AH 1.5 General Relativity

Question 1

In the spacetime diagram below, the blue shaded region is called time like.

What do the world lines on this diagram represent?

Answer 1

Light, travelling at 3 x 10⁸ ms^-1.

Question 2

What is meant by inertial mass ?

Answer 2

Inertial mass measures a body’s resistance to being accelerated (i.e. have its momentum changed) and is the ‘m’ in mv = momentum.

Notice that inertial mass makes no reference to gravity.

Question 3

What is the relationship between gravitational mass and inertial mass ?

Answer 3

Inertial and gravitational mass are equivalent

Question 4

Explain what is meant by the equivalence principle.

Answer 4

The Equivalence Principle (EP) states that inertial mass is equal to gravitational mass. Or, put another way, there is no experiment that will discern the difference between the effect of gravity and the effect of acceleration.

Question 5

Einstein used the equivalence principle to show that the path of a ray of ligth woud be affected by what?

Answer 5

Einstein used the equivalence principle to show that the path of a ray of ligth woud be affected by gravity.

Question 6

Copy and complete this statement:

Einstein used the equivalence principle to show that the stronger the ……………………. …………………………. the more ……………………. a clock would run.

Answer 6

Einstein used the equivalence principle to show that the stronger the gravitational field the more slowly a clock would run.

Question 7

Which clock runs mores slowly - one at sea level or one at the top of Ben Nevis? Explain your answer.

Answer 7

The clock at sea level runs more slowly.

This is because it is closer to the Earth’s centre (i.e. at lower altitude) and so is in a stronger gravitational field.

The stronger the gravitational field, the more slowly a clock runs.

Question 8

Look at the spacetime diagram below and identify the motions of each line (a), (b) and (c).

Answer 8

(a) stationary
(b) uniform speed
(c) iniform acceleration

Question 9

What are the lines drawn on a spacetime diagram called?

Answer 9

World lines (or geodesics).

Question 10

Copy and complete the following statement.

Euclidean space has co-ordinates which uniquely specify a point in space – a location.

Spacetime locates not points in space but ………………………… i.e. not just ……………………… something happens but also ……………………..

Answer 10

Spacetime locates not points in space but events i.e. not just where something happens but also when.

Question 11

How can gravity be explained ith refrenece to spacetime?

Answer 11

The presence of a massive object, like the earth, curves the ‘fabric’ of space-time. So gravity warps spacetime, as shown below.

Question 12

By referring to spacetime, explain what is happening in the diagram of the Earth and the Sun below.

Answer 12

The Earth follows the curvature of the spacetime which has been curved by the mass of the sun.

Question 13

Look at the diagram below.

Explain what is going on.

Answer 13

General Relativity suggests that photons are affected by gravity, meaning that light from distant stars is bent as it passes a massive object like the sun.

This results in the apparent (observed) position of the star being different form its actual position in the sky.

Question 14

With the aid of a diagram, explain how gravitational lensing occurs.

Answer 14

Massive cosmic objects, such as a galaxy bend and focus the light that flows around them using their gravity, acting like giant magnifying glasses. This effect is called gravitational lensing.

The diagram below shows the effect.

Question 15

Describe in words what is meant by a black hole.

Answer 15

A black hole is a star of such high density that it’s escape velocity is greater than the speed of light. So, not even light can escape it’s gravitational pull.

Question 16

Starting with the density equation ρ = m / V, show that, for any spherical star mass m, radius r

m /r = 4/3 ρ π r²

Note that this is not a required derivation, but you should be able to do it!

Answer 16

ρ = m / V

ρ = m / 4/3 π r³

ρ = m / (r x 4/3 π r²)

m /r = 4/3 ρ π r²

Question 17

Given that the escape velocity may be given by

v_esc = √ (8Gπr²ρ/3)

…find the escape velocity from a star which has

mass = 8.2×10³⁶ kg

radius = 1.230 x 10¹⁰ m

volume 7.795 x 10³⁰ m³

Is this star a black hole?

