IB Physics/Relativity

H.1 Introduction to relativity

H.1.1 Describe what is meant by a frame of reference

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 Transformation

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

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

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 relativity

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

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 relativity

• 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 simultaneity

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 Kinematics

H.3.1 Describe the concept of a light clock

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 interval

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 formula

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

l²+(vt')² = (l')²

t'² = (l')²/c²

t'² = (l²+(vt')²)/(c²)

After rearranging, t' = 1/( 1-v²/(c²) ) x t

H.3.6 Define proper length

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

H.4 Some consequences of special relativity

H.4.2 Discuss the Hafele-Keating experiment

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.4.5 Define rest mass

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

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

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 particle

The total energy E, momentum p, and the rest energy E₀ (=m₀c²) have a relationship E² = p²c² + E₀².

H.7 General relativity

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

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 equivalence

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.7.5 Describe the concept of spacetime

Spacetimeis 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 spacetime

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 matter

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 holes

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 radius

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 radius

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

H.8 Evidence to support general relativity

H.8.2 Describe gravitational lensing

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-shift

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.