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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