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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:
x^{μ} = {ct, x}
Four Velocity:
u^{μ} = dx^{μ}/dτ = {γc, γv}
Four Acceleration (less straightforward):
^{ } . .
a^{μ} = du^{μ}/dτ = {γγc, γγv + γ^{2}a}
Four Momentum:
p^{μ} = mu^{μ} = {γmc, γmv}
Four Force:
^{ } . .
F^{μ} = ma^{μ} = {mγγc, mγγv + mγ^{2}a}
Four Gradient:
The gradient of a scalar is vector with covariant
components.
Proof: Consider a scalar field φ. Let dx^{μ}
represent the distance between 2 points in the
field. The change in φ w.r.t. x^{μ} is given by:
dφ = (∂φ/∂x^{μ})dx^{μ}.
If LHS is a scalar then RHS must be a scalar.
Thus, ∂φ/∂x^{μ} must be equivalent to dx_{μ}. Also,
dx^{μ} = (∂x^{μ}/∂φ)dφ
Therefore,
dx_{μ}dx^{μ} = (∂φ/∂x^{μ})(∂x^{μ}/∂φ)dφ
= dφ
With this is mind:
∂_{μ}φ = ∂φ/∂x^{μ}
= {∂φ/∂t, ∇φ} = {(1/c)∂φ/∂t, ∂φ/∂x^{1}, ∂φ/∂x^{2}, ∂φ/∂x^{3}}
∂^{ν}φ = η^{μν}∂_{μ}φ
= {∂φ/∂t, -∇φ} = {(1/c)∂φ/∂t, -∂φ/∂x^{1}, -∂φ/∂x^{2}, -∂φ/∂x^{3}}
∂^{μ}∂_{μ}φ = (1/c^{2})∂φ^{2}/∂t^{2} - ∇^{2}φ
where,
(1/c^{2})∂^{2}/∂t^{2} - ∇^{2} = □ ... the d'Alembert operator