Special Relativity

Time Dilation

The time experienced by an observer in its inertial frame is called its proper time. Something interesting happens when we compare the proper time of a body at rest to the proper time of a body in motion within the same inertial frame that teaches us an important lesson about the nature of time.

Cite

Time intervals have different values when measured in different inertial frames.

Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).

The proper time interval between two events is the time interval measured by an observer for whom both events occur at the same location.

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Consider a “light clock” that consists of a photon that bounces between two mirrors set a distance apart. We count one “tick” of the clock when the photon makes one full round trip. Since the speed of light is constant in all inertial reference frames, an observer will always observe the photon moving at the speed of light, , regardless of the motion of its source.

Example

An astronaut observes a photon in the light clock on a spaceship moving at a constant velocity, , relative to the Earth. An astronomer on Earth also observes the same photon in the light clock. Both observers measure the time it takes for the photon to make one full round trip.

  • The astronaut observes the photon moving at exactly the speed of light, .
  • The astronomer observes the photon moving at exactly the speed of light, .
  • The astronaut and astronomer disagree on the time it takes for the photon to make one full round trip in the light clock.
    • The astronaut observes the photon completing one round trip in .
    • The astronomer observes the photon completing one round trip in .

Problem: Whose round-trip duration measurement is correct?

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Solution: The astronaut and astronomer are both correct! It’s not a trick question, this time—we can solve it with 8th-grade math and disciplined logic.

In the astronaut’s inertial frame:

  • The photon moves at speed .
  • The light clock is at rest so the photon travels the distance between the mirrors in the time it takes to complete one round trip. This is our intuitive understanding of time in the everyday sense.
  • The time it takes for the photon to make one full round trip is .

In the astronomer’s inertial frame:

  • The photon moves at speed .
  • The light clock is moving relative to the astronomer so the photon travels a longer distance in the time it takes to make one full round trip in the clock.
    • The photon travels sideways a distance in addition to the distance it travels up and down between the mirrors.
    • Despite moving additional distance in this reference frame compared to the astronaut’s frame, the photon travels at speed in both reference frames.
    • This means that in the astronaut’s frame, the photon travels more distance while moving at the same speed.
  • The only way this can be true is if the elapsed time experienced by the astronaut, , is longer than the elapsed time experienced by the astronomer, , when observing the same photon in the astronomer’s reference frame.

We see here that both of Einstein’s postulates are satisfied, and we have not violated any laws of physics. Yet two observers measured the photon taking different durations to make the same trip in space. How can this be?

Help

Remember that physics and science do not describe the truth, they describe math and logical axioms that are consistent with the observed data. As counterintuitive as it may be, special relativity overwhelmingly agrees with scientific observations.

In the astronomer’s reference frame, the photon moves across more space over the course of one “tick” of the clock. The only way the speed of the photon, , can be the same in all reference frames and yet travel across more distance in one reference frame than in another is if the elapsed time experienced by a moving object, , “dilates” compared to the elapsed time experienced by an observer at rest in the reference frame, .

time dilation

Therefore, the time dilation equation is: