relativity

The Special Theory of Relativity

E = mc2

Galilean-Newtonian Relativity

Inertial Reference Frame:

One in which Newton’s first law is valid.

Earth is rotating and therefore not an inertial reference frame, but can treat it as one for many purposes.

A frame moving with a constant velocity with respect to an inertial reference frame is itself inertial.

Relativity Principle:

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


If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system.

This is referred to as the Newtonian principle of relativity or Galilean invariance.

If the axes are also parallel, these frames are said to be Inertial Coordinate Systems


vx x t

y y

z z

t t

vx x t

y y

z z

t t

1. Parallel axes

2. S’ has a constant relative velocity (here in the x-direction) with respect to S.

3. Time (t) for all observers is a Fundamental invariant, i.e., it’s the same for all inertial observers.

The Inverse Relations

For a point P:

In one frame S:

P = (x, y, z, t)

In another frame S’:

P = (x’, y’, z’, t’)

The Galilean Transformation

This principle works well for mechanical phenomena.

However, Maxwell’s equations yield the velocity of light; it is 3.0 x 108 m/s.

So, which is the reference frame in which light travels at that speed?

Scientists searched for variations in the speed of light depending on the direction of the ray, but found none.


The Michelson–Morley Experiment

This experiment was designed to measure the speed of the Earth with respect to the ether.

The Earth’s motion around the Sun should produce small changes in the speed of light, which would be detectable through interference when the split beam is recombined.

The Michelson interferometer is sketched here, along with an analogy using a boat traveling in a river.


This interferometer was able to measure interference shifts as small as 0.01 fringe, while the expected shift was 0.4 fringe.

However, no shift was ever observed, no matter how the apparatus was rotated or what time of day or night the measurements were made.

The possibility that the arms of the apparatus became slightly shortened when moving against the ether was considered, but a full explanation had to wait until Einstein came into the picture.


2 2

v

1 v /

x tx

c

2

2 2

v /

1 v /

t x ct

c

y y

z z

Lorentz Transformation Equations

Einstein’s Special Theory of Relativity

Postulates of the Special Theory of Relativity

This solves the problem of Maxwell’s prediction of the invariance of the speed of light in vacuum – the speed of light is in fact the same in all inertial reference frames.

1. The laws of physics have the same form in all inertial reference frames.

2. Light propagates through empty space with speed c independent of the speed of source or observer.

It was impossible to achieve the kinds of speeds necessary to test his ideas (especially while working in the patent office…), so Einstein used Gedanken experiments or thought experiments.

Young Einstein

Gedanken (Thought) Experiments

Simultaneity

One of the implications of relativity theory is that time is not absolute. Distant observers do not necessarily agree on time intervals between events, or on whether they are simultaneous or not.

In relativity, an “event” is defined as occurring at a specific place and time.

Thought experiment: Lightning strikes at two separate places. One observer believes the events are simultaneous – the light has taken the same time to reach her – but another, moving with respect to the first, does not.

Simultaneity

Here, it is clear that if one observer sees the events as simultaneous, the other cannot, given that the speed of light is the same for each.

Simultaneity

Time Dilation

A different thought experiment, using a clock consisting of a light beam and mirrors, shows that moving observers must disagree on the passage of time.

Calculating the difference between clock “ticks,” we find that the interval in the moving frame is related to the interval in the clock’s rest frame:

Time Dilation

“Clocks moving relative to an observer are measured by that observer to run more slowly (as compared to clocks at rest)”.

The factor multiplying t0 occurs so often in relativity that it is given its own symbol, called Lorentz factor.

Time Dilation

The ratio v/c is sometimes replaced by the symbol called the speed factor.

The time t0 is called the proper time. It represents the time interval between two events in a reference frame where the two events occur in the same point in space.

Measurements of the same time interval from any other inertial reference frame are always greater. This time, t, is called the relativistic time.

Time dilation is reciprocal: observers in frames that are at rest see time travel faster than for those in motion. And vice versa!

Time Dilation

Time Dilation

Some values of :

Time Dilation

Example 1: Time dilation at 100 km/h.

Let us check time dilation for everyday speeds. A car traveling covers a certain distance in 10.00 s according to the driver’s watch. What does an observer at rest on Earth measure for the time interval?

Time Dilation

Example 2: Time dilation at very high speed.

A passenger on a high-speed spaceship traveling between Earth and Jupiter at a steady speed of 0.75c reads a magazine which takes 10.0 min according to her watch. (a) How long does this take as measured by Earth-based clocks? (b) How much farther is the spaceship from Earth at the end of reading the article than it was at the beginning?

