Minkowski Spacetime and Four-Vector Formalism

Manual: General · Subject: Physics

Introduces the geometric language of special relativity using four-vectors, tensors, and invariant intervals.

Geometry of Spacetime

Minkowski Metric

In flat spacetime, the interval is written as $ds^2 = \eta_{\mu\nu}dx^\mu dx^\nu,$ with metric signature conventions commonly chosen as $\eta_{\mu\nu} = \mathrm{diag}(1,-1,-1,-1)$ or $(-1,1,1,1)$ . The choice of signature is conventional; invariants do not depend on it.

Four-Vectors

A spacetime event is represented by $x^\mu = (ct, x, y, z)$ . The four-velocity is $u^\mu = \frac{dx^\mu}{d\tau},$ and the four-momentum is $p^\mu = mu^\mu = \left(\frac{E}{c}, \mathbf{p}\right).$

Invariant Norms

The four-velocity satisfies $u^\mu u_\mu = c^2$ under the $(1,-1,-1,-1)$ signature, while the four-momentum satisfies $p^\mu p_\mu = m^2c^2.$ These relations encode the invariant rest mass.

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Why Geometry Matters

Many special-relativistic calculations become one-line statements when expressed as invariant contractions of tensors.

What is the geometrical significance of the spacetime interval?

It is frame-dependent and changes under boosts It is invariant under Lorentz transformations It equals the spatial distance only It is meaningful only for lightlike trajectories

Write the energy-momentum four-vector in standard form.

Common Four-Vectors

Object	Components	Invariant
Position	$x^\mu = (ct, x, y, z)$	$ds^2$
Four-velocity	$u^\mu = dx^\mu/d\tau$	$u^\mu u_\mu = c^2$
Four-momentum	$p^\mu = (E/c, \mathbf{p})$	$p^\mu p_\mu = m^2c^2$
Four-current	$J^\mu = (c\rho, \mathbf{J})$	$\partial_\mu J^\mu = 0$

Why is proper time especially useful in relativistic dynamics?

It depends on the coordinate system chosen by the observer It is an invariant parameter along a timelike worldline It is identical to coordinate time in all frames It is defined only for photons

Four-Vector Workflow

1

Step 1: Package physical quantities into covariant objects.
2

Step 2: Use metric contractions to identify invariants.
3

Step 3: Translate scalar relations into frame-independent statements.
4

Step 4: Recover component formulas only at the end if needed.

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