## Contents

- 1 H.1 Introduction to relativity
- 2 H.2 Concepts and postulates of special relativity
- 3 H.3 Relativistic Kinematics
- 3.1 H.3.1 Describe the concept of a light clock
- 3.2 H.3.2 Define proper time interval
- 3.3 H.3.3 Derive the time dilation formula
- 3.4 H.3.4 Sketch and annotate a graph showing the variation with relative velocity of the Lorentz factor
- 3.5 H.3.5 Solve problems involving time dilation
- 3.6 H.3.6 Define proper length
- 3.7 H.3.7 Describe the phenomenon of length contraction
- 3.8 H.3.8 Solve problems involving length contraction

- 4 H.4 Some consequences of special relativity
- 4.1 H.4.1. Describe how the concept of time dilation leads to the "twin paradox"
- 4.2 H.4.2 Discuss the Hafele-Keating experiment
- 4.3 H.4.3 Solve one-dimensional problems involving the relativistic addition of velocities
- 4.4 H.4.4 State the formula representing the equivalence of mass and energy
- 4.5 H.4.5 Define rest mass
- 4.6 H.4.6 Distinguish between the energy of a body at rest and its total energy when moving
- 4.7 H.4.7 Explain why no object can ever attain the speed of light in a vacuum
- 4.8 H.4.8 Determine the total energy of an accelerated particle

- 5 H.5 Evidence to support special relativity
- 5.1 H.5.1 Discuss muon decay as experimental evidence to support special relativity
- 5.2 H.5.2 Solve problems involving the muon decay experiment
- 5.3 H.5.3 Outline the Michelson-Morley experiment
- 5.4 H.5.4 Discuss the result of the Michelson-Morley experiment and its implication
- 5.5 H.5.5 Outline an experiment that indicates that the speed of light in vacuum is independent of its source

- 6 H.6 Relativistic momentum and energy
- 7 H.7 General relativity
- 7.1 H.7.1 Explain the difference between the terms gravitational mass and inertial mass
- 7.2 H.7.2 Describe and discuss Einstein's principle of equivalence
- 7.3 H.7.3 Deduce that the principle of equivalence predicts bending of light rays in a gravitational field
- 7.4 H.7.4 Deduce that the principle of equivalence predicts that time slows down near a massive body
- 7.5 H.7.5 Describe the concept of spacetime
- 7.6 H.7.6 State that moving objects follow the shortest path between two points in spacetime
- 7.7 H.7.7 Explain gravitational attraction in terms of the warping of spacetime by matter
- 7.8 H.7.8 Describe black holes
- 7.9 H.7.9 Define the term Schwarzschild radius
- 7.10 H.7.10 Calculate the Schwarzschild radius
- 7.11 H.7.11 Solve problems involving time dilation close to a black hole
- 7.12 H.7.12 Describe the concept of gravitational red-shift
- 7.13 H.7.13 Solve problems involving frequency shifts between different points in a uniform gravitational field
- 7.14 H.7.14 Solve problems using the gravitational time dilation formula

- 8 H.8 Evidence to support general relativity

## H.1 Introduction to relativityEdit

### H.1.1 Describe what is meant by a frame of referenceEdit

A frame of reference refers to a point of view. Physics refers to observational frames of reference, i.e. what is heard, seen, touched, smelt or tasted from a certain point of view.

e.g. you have a frame of reference (in a computer chair) that views the world as stationary. A frame of reference with the sun as stationary would view you as moving around the earth.

### H.1.2 Describe what is meant by a Galilean TransformationEdit

Galilean is what seems intuitive. The equations do not involve relativity.

### H.1.3 Solve problems involving relative velocities using the Galilean transformation equationsEdit

This should be easy. position equation: x'=x-vt

Velocity equation: u'=u-v

e.g. You are sitting on a park bench. You see a bike and car moving away from each other. The bike is moving at 5m/s, the car is moving at 20 m/s. How fast is the bike moving in the cars reference frame? (Ans: 25m/s)

## H.2 Concepts and postulates of special relativityEdit

### H.2.1 Describe what is meant by an inertial frame of referenceEdit

A reference frame that is moving with a constant velocity. This really isn't that essential but you might see a fundamental flaw in this definition (just ignore it if you do).

