Simultaneity And Time Dilation

Consider how we measure elapsed time. If we use a stopwatch, for example, how do we know when to start and stop the watch? One method is to use the arrival of light from the event, such as observing a light turning green to start a drag race. The timing will be more accurate if some sort of electronic detection is used, avoiding human reaction times and other complications.

Now suppose we use this method to measure the time interval between two flashes of light produced by flash lamps. (See Figure 13.3.2.) Two flash lamps with observer A midway between them are on a rail car that moves to the right relative to observer B. Observer B arranges for the light flashes to be emitted just as A passes B, so that in B’s frame, both A and B are equidistant from the lamps when the light is emitted. Observer B measures the time interval between the arrival of the light flashes. According to postulate 2, the speed of light is not affected by the motion of the lamps relative to B. Therefore, light travels equal distances to him at equal speeds. Thus, observer B measures the flashes to be simultaneous. Since in B’s frame, A is moving to the right, B will measure the light from the right bulb arrive at observer A before the light from the left bulb arrives at observer A.

Now consider what observer A sees happening. As observer B also notes, she receives the flash from the right-hand bulb before the flash from the left-hand bulb. Since the lamps are equally distant from her and light travels at the same speed in both directions in her frame (postulate 2), she concludes that the right-hand bulb emitted its flash first. Here, a relative velocity between observers affects whether two events are observed to be simultaneous. Simultaneity is not absolute.

This thought experiment shows how the postulates of special relativity lead to a counterintuitive conclusion. We might expect two flashes emitted simultaneously for one observer to be emitted simultaneously for everyone. But the second postulate requires both observers to measure the same speed of light, despite their relative motion. Even after accounting for light-travel times, they disagree about whether the flashes were emitted simultaneously. Neither inertial reference frame is privileged (postulate 1), so we must give up the idea of absolute simultaneity.

We can represent the train-and-flashes experiment on a space-time diagram, a graph of events according to their positions and times in a chosen reference frame. In Figure 13.3.3, position $x$ increases to the right and time increases upward. The vertical axis is labeled $ct$: multiplying time by the speed of light gives a distance, so both axes can use the same scale. Each point represents an event, and all coordinates in this diagram belong to the station frame. The flashes are emitted when the train observer A passes the station observer B at the origin.

An object’s path on this graph is called its worldline. B stays at one position, so his worldline is vertical. A moves steadily to the right, so her worldline slopes to the right as time passes. Light travels at speed $c$, so its worldlines make 45° angles with the horizontal when the axis scales are equal. Intersections mark events at which light reaches an observer. Reading upward, we see the right flash reach A, both flashes reach B together, and finally the left flash reach A. A horizontal line joins events that are simultaneous in the station frame.

The diagram shows the different arrival times using station-frame coordinates. To understand the emission times assigned by A, return to her description of the experiment: she and the lamps are at rest relative to one another, and she is midway between the lamps. Light travels equal distances to her at the same speed in both directions, so the two travel times in her frame are equal. Because the right flash arrives first, she concludes that the right lamp emitted its flash first. Unequal arrival times alone would not establish unequal emission times; the equal distances and equal light speeds in A’s frame are essential to her conclusion.

The consideration of the measurement of elapsed time and simultaneity leads to an important relativistic effect.

Suppose, for example, an astronaut measures the time it takes for light to cross her ship, bounce off a mirror, and return. (See Figure 13.3.4.) How does the elapsed time the astronaut measures compare with the elapsed time measured for the same event by a person on the Earth? Asking this question (another thought experiment) produces a profound result. We find that the elapsed time for a process depends on who is measuring it. In this case, the time measured by the astronaut is smaller than the time measured by the Earth-bound observer. The passage of time is different for the observers because the distance the light travels in the astronaut’s frame is smaller than in the Earth-bound frame. Light travels at the same speed in each frame, and so it will take longer to travel the greater distance in the Earth-bound frame.

To quantitatively verify that time depends on the observer, consider the paths followed by light as seen by each observer. (See Figure 13.3.4(c).) The astronaut sees the light travel straight across and back for a total distance of $2D$, twice the width of her ship. The Earth-bound observer sees the light travel a total distance $2s$. Since the ship is moving at speed $v$ to the right relative to the Earth, light moving to the right hits the mirror in this frame. Light travels at a speed $c$ in both frames (postulate 2), and because time is the distance divided by speed, the time measured by the astronaut is

This time has a separate name to distinguish it from the time measured by the Earth-bound observer.

