unit 13 relativity

Unit 13Relativity

OutlineClassical Relativity and the Michelson–Morley Experiment

The Postulates of Special Relativity and the Relativity of Simultaneity

The Relativity of Length and Time: Time Dilation and Length Contraction

Relativistic Kinetic Energy, Momentum, Total Energy, and Mass–Energy Equivalence

The General Theory of Relativity

Relativistic Velocity Addition

Classical Relativity and the Michelson–Morley Experiment

We expect that the laws of physics should be the same in any inertial reference frame—that is, any frame that is not accelerating.

In an accelerating frame, this may not be true; accelerating frames have “pseudoforces” that replace the acceleration, such as the centrifugal force in circular motion.


This is the principle of classical relativity, known to Newton:

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


When the equations of electromagnetism were found to predict a specific speed for electromagnetic waves, classical relativity had a problem—this could not be true in all inertial reference frames.


In addition, there was no obvious medium that propagated electromagnetic waves.

Both of these problems would be solved by the existence of a “luminiferous ether.” This all-permeating substance would carry the electromagnetic waves, and would also establish a preferred reference frame in which the speed of light would have its calculated value.


Michelson and Morley devised an experiment to detect the ether by measuring the difference in the speed of light in two perpendicular directions, produced by the Earth’s motion through the ether.

All their measurements showed no difference in the speed of light in any direction.


Einstein thought that all the laws of physics, not just the laws of mechanics, should be the same in all inertial reference frames. This led him to his postulates of relativity.

Postulate I (principle of relativity): All the laws of physics are the same in all inertial reference frames.

Postulate II (constancy of the speed of light): The speed of light in a vacuum has the same value in all inertial systems.

The Postulates of Special Relativity and the Relativity of SimultaneityThe first postulate is classical relativity, extended to all laws of physics.

The second seems counterintuitive—how could the speed of light not add the same way as other speeds?


However, in science observation trumps theory, and the constancy of the speed of light has been confirmed many times over.

So, what are the consequences of the speed of light being the same in all reference frames?


First, events that are simultaneous in one frame are not necessarily simultaneous in another.

The Postulates of Special Relativity and the Relativity of SimultaneityWe see events occurring simultaneously in our reference frame. However, another observer moving with respect to us will not see the events as simultaneous—the light from one will reach them before the light from the other.

Events that are simultaneous in one inertial reference frame may not be simultaneous in a different inertial frame.


These effects are vanishingly small at ordinary speeds; this is why the constancy of the speed of light is not part of our everyday experience.


A clock may be made from a light beam and mirrors. As the speed of light is the same for all observers, an observer in another reference frame will see the clock running slow.


The time between pulses as seen by a moving observer is:

This effect is called time dilation. The time as measured in an object’s own reference frame is called its proper time.


The equations of special relativity become much less cumbersome to write if we define the following shorthand:

Using this, the equation for time dilation becomes:


The value of γ becomes significantly different from 1 only when the relative speed is a substantial fraction of the speed of light.


Suppose we made a clock that operated in a way that had nothing to do with the speed of light. Would time dilation still occur?

Yes, it would—otherwise different clocks in the same reference frame would behave differently depending on that frame’s speed with respect to another, violating the first principle of relativity.


How do we measure the length of an object that is moving with respect to us?

One way is to mark both ends simultaneously and then measure the distance between them; another is to measure the time it takes the entire object to pass by a given point.

However, we already know that moving observers disagree on simultaneity and on time intervals!


Therefore, we measure the length to be shorter than in the object’s own reference frame.


An object’s length is largest when measured by an observer at rest with respect to it (the “proper” observer). If the object is moving relative to an inertial observer, that observer measures a smaller length than the proper observer.


The contracted length is given by:


The twin paradox: Suppose one of a pair of identical twins goes on a spaceship ride close to the speed of light. When she returns to Earth, after 20 years have passed, she has aged only two years due to time dilation.

But wait! From her frame, the Earth was moving with respect to her, and her Earth twin should be the younger one.

What’s wrong here?


The answer is that the traveling twin, in order to return to Earth, has to turn around and come back. Therefore, her frame is not inertial, and she is indeed younger than her twin when she comes home.

Relativistic Kinetic Energy, Momentum, Total Energy, and

Mass–Energy Equivalence

Calculations show that relativistic kinetic energy is given by:

Relativistic Kinetic Energy, Momentum, Total Energy, and Mass–

Energy Equivalence

For speeds close to c, the relativistic kinetic energy increases rapidly, and approaches infinity as the speed approaches c.


Energy Equivalence

The relativistic momentum is then:

The momentum also becomes infinitely large as speeds approach c.


Energy EquivalenceThe total energy is then:

Note that this is not zero even when v = 0.


Energy Equivalence

When v = 0,

This equation (which you may recognize) gives the rest energy of an object, also called its mass energy. It is possible to convert this energy to other forms of energy, through nuclear reactions.


Energy Equivalence

So, when do we need to take relativistic effects into account, and when can we ignore them?

For speeds below 10% of the speed of light or kinetic energies less than 0.5% of an object’s rest energy, the error in using the nonrelativistic formulas is less than 1%, and it is then usually acceptable to use the nonrelativistic expressions.

General RelativityGeneral relativity applies to accelerating systems. It begins with the principle of equivalence:

An inertial reference frame in a uniform gravitational field is physically equivalent to a reference frame that is not in a gravitational field, but that is in uniform linear acceleration.

Equivalently,

No experiment performed in a closed system can distinguish between the effects of a gravitational field and the effects of an acceleration.

General Relativity

This is true for a spaceship that is either on the ground or accelerating with a = g.

General Relativity

It is also true for either an isolated spaceship or one in free fall near a massive object.

General RelativityThis means that light must bend in a gravitational field, just as it bends while accelerating.

General Relativity

This bending was first observed in the early 20th century during a total eclipse of the Sun.

General Relativity

This bending also means that an intervening mass can serve as a “lens” for a more distant object. Numerous examples have been observed in astronomical photographs.

General RelativityIt is possible to have an object with a gravitational field strong enough that the escape speed is equal to, or greater than, the speed of light. Such an object is called a black hole.

The radius within which the escape speed is c or greater is called the Schwarzschild radius:

An object is a black hole only if the mass M is within the radius R.

Relativistic Velocity AdditionClearly, the relativistic addition of velocities cannot be as simple as it is classically, if all speeds are to remain less than the speed of light.

Relativistic Velocity Addition

The correct form for velocity addition is given by:

If both v and u’ are small compared to c, this gives the classical formula u = v + u’.

Review

Classical relativity supported the existence of an “ether” to carry electromagnetic waves. No evidence for the ether was ever found.

Principle of relativity: All the laws of physics are the same in all inertial reference frames.

Principle of the constancy of the speed of light: The speed of light in a vacuum has the same value in all inertial systems.

Review

Relativistic effects in moving systems include time dilation, length contraction, and the relativity of simultaneity.

Relativistic energy:

Rest energy:

Review

Principle of equivalence:

An inertial reference frame in a uniform gravitational field is physically equivalent to a reference frame that is not in a gravitational field, but that is in uniform linear acceleration.

General relativity predicts black holes and the bending of light in a gravitational field.

unit 13 relativity

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