### H.2.2 State the two postulates of the special theory of relativityEdit

- the speed of light in a vacuum is constant for all inertial observers.
- The laws of physics are the same for all inertial observers.

### H.2.3 Discuss the concept of simultaneityEdit

Consult a textbook (or youtube "simultaneity"), best explained with diagrams.

In summary though, simultaneous events that take place at the same point in space will be simultaneous to all observers. However, events that take place at different points in space can be simultaneous for one observer but not for another.

## H.3 Relativistic KinematicsEdit

### H.3.1 Describe the concept of a light clockEdit

**Best described by diagram.**

Imagine a clock where light is bounced off two mirrors. mirror| light beam --> mirror |

Each time it hits a mirror a "tick" is registered. Since light is the same for all reference frames, this is the most accurate type of clock.

This seams simple for a stationary clock, HOWEVER in a reference frame where the clock moves, it must travel a diagonal distance which is greater than that for a stationary time frame. In this case a "tick" would take longer for one reference frame than the other. Therefore, time is shorter for one reference frame than another.

### H.3.2 Define proper time intervalEdit

Proper time interval is the elapsed time measured between two events when the observer is in an inertial frame of reference.

### H.3.3 Derive the time dilation formulaEdit

The time dilation formula can be derived using pythagoras's theorem. The length of the light clock is l. According to the observer's clocks, the clock in the light clock travels longer (l' = ct'). Horizontal distance traveled by the light clock: vt'. t = l/c

l^{2}+(vt')^{2} = (l')^{2}

t'^{2} = (l')^{2}/c^{2}

t'^{2} = (l^{2}+(vt')^{2})/(c^{2})

After rearranging, t' = 1/( 1-v^{2}/(c^{2}) ) x t

### H.3.4 Sketch and annotate a graph showing the variation with relative velocity of the Lorentz factorEdit

### H.3.5 Solve problems involving time dilationEdit

### H.3.6 Define proper lengthEdit

Length of an object as defined in a reference frame which is in rest with the object (inertial reference frame)

### H.3.7 Describe the phenomenon of length contractionEdit

### H.3.8 Solve problems involving length contractionEdit

## H.4 Some consequences of special relativityEdit

### H.4.1. Describe how the concept of time dilation leads to the "twin paradox"Edit

### H.4.2 Discuss the Hafele-Keating experimentEdit

Testing the theory of special relativity, scientists flew two clocks around the world in opposite directions and compared them to a clock that was stationary relative to the earth's surface. One plane flew eastward and another went westward. When the planes returned, it was found that that for the plane flying East, it's clock was slower than the clock on the earth's surface (it was behind by 59 ns). Since the earth spins to the east and the plane was travelling east relative to the earth's atmosphere, this clock had a greater velocity relative to the stationary frame of reference: the centre of the Earth and so it's clock ticked a little slower. The westward clock ran faster than the clock that remained on the earth's surface as it was moving more slowly relative to the centre of the earth. This experiment provided evidence of time dilation that was in excellent agreement with relativistic predictions.

### H.4.3 Solve one-dimensional problems involving the relativistic addition of velocitiesEdit

### H.4.4 State the formula representing the equivalence of mass and energyEdit

### H.4.5 Define rest massEdit

**Rest mass** of an object is defined to be the energy required to produce the object at rest.

### H.4.6 Distinguish between the energy of a body at rest and its total energy when movingEdit

### H.4.7 Explain why no object can ever attain the speed of light in a vacuumEdit

According to the classical mechanics, F = ma therefore if the force is applied to an object constantly over a very long time period, the speed of the object must increase without limit. However, this is not true; as the speed of the object increases, *the relativistic mass of the object increases*, therefore acceleration will gradually decrease. Only the particles with no rest mass (such as photon) can travel with the speed of light.

### H.4.8 Determine the total energy of an accelerated particleEdit

The total energy E, momentum p, and the rest energy E_{0} (=m_{0}c^{2}) have a relationship E^{2} = p^{2}c^{2} + E_{0}^{2}.

