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Last modified: March 1, 2022 ✓

4-vectors
---------

In Minkowski space one needs 4 real numbers (three 
space coordinates and one time coordinate) to refer 
to a point at a particular instant of time. This point 
at a particular instant of time, specified by the 
four coordinates, is called an event (or 4-vector). 
The distance between two different events is called 
the spacetime interval.  A path through Minkowski 
space, is called a world line. Since it specifies 
both position and time, a particle having a known 
world line has a completely determined trajectory 
and velocity. This is just like graphing the 
displacement of a particle moving in a straight 
line against the time elapsed. The curve contains 
the complete motional information of the particle.

4-Position:

xμ = {ct,x}

4-Velocity:

uμ = dxμ/dτ  = {γc,γv}

4-Acceleration (less straightforward):
                 .   .     
aμ = duμ/dτ =  {γγc,γγv + γ2a}

4-Momentum:

pμ = muμ = {γmc,γmv}

4-Force:
              .     .   
Fμ = maμ = {mγγc,mγγv + mγ2a}

4-Current:

jμ = (cρ,jx,jy,jz)

4-Gradient:

∂μ = (∂t/c,∂x,∂y,∂z)

The gradient of any 4-vector is found by applying 
the 4-gradient to the 4-vector in question, i.e.

∂μjμ = ∂ρ/∂t + ∂jx/∂x + ∂jy/∂y + ∂jz/∂z = 0

Example:

The gradient of a scalar is vector with covariant 
components.

Proof: Consider a scalar field φ.  Let dxμ 
represent the distance between 2 points in the 
field.  The change in φ w.r.t. xμ is given by:

    dφ = (∂φ/∂xμ)dxμ.  

If LHS is a scalar then RHS must be a scalar.  
Thus, ∂φ/∂xμ must be equivalent to dxμ.  Also, 

dxμ = (∂xμ/∂φ)dφ

Therefore,

dxμdxμ = (∂φ/∂xμ)(∂xμ/∂φ)dφ

       = dφ

With this is mind:

∂μφ = ∂φ/∂xμ 

    = {∂φ/∂t,∇φ} = {(1/c)∂φ/∂t,∂φ/∂x1,∂φ/∂x2,∂φ/∂x3} 

∂νφ = ημν∂μφ 

    = {∂φ/∂t,-∇φ} = {(1/c)∂φ/∂t,-∂φ/∂x1,-∂φ/∂x2,-∂φ/∂x3} 

∂μ∂μφ = (1/c2)∂φ2/∂t2 - ∇2φ

where,

(1/c2)∂2/∂t2 - ∇2 = □ ... the d'Alembert operator

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