Relativity Option

Lessons:

An Introduction to Special Relativity

Relativity was born out of the frustrations of a 26 year old Physicist called Albert Einstein who was searching for a way to reconsile the apparently contradictory laws of Newton (mechanics) and Maxwell (electromagnetism).

Newton's laws stated that all motion is relative- there is no such thing as absolute motion.

Under Newtonian Physics it is possible to determine a relationship between different frames of reference using Galilean Transformations.

x' = x -vt

y' = y

z' = z

t' = t

So using this transformation (the Galilean transformation), we can agree on the location of a stationary object between two frames of reference, where one is moving (at a velocity, v) relative to the other.

And we can measure the speed of an object in one frame of reference if we know its speed in the other...

Measurements of speed in the y and z directions will agree between the two frames in our example.

In the x direction...

speed = distance/ time

x'/t = x/t - vt/t

u' = u - v

If the moving frame is moving at 4m/s relative to the stationary frame, and an object is moving at 15m/s in the stationary frame, in the same direction as the moving frame, then it will be seen as moving at 11m/s in the moving frame.

Maxwell's laws suggested that the propogation of electromagnetism was dependent on there being a single absolute frame of reference.

As you know electromagnetic waves are the combined oscillation of an electric and a magnetic field.

The speed at which an e-m wave can propogate depends on the material it is travelling in, and it is fastest when it is travelling in a vacuum.

Maxwell's equations gave the speed of light as...

e0 and m0 t gives the speed of light in vacuum to be 299 792 458 m/s.

Maxwell implies by this that the speed of light is independent of the speed of the source that the light is emitted from, and is thus the same for all observers.

Now watch these to consolidate the point:

Galileo's transformation

Einstein's Solution with Special Relativity...

Einstein made two postulates to reconsile the differences between Newtonian and Maxwellian physics. These are the postulates of special relativity:

• All laws of nature are the same in all uniformly moving (inertial) frames of reference
• The speed of light in free space is the same for all observers, regardless of the speed of the source, the speed of the observer

Initially it doesn't look like any reconsilliation has happened does it?

The trick is in the need to start thinking about time and space (ideas that we have previously held as absolute) as being relative, and the only absolutes as being the two postulates mentioned above.

Questions

Kirk pp171-172, 177.

Consequences of Special Relativity: Time

The postulates of special relativity:

• All laws of nature are the same in all uniformly moving (inertial) frames of reference
• The speed of light in free space is the same for all observers, regardless of the speed of the source, the speed of the observer

Simultaneity

As a result of the two postulates several consequences arise that demand us to reconsider our ideas of space and time. The first of which is simultaneity.

Two events that are simultaneous in one frame of reference need not be simultaneous in a frame moving relative to the first.

Take the exampe of observers inside and outside a fast moving space craft when a light at its centre is switched on...

 Light signals reach simultaneously both ends of the chamber of the ship as seen by the traveling observer From the point of view of the static observer the signals do not reach simultaneously the ends of the chamber because the ship is moving while light moves at the same speed in either frame

This is a relativistic result - a consequence of light having the same speed for all observers independent of their frame of reference.

Questions:

1. How is the nonsimulaneity of hearing thunder after seeing lightening similar to relativistic nonsimultaneity?

2. Suppose that the observer standing on a planet in the above diagram sees a pair of lightening bolts simultaneously strike the front and rear ands of the high-speed rocket ship compartment. Will the lightening strikes be simultaneous to an observer in the middle of the rocket ship compartment? (We assume here that an observer can detect any slight differences in time for light to travel from the ends to the middle of the compartment).

Solutions

Time dilation

Watch Einstein explain 'The Discovery of Slowness'

We have to start considering time as the 4th dimension now, because relativistic effects apply to time just as they do to space. Just picture it as another axis on your graph.

So see how relative movement can affect time, we must consider a light clock.

It can be seen that time appears to pass more quickly to Jack who is stationary, than it does to Jill who is moving.

We have to define proper time, t0, as the time recorded in the frame of referece at rest compared to the event being measured.

In this case Jill records the proper time.

You can see that the amount of time dilation depends on the speed of the frames relative to one another. We can calculate this simply...

This equation is important as it gives us the Lorentz factor.This comes up again and again in relativity, so we give it the shorthand, g.

Exercise: Plot a sketch graph showing how the Lorentz factor, g, varies as a function of speed, v.

