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/s100 m/s

100 m/s

A

B

C

Einstein’s Thought Experiments

Einstein embarked on an intellectual journey to determine what the speed of light would appear to be

from the reference frame of various observers.

Source emits light Observer measures speed of light

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 LawsThe laws of physics are the same in all inertial frames of reference.

2) Constancy of the Speed of LightThe 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 LawsThe 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.

LightboardWhat would each of the following observers measure

for the speed of light emitted by the bulb?

v = 0

v = 0.9c

v = 0.9c

The speed of light is constant for all observers!

v = 0

v = 0.9c

v = 0.9c

They will ALL measure the speed of light as c.

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!

Then what about this case?

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!

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

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.

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

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.

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.

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!

What will the observer that is outside the train (moving relative to the light source) determine?

v

ctRemember! All observers see light traveling at speed c. d

vt (where v is the speed of the train)

ct d

vt

v

t

Write an expression for d in terms of c, t and v.

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.

v

Stationary Observer Observer Moving Relative to Source

t

t0

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

Time elapsed according to an observer outside the train

Speed of the train

Speed of light

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…