Relativistic effects

Imagine a light clock with two mirrors: one tick is a pulse's round trip between them. To a passenger the light travels vertically; to an outside observer it also travels along the spacecraft's motion. If both measure the same light speed c, will they measure the same time between ticks?

In a spacecraft moving uniformly without rotation, a free object keeps moving in a straight line at constant velocity, just as in any other inertial frame. The principle of relativity says that the laws of physics are the same in all such frames. Neither Earth nor the spacecraft is privileged by its name or a feeling of being truly at rest. Compare particular observers and measurement procedures. An accelerating spacecraft no longer fits this simple inertial model.

In special relativity, all inertial observers measure the same vacuum light speed c≈3·10⁸ м/с, regardless of the source's motion. We cannot apply the familiar addition c+v to light unchanged. For a light clock, an outside observer measures a longer, slanted pulse path at the same light speed, so the interval between ticks is longer. From here on, the relative speed of the frames is constant and v<c.

The slanted path gives the factor \gamma=1/\sqrt{1-v^2/c^2}. For half of the round trip, the right triangle has hypotenuse cτ/2, horizontal side vτ/2 and vertical side cτ₀/2. Pythagoras gives τ=γτ₀. The dimensionless γ is at least one. At v=0 it is 1; for v≪c it is almost 1. As v approaches c it grows sharply, and familiar mechanics no longer gives an accurate answer.

What is τ₀ in this construction? It is the proper time between the events, measured by a clock present at both. Two ticks of the spacecraft clock occur at the same place in the spacecraft frame; that clock measures τ₀. In the outside frame the clock moves during the interval, so the events have different coordinates. First choose one pair of events, then identify which interval is the proper time.

For this pair of events, \tau=\gamma\tau_0=\tau_0/\sqrt{1-v^2/c^2}, with both times in seconds. In the frame where the clock moves, the coordinate-time interval between its ticks is longer. This uses synchronized clocks in that frame, not merely a delayed image caused by the distance to an observer. Swap the inertial frames and each can describe the other's clocks as slow: they disagree about which distant events are simultaneous.

Length also needs a measurement procedure. Proper length l₀ is measured in the rod's rest frame. In another frame, record both ends simultaneously according to that frame's clocks, then subtract their coordinates. If you record the front end now and the back end later, the rod moves between records; the result is not its length at one instant. Simultaneity of distant events depends on the frame, so the two length measurements do not use the same pair of endpoint events.

Along the relative motion, l=l_0/\gamma=l_0\sqrt{1-v^2/c^2}. With lengths in metres, l≤l₀. At v=0 the lengths agree; increasing v reduces the measured longitudinal length. Transverse dimensions do not contract by this rule. The rod is not mechanically compressed: in its rest frame it has its original length. What changes is the result of recording its ends simultaneously in another frame. A photograph, which depends on light-arrival times, is a different procedure again.

Rest mass m₀, in kilograms, characterizes the object and does not change merely because we choose a different observer. Modern notation usually calls it simply mass. Acceleration increases energy and momentum, where the relativistic factor matters; it does not add matter to the object. Compare masses at the same internal state: heating or changing stored energy is a different physical process from merely switching to a moving observer.

Some problems use the historical convention m=\gamma m_0=m_0/\sqrt{1-v^2/c^2}, called relativistic mass. It is measured in kilograms but depends on the frame, unlike invariant m₀. As v→0 the two coincide; as v increases, m under this convention increases. Use this equation when the notation is explicitly defined. Otherwise, it is clearer to describe the change through energy and momentum without calling a speed-dependent quantity mass.

The spacecraft clock measures 4 s between two ticks. The spacecraft moves relative to Earth at v=0,6c. What interval between these events does the Earth inertial frame measure?

Both events are ticks of the same spacecraft clock. They occur at the same place in its rest frame, so τ₀ = 4 с is the proper interval.

γ = 1/√(1 − 0,6²) = 1/√0,64 = 1/0,8 = 1,25.

For the same pair of events in the Earth inertial frame, τ = γτ₀ = 1,25 × 4 = 5 с.

Five seconds is longer than four: motion increases the interval measured by the outside frame between ticks. This is not the waiting time for a light signal at a single Earth observation point.

In the Earth inertial frame, τ = 5 с. During that interval the spacecraft clock measures 4 s between the chosen ticks.

The constant light speed links measurements of distance and time. A light clock reveals where the factor γ comes from.

Proper time is measured by a clock at both events. A moving rod's length is found by recording its ends simultaneously in the chosen frame.

Rest mass is invariant. Distinguish the historical quantity γm₀ from mass in modern notation.

Next we will use energy and momentum to describe light quanta.