Questions:

1) What trend can you see in the graph? Describe it.

2) What does this imply about time dilation as you approach the speed of light?

This can also lead to length contraction!

Read about the Twin Paradox as predicted by Special Relativity and write 300words to summarise it. Diagrams will be useful in explaining the ideas.

What was the Hafele-Keating Experiment? Find out by reading the paper about it.

Questions

Kirk pp173-174, 176

Consequences of Special Relativity: Space

Length Contraction

Time isn't the only thing that changes when you move quickly! Since we must now consider time and space along the same lines, it implies that space changes too.

Objects get smaller as you approach the speed of light.

Initailly Physicists thought this was due to the obect contracting, but Einstein asserted that it was a consequence of space itself contracting, making object within it appear smaller.

The amount an object appears to contract depends on its speed, and is given by the equation...

where L= the measured length of the moving object, L0= the measured length of the object at rest (proper length), and v is the speed of the moving object.

We see the object contract as it gets closer to the speed of light.

The mathematical proof.

Question:

Calculate the length of an object of proper length 1m, when it is travelling at...

a) v= 0.87c

b) v= 0.995c

c) v= 0.999c

d) v= c

NB: We only see length contraction in the direction that the object is moving in. Think how this is consistent with Galilean relativity.

In the frame of reference of our meterstick, its length is 1 metre. An observer in a moving frame sees our metrestick contacted, where we see their metrestick contracted. The effects of relativity are always attributed to the other guy!

Implications for space travel

This all has imporant implications for space travel. Remember that we said it would be impossible to ever journey in space? Well relativity brings back the possibility.

If astronauts go fast, then the distances between stars appears to contract. How fast would they need to travel to make a distance of 4 light years as measured from Earth seem like just 1 light year?

BUT...

Observed from Earth the distance the spaceship travels stays the same (4ly) but clocks aboard the spaceship appear to slow down to explain why the astronaut thinks the journey is shorter.

So... length contraction and time dilation are in effect two faces of the same phenomnon!

Question:

A rectangular billboard in space has the dimensions 10m x 20m. How fast and in what direction with respect to the billboard would a space traveller have to pass for the billboard to appear square?

As a consequence of SR, velocities cannot simply add together. If this were true we would end up with situations, like we have seen previously, where observers in moving frames of reference would observe speed of light in other frames greater then c.

To overcome this we use an improved equation for the addition of velocities:

Question

Conceptual questions to grapple with

Kirk pp175

Back to Relativity Option

Consequences of Special Relativity: Mass and Energy

E=mc2

We've dealt with this before, but not really discussed how it comes about.

An object, even one at rest and not interacting with anything else, has an "energy of being". This is called its rest energy. The amount of rest energy something has depends on it's mass and the two quantities are related by the formula...

E=mc2

Predictably they rest mass of an object is the mass equivilence of its rest energy.

We have seen from fission and fusion reactions that the mass of products in these reactions are less than the reactants, demonstrating that mass has been converted to energy.

We find the same thing in other types of exothermic reactions (eg: combusion in coal or oil powered powerstations). In fact any object at rest that undergoes an energy change, also undergoes a mass change! Although these changes are almost inperceptably small, they do exist and have been measured:

• A lit filament bulb has a greater mass than an unlit bulb.
• A wound up spring clock has a greater mass then an unwound clock.

So just as length and time are fundamentally related, so are mass and energy.

A thought experiment...

Take a 1kg mass and apply a 1N force to it. What happens?

Continue to apply this force, what happens?

Theoretically, according to the Physics we know the mass should continue to accelerate at a= F/m indefinately.

But...

We also know that nothing can travel faster than the speed of light, so we have a problem.

 Acceleration as predicted by Newtonian physics In practice as we get towards the speed of light the acceleration of the mass begins to reduce.

With a decreasing acceleration and still a constant force, this implies that at high speeds mass must increase.

And it does so inline with the Lorentz factor such that...

Mass-Energy

Mass and energy are equivilent.

But... because mass increases at high speeds, we can no longer use classical equations like Ek= 1/2mv2 or p= mv.

The energy of a particle at rest (its rest energy) is calculated from its rest mass: E0= m0c2

The energy of a particle is greater if it has a velocity, because its mass increases, so: E= mc2

m is greater than the rest mass, m0 (m= gm0).