It has been proposed that space travel could take advantage of time dilation – if an astronaut’s speed is close enough to the speed of light, a trip of 100 light-years could appear to the astronaut as having been much shorter.

The astronaut would return to Earth after being away for a few years, and would find that hundreds of years had passed on Earth.

Time Dilation and Space Travel

Time Dilation and the Twin Paradox

This brings up the twin paradox – if any inertial frame is just as good as any other, why doesn’t the astronaut age faster than the Earth traveling away from him?

Time Stops For a Light Wave

Because:

And, when v approaches c:

Thus, for anything traveling at the speed of light:

In other words, any finite interval at rest appears infinitely long at the speed of light.

Problems (pp 979)

5.] The mean lifetime of stationary muons is measured to be 2.2 s. The mean lifetime of high-speed muons in a burst of cosmic rays observed from Earth is measured to be 16 s. Find the speed of these cosmic-ray muons relatiive to Earth.

6.] An unstable high-energy particle enters a detector and leaves a track 1.05 mm long before it decays. Its speed relative to the detector was 0.992c. What is its proper time? That is how long would the particle have lasted before decay had it been at rest with respect to the detecto?

[Assignment: solution to the twin paradox, 7]

Length Contraction

If time intervals are different in different reference frames, lengths must be different as well.

Length Contraction

Length contraction is given by:

or

The length L0 is called the proper length. It is the length of the object – or distance between two points whose positions are measured at the same time – as measured by observers at rest with respect to it.

Length L will be measured by observers when the object travels past them at speed .

Length Contraction

Length contraction occurs only along the direction of motion.

“The length of an object is measured to be shorter when it is moving relative to the observer than when it is at rest”.

Length ContractionA moving object shrinks Relative point of view

Length contraction is also reciprocal.

Problems (pp 980)

10.] The length of a spaceship is measured to be exactly half its rest length. (a) What is the speed of the spaceship relative to the observer’s frame? (b) By what factor do the spaceship’s clocks run slow, compared to clocks in the observer’s frame?

14.] A spaceship of rest length 1.30 m races past a timing station at a speed of 0.74c. (a) What is the length of the spaceship as measured by the timing station? (b) What time interval will the station clock record between the passage of the front and back end of the ship?

[Assignment: 11, 15]

If v << c, i.e., β ≈ 0 and g ≈ 1, yielding the familiar Galilean transformation.

Space and time are now linked, and the frame velocity cannot exceed c.

2 2

v

1 v /

x tx

c

2

2 2

v /

1 v /

t x ct

c

y y

z z

2 2

v

1 v /

x tx

c

2

2 2

v /

1 v /

t x ct

c

y y

z z

Length contraction

Simultaneity problems

Time dilation



A more symmetrical form:

v / c

2 2

1

1 v / c

If v << c, i.e., β ≈ 0 and g ≈ 1, yielding the familiar Galilean transformation.

Space and time are now linked, and the frame velocity cannot exceed c.

Relativistic Mass and Momentum

Expressions for mass and momentum also change at relativistic speeds.

Mass:

m0 = rest mass

Momentum:

The Ultimate Speed

A basic result of special relativity is that nothing can equal or exceed the speed of light. This would require infinite momentum – not possible for anything with mass.

Mass and Energy

At relativistic speeds the formula for energy is modified as well.

Kinetic Energy:

m = relativistic massm0 = rest mass

Mass and Energy

Total Energy:

total energyrest energy

Combining the relations for energy and momentum gives the relativistic relation between them:

Energy and Momentum

All the formulas presented here become the usual Newtonian kinematic formulas when the speeds are much smaller than the speed of light.

There is no rule for when the speed is high enough that relativistic formulas must be used – it depends on the desired accuracy of the calculation.

Mass, Energy and Momentum

Problems (pp 982)

40.] How much work must be done to increase the speed of an electron from rest to (a) 0.50c, (b) 0.990c, (c) 0.9990c?

44.] What is the speed of an electron whose kinetic energy is 100MeV?

45.] A particle has a speed of 0.990c in a laboratory reference frame. What are its kinetic energy, its total energy, and its momentum if the particle is (a) a proton and (b) an electron?

[Assignment: 42, 48, 54]

The Impact of Special Relativity

The predictions of special relativity have been tested thoroughly, and verified to great accuracy.

The correspondence principle says that a more general theory must agree with a more restricted theory where their realms of validity overlap. This is why the effects of special relativity are not obvious in everyday life.

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