Time Dilation

Whiteboard Warmup! A speaker emits sound waves that travel outward in all directions. Relative to the speaker, the sound waves travel at 343 m/s. If each of the observers shown below measures the speed of sound relative to their own reference frame, what do they measure? 100 m/s 100 m/s A C 100 m/s B

Einstein’s Thought Experiments Source emits light Observer measures speed of light Einstein embarked on an intellectual journey to determine what the speed of light would appear to be from the reference frame of various observers.

Einstein’s Thought Experiment Would the detector observe light traveling at 1.5c?

Would the detector observe light traveling at 0.5c?

Would the detector observe light traveling at 2c??

The light would stay in place??? This doesn’t seem to make any sense!!!

Einstein’s Two Postulates 1) The Equivalence of Physical Laws The laws of physics are the same in all inertial frames of reference. 2) Constancy of the Speed of Light The speed of light in a vacuum, c = 3.00 x 108 m/s, is the same in all frames of reference, independent of the motion of the source and the receiver.

1) The Equivalence of Physical Laws The laws of physics are the same in all inertial frames of reference. There is no experiment (mechanics, electricity, magnetism, thermodynamics) that is affected by the motion of an inertial reference frame. If you are in a reference frame that is moving at a constant velocity, there is no way to find out whether or not you are moving (because you are not moving, in an absolute sense! All motion is relative!)

There is no absolute rest frame in the universe. All motion is relative. One could never say that a reference frame is “at rest” in an absolute sense. Now, on to this speed of light thing…

The source emits light at speed c. The detector will observe light traveling at speed c! Light always travels at speed c, regardless of the observer’s reference frame!

What would the detector measure? Light traveling at speed c!

The detector would still see light traveling at 3 x 108 m/s! This leads to some very bizarre results about the nature of space and time.

Lightboard What would each of the following observers measure for the speed of light emitted by the bulb? v = 0.9c v = 0.9c v = 0

The speed of light is constant for all observers! They will ALL measure the speed of light as c. v = 0.9c v = 0.9c v = 0

Then what about this case? This scenario is could never happen, because no source or observer could travel at the speed of light! It is fundamentally forbidden by the laws of physics!

According to physics as we currently know it, no object that has mass could ever travel at the speed of light or faster than the speed of light!

The detector will still measure the speed of light to be 3 x 108 m/s. No matter what the motion of the source or the observer, the observed speed of light will be 3 x 108 m/s. If this were not true, then it would be possible to determine an “absolute” frame of reference, which is quite simply not the case in the universe. 0.999c 0.999c The detector will still measure the speed of light to be 3 x 108 m/s. Craziness!

Whiteboard: Einstein’s Thought Experiment A moving train cart is rigged so that a pulse of light starts at its floor and is detected when it reaches the ceiling. First, let’s consider an observer that is on the train, at rest relative to the source of light detector Write an expression for d in terms of c and t0. t0 is the time that it takes the light to reach the ceiling from the reference frame of a person that is at rest (v = 0) relative to the source.

By the time the light reaches the ceiling, the clock that is at rest relative to the source will have elapsed by an amount t0. detector t0 is the time that it takes the light to reach the ceiling from the reference frame of a person that is at rest relative to the source.

Now you’re going to need to stretch your conception of reality. We generally think of time as moving forward at a constant rate, the same for all reference frames. However, by accepting Einstein’s second postulate, we reach some very surprising conclusions regarding the passage of time in a reference frame that is in motion relative to another. All that is required to achieve the result it Einstein’s second postulate and some algebra! Let’s see!

A little bit stranger now! Now, imagine that the train is moving to the right with constant speed v relative to an observer on the side of the tracks. The light pulse will still shine on the same part of the ceiling, but... v According to this observer, the light traveled a greater distance to get there!

Remember! All observers see light traveling at speed c. What will the observer that is outside the train (moving relative to the light source) determine? v ct Remember! All observers see light traveling at speed c. d vt (where v is the speed of the train)

Write an expression for d in terms of c, t and v. vt t is the time that it takes the light to reach the ceiling from the reference frame of a person that is in motion relative to the source (blue Ruggles).

ct d vt Where t is the amount of time that has elapsed on the observer’s clock that is not on the train.

Observer Moving Relative to Source Stationary Observer Observer Moving Relative to Source v t0 t Since the observers will certainly agree on the height of the train (but not the amount of time that it took light to travel those different distances!)…

This is known as the equation for time dilation. Rearranging some terms (I leave the algebra as an exercise for you to do on your own if you are interested in this stuff – you should be!) We end up with a relationship between the measurements of time that the each observer will make about the same event, from different reference frames. Time elapsed according to an observer inside the train Speed of the train Speed of light Time elapsed according to an observer outside the train This is known as the equation for time dilation.

What does this mean?? Since the speed of light is constant in all reference frames, but different observers will see light travel different distances based on their own relative motion, time itself must elapse at different rates for different observers. This means that the person on the tracks will measure a larger time interval than the person on the train for the same event! Give this some thought and we will explore it more deeply on the morrow! Oh, Einstein…