Therefore the kinetic energy a particle has is the energy is has through motion and rest, minus the energy it has through rest...

EK= E- E0= mc2-m0c2= gm0c2-m0c2 = (g-1)m0c2

EK= (g-1)m0c2

Question:

An electron is accelerated through a p.d. of 1.0 x 106V. Calculate its velocity.

(Ans: 0.94c)

Data analysis question on relativistic mass increase in Beta particles

Kirk pp179

Back to Relativity Option

Evidence for Special Relativity

The muon experiment- We would expect, given the muons very short half life, that almost no muons which are produced 10km above the Earth's surface to reach the surface, even though they are travelling very quickly. If we account for relativistic effects of time dilation and length contraction, we see a vastly increased flux of muons hitting the surface of the Earth. This is consistent with observation.

Michelson-Morley experiment - The important thing here is that the effect shown in the animation is not observed. Thus demonstrating that the speed of light is invarient (does not change with speed through the "aether") and verifying the second postulate of SR.

Pion decay experiment- Neutral pions, a type of particle made from two quarks, travelling at nearly the speed of light in a particle accelerator. When these pions decay, they produce gammas (high-frequency light). The speed of these gammas was measured directly, and was found to be c. Even though the pions were themselves travelling almost the speed of light, the light they emitted was still going c.

Relativistic Momentum and Energy

We saw in the last lesson the relationships between mass and energy...

E0= m0c2 and E= mc2

From these we can derive the equations for other physical quantities such as momentum and total energy. You don't need to derive these equations, just remember to use them if the situation is relativistic.

p=g m0v ,

for momentum, and because momentum affects an objects total energy...

E2=p2c2 + m02c4,

for total energy.

Units

You are familiar with the use of eV instead of J.

How many eV is one Joule?

If we use MeV to describe energy in relativistic mechanics, and we use the units for the speed of light to be c (ie. half the speed of light is denoted 0.5c rather than 1.5 x 108m/s), then what are the units we use commonly for mass and momentum?

• Mass has units of MeV c-2

• Momentum has units of MeV c-1

IB Questions on Special Relativity

Kirk pp180

Back to Relativity Option

General Relativity:The Equivilence Principle

The Principle of Equivilence

What is the difference between sitting in a gravitational field of 1g like we are doing now, and accelerating at a rate of 9.81ms-2 in deep space?

This is Einstein's Equivilence Principle, and the foundation of the General Theory of Relativity. It states:

There is no difference between an accelerating frame of reference and a gravitational field.

Watch the explanation of the Equivilence Principle using a lift and a rocket

Inertial and Gravitational Mass

Mass is a quantity which describes two properties of matter...

Inertial mass: Describes an objects response to a force (ie: how much acceleration it undergoes),

from F= ma

Gravitational mass: Describes how much atraction an object feels due to the presence of another mass,

from F= GMm/r2

We don't normally distinguish between these two quantities because for all intents and purposes they are the same (on Earth).Well the same is true with gravitational fields and acceleration.

Bending of Light by Gravity Keeping a spaceship as our laboratory, consider what happens to a ball thrown across it, according to an observer inside the spaceship, when it is stationary on it's launchpad...

Now because of the equivilence principle the same thing must be observed if the ball is thrown while the space ship is accelerating at a rate of a= g. This is fairly straight forward to appreciate.Can you imagine what an observer external to the spaceship frame of reference sees?

Now consider light.In an accelerating spaceship, a beam of light shone across it must suffer some deflection to an observer inside the spaceship in the same way the ball did.

The only difference is that the deflection is very small, because light travels from one side of the spaceship to the other very quickly.The equivilence principle therefore demands that light would suffer a similar defection in a gravitational field.

Now the big deduction.Because we assume light to travel in a straight line through space, and gravity bends light, this must mean that gravity bends the space(time) through which light travels. This is the basic idea of General Relativity.

Thought Experiment to demonstrate the effect of bent space-time due to gravity.

Imagine you and a friend both start out on a long D of E expedition to walk from the equator to the north pole. One of you starts in Indonesia and the other starts in Uganda. (Ignoring the fact that there is sea in the way) You both walk north (parallel to each other) untill you get there, and you meet. This could be explained in two ways. As you travel north you get closer together as a result of the curvature of space (the surface of the Earth) or as a result of a force of attraction (gravity) between you. Although in this case you know it is as a result of the Earth being curved, but can you see how gravity can bring about a similar effect?

