From Special Relativity to General Relativity

Special Relativity¶

History¶

The problem with classical mechanics¶

In the 19th century, Maxwell’s equations were established.
A fundamental problem appeared: Maxwell’s equations are not invariant under Galilean transformations, while classical mechanics is.

This led to three logical possibilities:

Maxwell’s equations are valid only in one special inertial frame
Classical mechanics is wrong
Maxwell’s equations are wrong

Experiments confirmed Maxwell’s equations with high precision, so option (3) is excluded.

Option (1) introduced the ether hypothesis, a preferred inertial frame in which light propagates.
The Michelson–Morley experiment showed no evidence for such a frame, ruling out the ether.

The only remaining option is (2): classical mechanics must be modified.

Postulates of Special Relativity¶

Special relativity is based on two postulates:

Principle of relativity
The laws of physics have the same form in all inertial frames.
Constancy of the speed of light
Light propagates in vacuum with the same speed ( c ), independent of the motion of the source or the observer.

Consequences¶

Time and space are not absolute
Space and time are unified into spacetime
Physical laws are invariant under Poincaré transformations

Spacetime and the Minkowski Metric¶

Events are described by spacetime coordinates:

x^\mu = (ct,\,x,\,y,\,z)

(1)

The geometry of spacetime is defined by the Minkowski metric:

\eta_{\mu\nu} = \mathrm{diag}(-1,\,1,\,1,\,1)

(2)

The invariant spacetime interval between two infinitesimally close events is:

ds^2 = \eta_{\mu\nu}\,dx^\mu\,dx^\nu = -c^2 dt^2 + dx^2 + dy^2 + dz^2

(3)

This quantity is invariant in all inertial frames.

Causal Structure¶

For two events separated by an interval ( ds^2 ):

$ds^2 < 0$ : timelike separation (causal connection possible)
$ds^2 = 0$ : lightlike separation
$ds^2 > 0$ : spacelike separation (no causal connection)

Poincaré Group and Lorentz Transformations¶

Poincaré transformations¶

The most general symmetry of special relativity is:

x'^{\mu} = \Lambda^{\mu}{}_{\nu} x^{\nu} + c^{\mu}

(4)

where:

$\Lambda^{\mu}{}_{\nu}$ is a Lorentz transformation
$c^\mu$ is a constant spacetime translation

Lorentz transformations¶

Lorentz transformations satisfy:

\eta_{\rho\sigma} \Lambda^{\rho}{}_{\mu} \Lambda^{\sigma}{}_{\nu} = \eta_{\mu\nu}

(5)

This condition guarantees the invariance of the spacetime interval:

ds'^2 = ds^2

(6)

Hence, all inertial observers agree on the causal structure of spacetime.

From Special Relativity to General Relativity¶

Tensors in Flat Spacetime¶

In special relativity, spacetime is flat and described by the Minkowski metric (\eta_{\mu\nu}).
Under a Lorentz transformation:

x^\mu \to x'^\mu = \Lambda^\mu{}_\nu x^\nu

(7)

A tensor of rank (n) transforms as:

T^{\mu_1 \dots \mu_n} \to T'^{\mu_1 \dots \mu_n} = \Lambda^{\mu_1}{}_{\nu_1} \dots \Lambda^{\mu_n}{}_{\nu_n} T^{\nu_1 \dots \nu_n}

(8)

Rank 0: Scalars¶

A scalar field (\phi(x)) is invariant under Lorentz transformations:

\phi'(x') = \phi(x)

(9)

The Minkowski interval and the proper time of a particle are scalars:

ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu = -c^2 d\tau^2

(10)

Rank 1: 4-Vectors¶

4-position: $x^\mu = (ct, \vec{x})$
4-velocity:

u^\mu = \frac{dx^\mu}{d\tau} = \gamma (c, \vec{v})

(11)

with $\gamma = \frac{dt}{d\tau} = \frac{1}{\sqrt{1-v^2/c^2}}$

4-acceleration: $a^\mu = \frac{du^\mu}{d\tau}$

Derivative Operators¶

Covariant derivative in flat spacetime:

\partial_\mu = \frac{\partial}{\partial x^\mu}, \quad \partial^\mu = \eta^{\mu\nu} \partial_\nu

(12)

Components:

\partial_\mu = \left(\frac{\partial}{\partial ct}, \nabla \right), \quad \partial^\mu = \left(-\frac{\partial}{\partial ct}, \nabla \right)

(13)

D’Alembertian (wave) operator:

\square = \partial_\mu \partial^\mu = -\frac{\partial^2}{\partial t^2} + \nabla^2

(14)

This operator is Lorentz invariant.

Rank 2: Tensors¶

Covariant equations are built using tensors to ensure laws are the same in all inertial frames.
Example: Klein-Gordon equation for a scalar field (\phi):

\square \phi - m^2 \phi = 0

(15)

Maxwell Equations as Covariant Tensor Equations¶

Electromagnetic fields are described by the field strength tensor (F_{\mu\nu}):

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu

(16)

Maxwell’s equations in vacuum:

\partial_\mu F^{\mu\nu} = \mu_0 J^\nu, \quad \partial_\alpha F_{\beta\gamma} + \partial_\beta F_{\gamma\alpha} + \partial_\gamma F_{\alpha\beta} = 0

(17)

These are manifestly covariant under Lorentz transformations.

Why Covariance Matters¶

Equations that transform like tensors under Lorentz transformations are covariant, meaning all inertial observers see the same physics. This is crucial for building relativistic laws of nature.

Lagrangian and Action Principle¶

Classical Mechanics¶

For a particle with coordinate (q(t)):

L(q, \dot{q}) = T - V = \frac{1}{2} m \dot{q}^2 - V(q)

(18)

Action:

S[q] = \int dt\, L(q, \dot{q})

(19)

Euler-Lagrange equation:

\frac{d}{dt} \frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0

(20)

Action in Special Relativity¶

The Lagrangian must be a scalar to produce covariant equations of motion.
Free particle:

S = -m c \int d\tau = -m c \int \sqrt{-\eta_{\mu\nu} dx^\mu dx^\nu}

(21)

Free scalar field:

\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2

(22)

Electromagnetic field (Maxwell):

\mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} - J_\mu A^\mu

(23)

These Lagrangians are scalars, ensuring covariant equations under Lorentz transformations.

Summary¶

Scalars, vectors, and tensors provide a language for relativistic physics.
Proper choice of Lagrangian guarantees covariant laws of motion.
Maxwell equations, Klein-Gordon equations, and free particles are tensor equations, valid in all inertial frames.