Answer 17

The stellar density is

ρ = m/V = 8.2×10³⁶ / 7.795 x 10³⁰ = 1.05195 x 10⁶ kg m^-3

v_esc = √ (8Gπr²ρ/3)

v_esc = 5.17 x 10⁸ ms^-1

As v_esc > c then this is indeed a black hole.

Question 18

Name ‘A’ and ‘B’ in the diagram below.

Answer 18

A singularity

B event horizon

Question 19

Insert ther missing words.

An object far enough above the surface of a the singularity feels a …………………… gravitational pull than on its surface – so the escape velocity is …………………. up there.

At a certain (big enough) radius the escape velocity drops to ………. …………. ….. ……………………..

At or beyond this level light can ……………………… from the black hole.

This boundary is called the ……………… …………………

Answer 19

An object far enough above the surface of a the singularity feels a weaker gravitational pull than on its surface – so the escape velocity is smaller up there.

At a certain (big enough) radius the escape velocity drops to the speed of light.

At or beyond this level light can escape from the black hole.

This boundary is called the event horizon.

Question 20

Explain what is meant by the event horizon of a black hole.

Answer 20

The event horizon is the boundary defining the region of space around a black hole from which nothing, not even light can escape, as shown in the diagram below.

Question 21

Look at the diagram of a black hole below.

a) Would light originiating at point X escape from the black hole? Explain your answer.
b) Would light originiating at point Y escape from the black hole? Explain your answer.

Answer 21

a) No. This light is closer to the singularity than the event horizon. The gravitational field is so strong here not even light can escape.
b) Yes. This light is further from the singularity than the event horizon. The gravitational field is less strong here and light can escape.

Question 22

With the aid of a diagram, explain what happens to the passage of time as one approaches a black hole.

Answer 22

Time runs more slowly as one approaches a black hole.

Explanation:

Gravity curves spacetime.

So, whereas light would travel in a straight line in the absence of a gravitaitonal field ( a —> b below), it travels in a curved path when there is a gravitatonal field (c —> d below).

Light travels at a constant speed so must take longer to travel from c to d than from a to b.

So time runs slower wherever gravity is strongest, because gravity curves space-time.

This is gravitational time dilation

Question 23

a) Name r_s in the equation below.

r_s = 2Gm/c²

b) Explain what r_s is.

Answer 23

a) r_s = the Schwarzschild radius.
b) The Schwarzschild radius (r_s) is the radius of the event horizon surrounding a (non-rotating) black hole, as shown in the diagram below.

Question 24

Calculate the Schwarzchild Radius r_s of a star of mass

2.8 × 10³⁰ kg

(r_s = 2Gm/c²)

Answer 24

r^s = 4.2 × 10³ m

Question 25

Describe the three key pieces of evidence which brought experimental confirmation of Eisntein’s General Theory of Relativity.

Answer 25

1. The bending of light (photons) by gravity, confirmed by obervations of gravitational lensing.
2. The precission of Mercury’s orbit - this could not be explained using Newtonian graviatational theories.
3. Gravitational Redshift.

Question 26

With the aid of a diagram, explain gravitational redshift.

Answer 26

Question 27

Copy and complete

Inertial reference frames move with ………………………………………….. ……………………………………………..

with respect to each other.

Answer 27

Inertial reference frames move with uniform velocity

with respect to each other.

Question 28

Copy and complete

A non-Inertial reference frames is …………………………………………….. with respect to an inertial reference frame.

Answer 28

A non-Inertial reference frames is accelerating with respect to an inertial reference frame.

Question 29

A spaceship is accelerating in space. Is this an inertial or a non-inertial reference frame?

Answer 29

Non-inertial (as it is accxelerating)

Question 30

A spaceship is travelling with uniform velocity in space. Does the spaceship constitute an inertial or a non-inertial reference frame?

Answer 30

Inertial as it is moving with uniform velocity (presumably with respect to another inertial refrence frame).

Question 31

In which reference frame, inertial or non-inertial, would a body appear to be subject to a force?

Answer 31

Non-interial reference frame.

Question 32

When we carry out Physics experiments in a uniform gravitational field we make the same observations as we do when carrying out the same experiments in a non-inertial (accelerating) reference frame. This effect is known as the ………………………… …………………..

Answer 32

Equivalence Principle