Einstein's Postulates & Spacetime

In 1905 Einstein resolved the contradiction between Newtonian mechanics and Maxwell's electrodynamics with just two postulates: the laws of physics are the same in all inertial frames, and the speed of light in vacuum is the same for all inertial observers regardless of the motion of the source. These postulates force a radical revision of space and time.

Key Concepts

First Postulate

The laws of physics (including Maxwell's equations) take the same form in all inertial reference frames. No inertial frame is preferred — there is no absolute rest.

Second Postulate

The speed of light in vacuum,

c=2.998\times10^8

m/s, is the same for all inertial observers regardless of the motion of the source or observer. This is the key contradiction with Galilean relativity.

Relativity of Simultaneity

Two events that are simultaneous in frame

S

are generally not simultaneous in frame

S'

moving relative to

S

. There is no universal "now" — simultaneity is frame-dependent.

Spacetime Interval

s^2 = c^2t^2 - x^2 - y^2 - z^2

. This combination is invariant — the same in all inertial frames. Positive: timelike; zero: lightlike; negative: spacelike separation.

Key Equations

Speed of light

c = 2.998\times10^8\text{ m/s} \approx 3.0\times10^8\text{ m/s}

The universal speed limit; same for all inertial observers.

Spacetime interval

s^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2

Lorentz invariant; positive = timelike, zero = lightlike, negative = spacelike.

Relativistic Doppler

f_{\rm obs} = f_{\rm source}\sqrt{\frac{1-\beta}{1+\beta}}

For source receding (β = v/c > 0); redshift when moving away.

Worked Example

Relativity of Simultaneity

Problem

Two lightning strikes hit both ends of a train (length $L_0=300$ m in its rest frame) simultaneously in the ground frame. The train moves at $v=0.60c$ . Are the strikes simultaneous for a train observer?

Solution

In the ground frame ( $S$ ): $\Delta t=0$ , $\Delta x=L=L_0/\gamma$ (contracted length).

For the train frame ( $S'$ ) using the Lorentz transform: $\Delta t' = \gamma(\Delta t - v\Delta x/c^2)$ .

With $\gamma=1.25$ , $\Delta t=0$ , $\Delta x=L_0/\gamma=240$ m:

\Delta t' = 1.25\left(0 - \frac{0.60c \times 240}{c^2}\right) = 1.25\times\frac{-0.60\times240}{c} = \frac{-180}{c} = -600\text{ ns}

The front strike happens 600 ns before the rear strike in the train frame — not simultaneous.

Answer No — the front strike precedes the rear strike by 600 ns in the train's frame.

Practice

Exercises

7 problems

1 of 7

A photon is emitted by a source moving at $0.80c$ relative to you. What is the photon's speed (in m/s) as measured by you?

Your answer

m/s

2 of 7

Two events are separated by $\Delta x=9.0\times10^8$ m and $\Delta t=4.0$ s. Find the spacetime interval $s$ (in m, use $c=3\times10^8$ m/s). $s^2=c^2\Delta t^2-\Delta x^2$ .

Your answer

3 of 7

Light from a star 4.0 light-years away is observed on Earth. How many years does it take to arrive?

Unlock Exercise 3