37-1 Simultaneity and Time Dilation#
Prompts
State the two postulates of special relativity. Why is the speed of light \(c\) called the “ultimate speed”?
Two events are simultaneous in one inertial frame. Will they be simultaneous in another frame moving relative to the first? Explain.
What is proper time \(\Delta t_0\)? When can a single clock measure it? Why does an observer in another frame measure a larger interval \(\Delta t\)?
Derive or apply the time dilation formula \(\Delta t = \gamma \Delta t_0\). What are \(\beta\) and \(\gamma\)? When is \(\gamma \approx 1\)?
A muon at rest decays in 2.2 \(\mu\)s. If it travels at 0.99\(c\) through a lab, how far does it go before decaying (in the lab frame)? Use both classical and relativistic time.
Lecture Notes#
Overview#
Special relativity applies to inertial reference frames (Newton’s laws hold). It rests on two postulates: (1) the laws of physics are the same in all inertial frames; (2) the speed of light in vacuum is \(c\) in all inertial frames.
Simultaneity is relative: two events simultaneous in one frame are generally not simultaneous in another frame moving relative to it.
Time dilation: when two events occur at the same place in one frame, the time interval \(\Delta t_0\) (proper time) measured there is shorter than the interval \(\Delta t\) measured in any other frame: \(\Delta t = \gamma \Delta t_0\), with \(\gamma > 1\).
Space and time are entangled—their separations depend on who measures them and how frames move relative to each other.
The postulates#
1. Relativity postulate: The laws of physics are the same for observers in all inertial reference frames. No frame is preferred. (Galileo assumed this for mechanics; Einstein extended it to all physics, including electromagnetism.)
2. Speed of light postulate: The speed of light in vacuum has the same value \(c\) in all directions and in all inertial frames.
This implies an ultimate speed \(c\)—no entity carrying energy or information can exceed it. Particles with mass cannot reach \(c\) no matter how long they are accelerated. The exact value is \(c = 2.997\,924\,58\times 10^8\) m/s.
Within a frame
Within a single inertial frame, we still use classical kinematics and Newtonian mechanics. Relativity matters when we transform measurements between frames in relative motion.
Simultaneity is relative#
An event is specified by three space coordinates and one time coordinate. To measure when and where an event occurs without ambiguity, we imagine a grid of synchronized clocks and measuring rods filling each frame. (This eliminates the need to correct for signal travel time.)
Key result: If two observers are in relative motion, they generally will not agree on whether two events are simultaneous. Events that are simultaneous in one inertial frame need not be simultaneous in another.
The reason is the second postulate: light has the same speed \(c\) in both frames. In a classic thought experiment, two flashes occur at the ends of a moving train. An observer on the train (midpoint) sees them arrive simultaneously and concludes the flashes were simultaneous. A ground observer sees the flash from the rear arrive first (the train has moved), and concludes the rear flash occurred earlier. Both are correct in their own frames.
Proper time and time dilation#
When two events occur at the same location in an inertial frame, a single clock at that location can record both. The time interval between them, measured in that frame, is called the proper time \(\Delta t_0\).
Observers in other frames (moving relative to that location) must use two different clocks—one at each event—because the events occur at different places in their frame. They always measure a larger time interval \(\Delta t\):
where \(v\) is the relative speed between the frame where the events occur at the same place (the “rest frame” of the clock) and the frame where \(\Delta t\) is measured. The time dilation effect: moving clocks run slow (as measured by the other frame).
Speed parameter and Lorentz factor:
\(\beta < 1\) always; \(\gamma \geq 1\), with \(\gamma \to \infty\) as \(v \to c\).
For \(v \ll 0.1c\), \(\gamma \approx 1\) and classical physics suffices. For higher speeds, relativity is essential.
Why the other observer measures more
The observer who measures \(\Delta t_0\) (proper time) uses one clock. The other observer uses two clocks. Because of the relativity of simultaneity, the two observers disagree on whether those two clocks are synchronized. From the proper-time observer’s view, the other’s clocks were set wrong—one was ahead—so the other reads a larger interval.
Tests of time dilation#
Muons: Unstable particles with proper lifetime \(\Delta t_0 \approx 2.2\,\mu\)s. When moving at \(v = 0.9994\,c\) through a lab, \(\gamma \approx 29\); the dilated lifetime is \(\Delta t \approx 64\,\mu\)s. Muons produced in the upper atmosphere survive long enough to reach the ground—consistent with time dilation.
Macroscopic clocks: Hafele and Keating (1971) flew atomic clocks around the world; the time difference between flying and ground clocks matched relativistic predictions. GPS satellites must correct for both special and general relativistic effects.
Example: relativistic traveler
A starship passes Earth at \(v = 0.999c\), travels 10 y (ship time) to a lookout post, then returns at the same speed, taking another 10 y (ship time). Total ship time: 20 y. For each leg, \(\gamma \approx 22.4\), so Earth measures \(\Delta t \approx 224\) y per leg. Round trip: 20 y for the traveler, ~448 y on Earth.
Summary#
Postulates: Same laws in all inertial frames; \(c\) is invariant. No speed exceeds \(c\).
Simultaneity is frame-dependent; observers in relative motion disagree.
Proper time \(\Delta t_0\): interval between two events at the same place, measured by one clock.
Time dilation \(\Delta t = \gamma \Delta t_0\); \(\gamma = 1/\sqrt{1 - v^2/c^2}\). Moving clocks run slow.