Physic Labs

Theory of relativity

Einstein’s postulates and the Lorentz transformation

Einstein’s two postulates lead to Lorentz transformations: physical laws are the same in all inertial frames, and light in vacuum has the same speed c.

Let S′ move along +x at speed v relative to S. To preserve the same vacuum light speed, the coordinate change mixes space and time: the two frames generally assign different positions and times to the same event.

x′=γ(x−vt),t′=γ(t−vxc2),y′=y, z′=z,γ=11−v2/c2x′=γ(x−vt),\quad t′=γ\left(t−\frac{vx}{c^2}\right),\quad y′=y,\ z′=z,\quad γ=\frac{1}{\sqrt{1−v^2/c^2}}

Definition: Definition

The factor γ=(1−v2/c2)−1/2γ=(1−v^2/c^2)^{-1/2} measures the strength of relativistic effects. The inverse transformation replaces v by −v. These relations assume inertial frames with parallel axes in uniform relative motion along x.

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Physical meaning

A light pulse following x=ct also satisfies x′=ct′ after transformation, so its speed remains c. For v≪c, γ≈1 and the results approach Galilean kinematics.

Example: Worked example

In S, an event has x=300,000 km and t=2.0 s. For v=0.6c, find its coordinates in S′.

Solution

γ=1.25 and c=300,000 km/s, so x′=1.25(300,000−360,000)=−75,000 km and t′=1.25(2−0.6)=1.75 s.

Lorentz transformations have a group structure: two collinear boosts combine into one boost with velocity u=(v+w)/(1+vw/c2)u=(v+w)/(1+vw/c^2). The denominator prevents the result from exceeding cc; for v,w≪cv,w\ll c, the rule reduces to Galilean addition. Ordinary addition therefore cannot combine relativistic velocities.

Example: Relativistic velocity addition

A spacecraft moves at 0.8c0.8c relative to Earth and launches a probe forward at 0.7c0.7c relative to the craft. What is the probe’s speed relative to Earth? Use the Lorentz velocity-addition law.

Solution

u=(0.8c+0.7c)/(1+0.8⋅0.7)=1.5c/1.56≈0.962cu=(0.8c+0.7c)/(1+0.8\cdot0.7)=1.5c/1.56\approx0.962c. The result remains below cc, as required by invariant light speed.

The postulates also constrain simultaneity: lightning strikes recorded simultaneously at both ends of a platform need not be simultaneous in a passing train’s frame. The term −vx/c2-vx/c^2 in the time transformation encodes this relativity of simultaneity. There is no universal time shared by all inertial observers; Lorentz transformations preserve causal relations without making every time measurement identical.

Quick check

For v=0.6c, what is the Lorentz factor γ?

Which quantity is invariant under Lorentz transformations?

References

  1. Edwin F. Taylor, John Archibald Wheeler (1992). Spacetime Physics
  2. Robert Resnick (1968). Introduction to Special Relativity