Special Theory of Relativity and 4-Tensors

Two Postulates of Einstein

In 1905, Einstein proposed a radically new view of space and time, based on two postulates:

1. Principle of Relativity: The laws of physics are the same in all inertial reference frames
2. Constancy of the Speed of Light: The speed $c = 3 \times 10^8$ m/s is the same in all inertial frames, regardless of the motion of the source or observer

The second postulate seems paradoxical: if I run at speed $v$ towards the light, shouldn’t the light move relative to me at $c + v$? No — and all experiments confirm this. The consequence: our intuitive ideas about time and space are mistaken.

Minkowski Spacetime

Instead of separate space and time, Minkowski proposed a unified 4-dimensional spacetime. An event is a point $(t, x, y, z)$. The interval between two events:

$ ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 = \eta^{\mu\nu} dx^\mu dx^\nu $

Minkowski metric: $\eta^{\mu\nu} = \mathrm{diag}(+1, -1, -1, -1)$. Symbols: $dx^\mu = (c,dt, dx, dy, dz)$ — coordinates of a 4-vector (upper index — contravariant). The sum with the metric tensor $\eta^{\mu\nu}$ is the scalar product in spacetime.

Key property: $ds^2$ is an invariant (the same in all inertial frames). Depending on the sign: $ds^2 > 0$ — “timelike” interval (causal connection possible); $ds^2 < 0$ — “spacelike” (simultaneous in some frame); $ds^2 = 0$ — “lightlike” (worldline of a photon).

Proper time: $d\tau^2 = ds^2/c^2 = dt^2(1 - v^2/c^2)$. This is the time measured by a clock moving with the particle. $\tau$ is invariant. $\gamma = 1/\sqrt{1 - v^2/c^2}$ — Lorentz factor.

Lorentz Transformations

A boost along $x$ with velocity $v$:

$ t' = \gamma (t - vx/c^2) \ x' = \gamma(x - vt) \ y' = y, \quad z' = z $

Consequences:

Time Dilation: $\Delta t = \gamma \Delta \tau > \Delta \tau$. Moving clocks run slower. Muons from cosmic rays are created at an altitude of 10 km at a speed of 0.998$c$. Their proper lifetime is 2.2 $\mu$s. Classically: they would travel only $0.998 \times 3 \times 10^8 \times 2.2 \times 10^{-6} \approx 660$ m. But $\gamma \approx 15$, and the laboratory lifetime is $\gamma \times 2.2,\mu\text{s} \approx 33,\mu\text{s}$. Distance traveled: 10 km — muons reach the Earth. Confirmed experimentally!

Length Contraction: $L = L_0/\gamma$. In the muon’s frame: the distance to Earth contracts from 10 km to $10/15 \approx 670$ m — exactly the distance a muon can travel.

Relativity of Simultaneity: Events simultaneous in one system ($dt=0$ with $dx \neq 0$) are not simultaneous in another.

4-Vectors and Tensors

4-Momentum: $p^\mu = m u^\mu = \left(\frac{E}{c}, \vec{p}\right)$. Invariant: $p^\mu p_\mu = m^2 c^2 \rightarrow E^2 = (pc)^2 + (mc^2)^2$.

At $m=0$ (photon): $E = pc$, $E = \hbar \omega$, $p = \hbar k$. For a fast particle as $v \to c$: $E \approx pc \rightarrow E = \hbar \omega$ (as for a photon). This is the boundary between the classical and quantum.

Transformation of a 4-Tensor $T^{\mu\nu}$: $T'^{\mu\nu} = \Lambda^\mu_\alpha \Lambda^\nu_\beta T^{\alpha\beta}$, where $\Lambda$ is the Lorentz matrix. Any physical law written in 4-tensor form is automatically covariant — it looks the same in all inertial frames.

Electromagnetic Tensor

$\vec{E}$ and $\vec{B}$ are unified in the antisymmetric Faraday tensor $F^{\mu\nu}$:

$ F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu \quad (\text{where } A^\mu = (\varphi/c, \vec{A}) \text{ is the 4-potential}) $

The four Maxwell equations are written as two compact covariant equations: $\partial_\nu F^{\mu\nu} = \mu_0 J^\mu$ and $\partial_{[\alpha} F_{\beta\gamma]} = 0$. The covariant form shows that Maxwell’s equations are originally relativistically invariant — Einstein simply did not alter them, but demonstrated their true geometric meaning.

Numerical Example: Pion Decay

A neutral pion $\pi^0$ is at rest and decays into two photons. Mass of the pion: $m_\pi = 135$ MeV/$c^2$. In the rest frame: both photons fly in opposite directions, each with energy $E = m_\pi c^2/2 = 67.5$ MeV.

If the pion moves at speed $v = 0.9c$ ($\gamma = 2.29$), and one photon flies forward, the other backward. In the laboratory frame: $E_1 = \gamma(1+\beta)\cdot 67.5,\text{MeV} \approx 286,\text{MeV}$; $E_2 = \gamma(1-\beta)\cdot 67.5,\text{MeV} \approx 16,\text{MeV}$. The asymmetry of photons is directly observed in particle detectors.

Special Theory of Relativity in Practical Systems

Relativistic effects are important not only for particle physics, but also for critical engineering systems. GPS satellites move at a speed of about 3.9 km/s and experience a clock slowdown of 7.2 $\mu$s/day due to special relativity; at the same time, the gravitational blue shift from general relativity adds plus 45.9 $\mu$s/day. The total correction of about 38 $\mu$s/day is accounted for in GPS receiver firmware: without relativistic corrections, positioning error would accumulate at about 10 km/day. In particle accelerators (LHC at CERN) protons are accelerated to Lorentz factor $\gamma \approx 7000$, and all calculations of trajectories, synchrotron radiation losses, and beam control systems are strictly done using relativistic mechanics. High-resolution mass spectrometers also use relativistic dynamics for precise determination of heavy nuclei and isotope masses. Electrons in accelerators for medical radiation therapy reach $\gamma \approx 20–100$, and treatment dose planning requires taking into account the relativistic angle of radiation.

Assignment: (a) A muon is created at an altitude of 10 km with a speed of 0.998$c$, $\tau_0 = 2.2,\mu$s. Will it reach the Earth? Compute the distance traveled in the laboratory frame and in the muon’s frame. (b) From the transformation of $F^{\mu\nu}$ under a boost along $x$ obtain: $E'_x = E_x$ (longitudinal $E$ is unchanged), $E'_y = \gamma(E_y - v B_z)$ (transverse $E$ mixes with $B$). Physical meaning of this mixing.