The astronaut observing the reflected light measures proper time. The time measured by the Earth-bound observer is

To find the relationship between $\mathrm{\Delta }{t}_0$ and $\mathrm{\Delta }t$, consider the triangles formed by $D$ and $s$. (See Figure 13.3.4(c).) The third side of these triangles is $L$, the distance the astronaut moves as the light goes across her ship. In the frame of the Earth-bound observer,

Using the Pythagorean Theorem, the distance $s$ is found to be

Substituting $s$ into the expression for the time interval $\mathrm{\Delta }t$ gives

We square this equation, which yields

Note that if we square the first expression we had for $\mathrm{\Delta }{t}_0$, we get ${(\mathrm{\Delta }{t}_0)}^2=\frac{{4D}^2}{c^2}$. This term appears in the preceding equation, giving us a means to relate the two time intervals. Thus,

Gathering terms, we solve for $\mathrm{\Delta }t$:

Thus,

Taking the square root yields an important relationship between elapsed times:

where

This equation for $\mathrm{\Delta }t$ is truly remarkable. First, as contended, elapsed time is not the same for different observers moving relative to one another, even though both are in inertial frames. Proper time $\mathrm{\Delta }{t}_0$ measured by an observer, like the astronaut moving with the apparatus, is smaller than time measured by other observers. Since those other observers measure a longer time $\mathrm{\Delta }t$, the effect is called time dilation. The Earth-bound observer sees time dilate (get longer) for a system moving relative to the Earth. Alternatively, according to the Earth-bound observer, time slows in the moving frame, since less time passes there. All clocks moving relative to an observer, including biological clocks such as aging, are observed to run slow compared with a clock stationary relative to the observer.

Note that if the relative velocity is much less than the speed of light ($v\ll c$), then $\frac{v^2}{c^2}$ is extremely small, and the elapsed times $\mathrm{\Delta }t$ and $\mathrm{\Delta }{t}_0$ are nearly equal. At low velocities, modern relativity approaches classical physics—our everyday experiences have very small relativistic effects.

The equation $\mathrm{\Delta }t={\gamma \mathrm{\Delta }t}_0$ also implies that relative velocity cannot exceed the speed of light. As $v$ approaches $c$, $\mathrm{\Delta }t$ approaches infinity. This would imply that time in the astronaut’s frame stops at the speed of light. If $v$ exceeded $c$, then we would be taking the square root of a negative number, producing an imaginary value for $\mathrm{\Delta }t$.

There is considerable experimental evidence that the equation $\mathrm{\Delta }t={\gamma \mathrm{\Delta }t}_0$ is correct. One example is found in cosmic ray particles that continuously rain down on the Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called muons. The half-life (amount of time for half of a material to decay) of a muon is $1.52\mu \text{s}$ when it is at rest relative to the observer who measures the half-life. This is the proper time $\mathrm{\Delta }{t}_0$. Muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It has been found that the muon’s half-life as measured by an Earth-bound observer ($\mathrm{\Delta }t$) varies with velocity exactly as predicted by the equation $\mathrm{\Delta }t={\gamma \mathrm{\Delta }t}_0$. The faster the muon moves, the longer it lives. We on the Earth see the muon’s half-life time dilated—as viewed from our frame, the muon decays more slowly than it does when at rest relative to us.

Another implication of the preceding example is that everything an astronaut does when moving at $\text{95}\text{.}\mathrm{0\%}$ of the speed of light relative to the Earth takes 3.20 times longer when observed from the Earth. Does the astronaut sense this? Only if she looks outside her spaceship. All methods of measuring time in her frame will be affected by the same factor of 3.20. This includes her wristwatch, heart rate, cell metabolism rate, nerve impulse rate, and so on. She will have no way of telling, since all of her clocks will agree with one another because their relative velocities are zero. Motion is relative, not absolute. But what if she does look out the window?

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to the Earth would age less than her Earth-bound twin. Imagine the astronaut moving at such a velocity that $\gamma =30\text{.}0$, as in Figure 13.3.6. A trip that takes 2.00 years in her frame would take 60.0 years in her Earth-bound twin’s frame. Suppose the astronaut traveled 1.00 year to another star system. She briefly explored the area, and then traveled 1.00 year back. If the astronaut was 40 years old when she left, she would be 42 upon her return. Everything on the Earth, however, would have aged 60.0 years. Her twin, if still alive, would be 100 years old.

The situation would seem different to the astronaut. Because motion is relative, the spaceship would seem to be stationary and the Earth would appear to move. (This is the sensation you have when flying in a jet.) If the astronaut looks out the window of the spaceship, she will see time slow down on the Earth by a factor of $\gamma =30\text{.}0$. To her, the Earth-bound sister will have aged only 2/30 (1/15) of a year, while she aged 2.00 years. The two sisters cannot both be correct.

During each constant-velocity leg of the journey, each twin measures the other’s clock as running slower. This symmetry does not extend to the whole round trip. The astronaut turns around and changes inertial frames, while the Earth-bound twin remains in approximately the same inertial frame. Changing frames also changes which event on Earth the astronaut assigns as simultaneous with her own: the outward and return frames assign different ages to the distant Earth-bound twin at the turnaround. Accounting for this change, special relativity predicts that the astronaut ages less when the twins reunite. The key is that the twins follow different paths through spacetime, so they need not experience the same elapsed time.

In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the Earth on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. The predictions included both the effects of motion from special relativity and gravitational time dilation from general relativity: clocks at the aircraft’s higher altitude run faster than clocks on the ground because of the difference in gravitational potential.