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Last modified: November 18, 2021 ✓

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.

Four Position:

xμ = {ct, x}

Four Velocity:

uμ = dxμ/dτ  = {γc, γv}

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

Four Momentum:

pμ = muμ = {γmc, γmv}

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

Four Gradient:

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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