-Vector : Four-Vector Aharonov-Bohm Effect Diagram
SRQM 4-Vector : Four-Vector Aharonov-Bohm Effect Diagram


SRQM 4-Vector : Four-Vector Josephson Junction Effect Diagram
SRQM 4-Vector : Four-Vector Josephson Junction Effect Diagram


SRQM 4-Vector : Four-Vector Hamilton-Jacobi vs Action, Josephson vs Aharonov-Bohm Diagram
Four-Vector Hamilton-Jacobi vs Action, Josephson vs Aharonov-Bohm Diagram


SRQM 4-Vector : Four-Vector Motion of Lorentz Scalar Invariants
SRQM 4-Vector : Four-Vector Motion of Lorentz Scalar Invariants


SRQM 4-Vector : Four-Vector Motion of Lorentz Scalar Invariants, Conservation Laws & Continuity Equations
SRQM 4-Vector : Four-Vector Motion of Lorentz Scalar Invariants, Conservation Laws & Continuity Equations


SRQM 4-Vector : Four-Vector Speed of Light (c)
SRQM 4-Vector : Four-Vector Speed of Light (c)


SRQM 4-Vector : Four-Vector Minimal Coupling Conservation of 4-TotalMomentum)
SRQM 4-Vector : Four-Vector Minimal Coupling Conservation of 4-TotalMomentum


SRQM 4-Vector : Four-Vector Relativistic Action (S) Diagram
SRQM 4-Vector : Four-Vector Relativistic Action (S) Diagram


SRQM 4-Vector : Four-Vector Relativistic Lagrangian Hamiltonian Diagram
SRQM 4-Vector : Four-Vector Relativistic Lagrangian Hamiltonian Diagram


SRQM 4-Vector : Four-Vector Relativistic Euler-Lagrange Equation
SRQM 4-Vector : Four-Vector Relativistic Euler-Lagrange Equation


SRQM 4-Vector : Four-Vector Relativistic EM Equations of Motion
SRQM 4-Vector : Four-Vector Relativistic EM Equations of Motion


SRQM 4-Vector : Four-Vector Einstein-de Broglie Relation hbar
SRQM 4-Vector : Four-Vector Einstein-de Broglie Relation hbar


SRQM 4-Vector : Four-Vector Quantum Canonical Commutation Relation
SRQM 4-Vector : Four-Vector Quantum Canonical Commutation Relation


SRQM 4-Vector : Four-Vector QM Schroedinger Relation
SRQM 4-Vector : Four-Vector QM Schroedinger Relation


SRQM 4-Vector : Four-Vector Quantum Probability
SRQM 4-Vector : Four-Vector Quantum Probability


SRQM 4-Vector : Four-Vector CPT Theorem
SRQM 4-Vector : Four-Vector CPT Theorem


SRQM 4-Vector : Four-Vector Lorentz Transforms Connection Map
SRQM 4-Vector : Four-Vector Lorentz Transforms Connection Map


SRQM 4-Vector : Four-Vector Lorentz Discrete Transforms
SRQM 4-Vector : Four-Vector Lorentz Discrete Transforms


SRQM 4-Vector : Four-Vector Lorentz Transforms - Trace Identification
SRQM 4-Vector : Four-Vector Lorentz Transforms - Trace Identification


SRQM 4-Vector : Four-Vector Lorentz Lorentz Transforms-Interpretations
CPT Symmetry, Baryon Asymmetry Problem Solution, Matter-Antimatter Symmetry Solution, Arrow-of-Time Problem Solution, Big-Bang!
SRQM 4-Vector : Four-Vector Lorentz Transforms-Interpretations, CPT Symmetry, Baryon Asymmetry Problem Solution, Matter-Antimatter Symmetry Solution, Arrow-of-Time Problem Solution, Big-Bang!


See SRQM: QM from SR - The 4-Vector RoadMap (.html)
See SRQM: QM from SR - The 4-Vector RoadMap (.pdf)


The Science Realm: John's Virtual Sci-Tech Universe
John's Science & Math Stuff: | About John | Send email to John
4-Vectors | Ambigrams | Antipodes | Covert Ops Fragments | Cyrillic Projector | Forced Induction (Sums Of Powers Of Integers) | FractaTM | Frontiers |
JavaScript Graphics | Kid Science | Kryptos | Photography | Prime Sieve | QM from SR | QM from SR-Simple RoadMap | Quantum Phase | Quotes |
RuneQuest Cipher Challenge | Scientific Calculator | Secret Codes & Ciphers | Science+Math | Sci-Pan Pantheist Poems | Stereograms | Turkish Grammar |

Quantum Mechanics is derivable from Special Relativity
See QM from SR-Simple RoadMap