## H.5 Evidence to support special relativityEdit

### H.5.1 Discuss muon decay as experimental evidence to support special relativityEdit

### H.5.2 Solve problems involving the muon decay experimentEdit

### H.5.3 Outline the Michelson-Morley experimentEdit

### H.5.4 Discuss the result of the Michelson-Morley experiment and its implicationEdit

### H.5.5 Outline an experiment that indicates that the speed of light in vacuum is independent of its sourceEdit

## H.6 Relativistic momentum and energyEdit

### H.6.1 Apply the relation for the relativistic momentum p = γm_{0}u of particlesEdit

### H.6.2 Apply the formula E_{k} = (γ-1)m_{0}c^{2} for the kinetic energy of a particleEdit

### H.6.3 Solve problems involving relativistic momentum and energyEdit

## H.7 General relativityEdit

### H.7.1 Explain the difference between the terms gravitational mass and inertial massEdit

Gravitational mass is the mass obtained from when a gravitational force acts upon an object. The inertial mass is the mass obtained from when an external force acts upon a body (the mass is resisting the acceleration). Evidently, the gravitational and inertial mass are exactly the same because uniform acceleration is indistinguishable from a gravitational field.

### H.7.2 Describe and discuss Einstein's principle of equivalenceEdit

The principle of equivalence states that there is no difference between an accelerating observer and an observer in a gravitational field. This principle coincides with General Relativity and the IB exams focus on light being bent by a gravitational field; Einstein stated this idea in respect to his closed elevator thought experiment. Light bent the same way regardless of whether the elevator was at rest or accelerating downward. The second part of the principle of equivalence is that time slows down when an object approaches a massive body such as black holes.

### H.7.3 Deduce that the principle of equivalence predicts bending of light rays in a gravitational fieldEdit

### H.7.4 Deduce that the principle of equivalence predicts that time slows down near a massive bodyEdit

### H.7.5 Describe the concept of spacetimeEdit

**Spacetime**is the 4-dimensional world with three space and one time coordinates.

### H.7.6 State that moving objects follow the shortest path between two points in spacetimeEdit

In the absence of any forces, a moving object will move in spacetime along path with the shortest length. This path is called **geodesics**.

### H.7.7 Explain gravitational attraction in terms of the warping of spacetime by matterEdit

Large masses will warp space-time in such a way that the shortest distance to be travelled between point A and B for a particle is now a curve around the large mass. As such, this curved path that the particle follows can be thought of as the gravitational attraction.

### H.7.8 Describe black holesEdit

**Black hole** is a singularity of space time, and is a point of infinite density. It causes extreme curvature of spacetime around it.

### H.7.9 Define the term Schwarzschild radiusEdit

**Schwarzschild radius** (R_{s}) is sometimes called **gravitational radius**, or **event horizon**. Within R_{s}, no object can escape from the gravitational field (because the escape velocity exceeds c within R_{s}).

### H.7.10 Calculate the Schwarzschild radiusEdit

R_{s} = 2GMc^{-2}, where G is the gravitational constant and M is mass of the black hole (or the star).

### H.7.11 Solve problems involving time dilation close to a black holeEdit

### H.7.12 Describe the concept of gravitational red-shiftEdit

### H.7.13 Solve problems involving frequency shifts between different points in a uniform gravitational fieldEdit

### H.7.14 Solve problems using the gravitational time dilation formulaEdit

## H.8 Evidence to support general relativityEdit

### H.8.1 Outline an experiment for the bending of EM waves by a massive objectEdit

### H.8.2 Describe gravitational lensingEdit

Assuming a massive galaxy has a large gravitational pull, light from a distant quasar is bent, thereby creating two quasars when viewed through a telescope. The galaxy acts a lens as it bends the incoming light from the quasar.

### H.8.3 Outline an experiment that provides evidence for gravitational red-shiftEdit

Pound-Rebka Experiment where they shot a photon from the ground floor of a building up to the attic. They found that the frequency of the photon was longer up at the attic then at the ground floor and thus, provides evidence for gravitational red-shift.