5.3 Time dilation  (Page 2/8)

Then we rearrange to obtain

Finally, solving for $\text{Δ}t$ in terms of $\text{Δ}\tau$ gives us

This is equivalent to

where $\gamma$ is the relativistic factor (often called the Lorentz factor ) given by

and v and c are the speeds of the moving observer and light, respectively.

Note the asymmetry between the two measurements. Only one of them is a measurement of the time interval between two events—the emission and arrival of the light pulse—at the same position. It is a measurement of the time interval in the rest frame of a single clock. The measurement in the earthbound frame involves comparing the time interval between two events that occur at different locations. The time interval between events that occur at a single location has a separate name to distinguish it from the time measured by the earthbound observer, and we use the separate symbol $\text{Δ}\tau$ to refer to it throughout this chapter.

The equation relating $\text{Δ}t$ and $\text{Δ}\tau$ is truly remarkable. First, as stated earlier, elapsed time is not the same for different observers moving relative to one another, even though both are in inertial frames. A proper time interval $\text{Δ}\tau$ for an observer who, like the astronaut, is moving with the apparatus, is smaller than the time interval for other observers. It is the smallest possible measured time between two events. The earthbound observer sees time intervals within the moving system as dilated (i.e., lengthened) relative to how the observer moving relative to Earth sees them within the moving system. Alternatively, according to the earthbound observer, less time passes between events within the moving frame. Note that the shortest elapsed time between events is in the inertial frame in which the observer sees the events (e.g., the emission and arrival of the light signal) occur at the same point.

This time effect is real and is not caused by inaccurate clocks or improper measurements. Time-interval measurements of the same event differ for observers in relative motion. The dilation of time is an intrinsic property of time itself. All clocks moving relative to an observer, including biological clocks, such as a person’s heartbeat, or aging, are observed to run more slowly compared with a clock that is stationary relative to the observer.

Note that if the relative velocity is much less than the speed of light $\left(v\text{<}\text{<}c\right),$ then $v^2\text{/}{c}^2$ is extremely small, and the elapsed times $\text{Δ}t$ and $\text{Δ}\tau$ are nearly equal. At low velocities, physics based on modern relativity approaches classical physics—everyday experiences involve very small relativistic effects. However, for speeds near the speed of light, $v^2\text{/}{c}^2$ is close to one, so $\sqrt{1-v^2\text{/}{c}^2}$ is very small and $\text{Δ}t$ becomes significantly larger than $\text{Δ}\tau .$

Half-life of a muon

There is considerable experimental evidence that the equation $\text{Δ}t=\gamma \text{Δ}\tau$ is correct. One example is found in cosmic ray particles that continuously rain down on 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 μs when it is at rest relative to the observer who measures the half-life. This is the proper time interval $\text{Δ}\tau .$ This short time allows very few muons to reach Earth’s surface and be detected if Newtonian assumptions about time and space were correct. However, 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 earthbound observer ( $\text{Δ}t$ ) varies with velocity exactly as predicted by the equation $\text{Δ}t=\gamma \text{Δ}\tau .$ The faster the muon moves, the longer it lives. We on Earth see the muon last much longer than its half-life predicts within its own rest frame. As viewed from our frame, the muon decays more slowly than it does when at rest relative to us. A far larger fraction of muons reach the ground as a result.