Thus gravity can be thought of as warping space-time.

Demonstration A ball and curved sheet

• mass 'tells' space how to curve
• space 'tells' mass how to move

Experimental evidence for the bending of space time.

The difficulty with measuring this effect is that light travels so fast, that it is difficult to measure the tiny amounts by which it is bent in standard laboratory conditions...

But we have had success with measuring and provding evidence for GR with large scale laboratories (ie: interstellar space!)

Gravitational Lensing

We can see the effect of light bending space by using a technique called gravitational lensing...

We see an image of an interstellar object displaced from it's original position due to the bending effect of a large gravitiational field between us and the object we are trying to observe.Initially this technique was used to verify the theory of GR (by Arthur Eddington during the solar eclipes of 1919).

More recently gravitational lensing has been used to predict and verify the existance and locations of dark massive objects (such as Black holes!)

Watch Einstein explain... 'Bent Space'

Questions:

Kirk pp181-182

General Relativity: Black Holes and Gravitational Red Shift

Gravitational Time Dilation

Consider 3 clocks, 2 placed on a rotating disk as shown, while one is at rest on a table next to it.

Clocks 1 and 3 will run at the same speed because they are not moving relative to each other (although clock 1 is rotating).

Clock 2 will appear to run slowly to an observer in clock 3's frame of reference, because it is moving with respect to clock 3, and hence experiences time dilation as a result of special relativity.

Since clocks 1 and 3 run at the same rate, then clock 2 will also appear to run slowly to an observer in clock 1's frame of reference.

BUT... Clock 2 doesn't appear to be moving to an observer in clock 1's frame of reference. Got it?

So how to explain it?

Clock 2 experiences a centripetal acceleration, and thus appears to run slowly to clock 1.

The principle of equivalence states that there is no difference between an accelerated frame of reference and one in a gravitational field.

Thus time should run slowly for an observer in a gravitational field - Gravitational Time Dilation.

Gravitational Red-Shift

We can use atom as clocks (hence atomic clocks!). As electrons in atoms orbit they emit a radiation, which is representative of the rate that they orbit.

The electrons in different atoms have different orbital velocities, and therefore different frequencies of radiation. But similar atoms (ie: of the same element) should emit similar frequency radiation. Effectively the radiation emitted by an an atomic electron acts like a clock.

Thus in a gravitational field if time is slowed down to an external observer, then an electron will appear to orbit more slowly, and thus emit a lower frequency of radiation than a similar electron viewed in the observer's own frame of reference.

Lower frequency --> Longer wavelength --> Red shift.

OR

A photon moving away from a gravitational field increases it gravitational potential energy (a negative quantity remember). Thus it decreases its total energy.Since E = hf, then it must too reduce its frequency.

Lower frequency --> Longer wavelength --> Red shift.

OR

You can simply imagine gravity acting on the wave of radiation itself (photons have energy, mass energy equivalence implies that gravity can act on energy as well as mass), thus stretching its wave length and red shifting the radiation.

Maths of Gravitational RedshiftYou need to be able to use this equation to determine the frequency shift (Df) for photons moving horizontal distances (Dh) into and out of a gravitational field of strength (g).

And finally to Black Holes

At masses greater than about 3 solar masses, then neutron degeneracy can no longer adequately oppose the collapsing force of gravity on a stellar remnant, and it becomes a black hole.We think that all the matter in a black hole is concentrated in a single point, which we call a singularity.

The gravitational influence of this intensely dense singularity is to warp space time considerably..

The bending of space time near to a black hole is so extreme that not even light can escape it. The closer light gets to a black hole the more it gets bent. Eventually there becomes a distance from the black hole at which light gets trapped and continues in a circular path around the black hole forever. This is called the photon sphere.

Moving further towards the singularity you come to the Schwartzchild radius. This is the distance from the black hole where the escape velocity,

, equals the speed of light.

Thus the Schwartzchild radius is expressed as...

Any problem with this?

We've used Newtonian equations to deduce this relativistic principle! Actually the relativistic derivation is much longer and complex, but pleasingly come out with the same result!

Given the massive warping of space that blackholes cause, and that we no longer consider space alone, but spacetime, this means that blackholes have a warping effect on time. They dilate it due to their massive gravitational effect.

The magnitude of the effect is determined by comparing the distance from the centre of the blackhole compared to the Schwartzchild radius...