Physics 6 Tutorial 1 - General Relativity

 Contents

Before you attempt this tutorial, look at Turning Points Tutorial 5 and Turning Points Tutorial 6.

Equivalence Principle

You feel gravity in the form of your weight.  As you know, weight is the force you feel as gravity attracts you to the floor.  It is the force that will cause you to accelerate at 9.81 m s-2, if a hole appears in the floor under you.   If you are in a sealed lift (i.e. with no windows) going upwards at a constant speed, you will feel your normal weight.

This would be confirmed if you stand on a bathroom scales.

Now suppose the cable to the lift breaks, and the lift plunges down the mineshaft.  Your weight will feel as if it were zero.  This is because you are accelerating at the same rate (9.81 m s-2 downwards) as the lift.  You would float.  A football with you would also float.

This feeling of weightlessness is because you, the lift, and the football are all accelerating down the mineshaft at 9.81 m s-2They all still have a weight.  However, relative to each other, there is zero acceleration, hence zero force (weight).  In your sealed lift cage would not be able to tell whether this was because gravity had been turned off, or that you were accelerating (towards your death).  The only way you would see that you were accelerating is if you could see the sides of the mineshaft rushing past you.

Now suppose (belatedly) the emergency braking system on the lift carriage works, you will see the football hit the floor at the same time as you do.

In your sealed system, you don't know if gravity has been turned back on, or that there is an upward force that is slowing the lift carriage down.  Of course, you are a physicist and you know that you can't turn off gravity.  But the effects are the same.

So this thought experiment led Albert Einstein in 1907 to come up with the Principle of Equivalence.  (He was sitting in his office at the Patents Office in Bern at the time.  He was definitely off task.)  The principle states:

In a sealed system, it is impossible to distinguish the physical effects due to gravity and acceleration.

We know that the gravitational field strength, g, is a force per unit mass (N kg-1) and it is an acceleration (m s-2).

Effect of Gravity on Light

We know that light travels in straight lines.  Now suppose we have another lift carriage.  This one works properly and can accelerate upwards.  In this lift, there is a small hole through which a beam of light can pass.  At exactly the same place on the other side, there is a second hole.

Therefore the ray of light will pass through the second hole as shown.  However, suppose we make the lift accelerate upwards, and we tracked a particular photon.  From outside the lift, we know that the photon will continue to travel in a straight line.  However if we were sealed in the lift, we would see that the photon would appear to move downwards in a parabolic path.

So what can we conclude from this?  Even though a photon is a particle with zero mass, from the point of view of the observer in the lift, its parabolic path suggests that its path is deviated by an acceleration.  By Einstein's Equivalence Principle it should therefore be subject to gravity and can be bent by the gravity of a very large object, like a big star, or a black hole.

Now, of course, this thought experiment is impossible to reproduce in real life:

• The acceleration would be massive (about 5 × 1033 m s-2) - or the lift impossibly wide;

• An individual photon, being a quantum being, is impossible to track.

Sorry to spoil a good story...

However astronomers have found things that are explained by this argument.

Space Time

Newton considered gravity to be a force.  And that is how we generally deal with it.  However Einstein produced a different model that treats gravity in a completely different way.  He used a geometry that replaced the traditional (Euclidian) notion that forces act along straight lines of least effort.  On a flat surface, we know that the shortest distance between two points is a straight line.  On a curved surface, however, the shortest distance is a geodesic.   This is important for pilots making long journeys in aeroplanes.  Consider the way that pilots fly from Australia to South America.  On the traditional Mercator projection, the shortest line is:

Image from Wikimedia Commons

Now if we plot the course across the spherical surface of the Earth we see:

Image:  Rolypolyman, Wikimedia Commons

Such courses are called Polar Routes.

Einstein considered that space was curved.  He proposed the idea of space-time, with a fourth dimension of time.  The diagram below shows the idea of dimensions.

So the shortest line between two points on the curved surface of space-time is a geodesic.

Gravity could be explained by depressions in space-time.  The best way of explaining this to someone who (like me) finds it hard to visualise is to model space-time as a rubber sheet.  Consider a rectangular trampoline frame:

We can put large objects onto the rubber sheet like this:

Viewed from the side:

If you push a smaller ball at an angle into the depression (or energy well) made by the heavier ball, you will find that the ball will make a spiral path into the depression.  The depression is sometimes called a warp.  If you fire the ball fast enough, it will go around the depression in an orbit, which is the path that uses the least energy as it runs along an equipotential.  If the speed is high enough, the small ball will come out of the depression.

The mathematical analysis is complex and will not be considered here.

There is no central frame of reference for the universe.  The most logical one to use is the frame of reference that is based on the Earth and its immediate environs.

So far we have discussed space-time as a flat surface.  The exact shape is matter of debate among cosmologists.  Some consider it to be flat, others curved, while others say that it's a sphere.  Theoretical Physicists consider that there are even more dimensions than the fourth suggested by Einstein.

Evidence

While we cannot see space-time directly, there are observations that have been made that cannot be explained without reference to space-time.

GPS

The Global Positioning System is a set of 24 satellites that was originally intended for military navigation.  Now it's used worldwide, not just for navigation in the car, for aeroplanes, but also for accurate positioning, vital for surveying for construction projects.  You can buy a hand-held device that will give you a precise positioning of latitude and longitude down to the nearest ten  metres.  Each satellite has a very accurate and precise atomic clock, accurate to 50 nanoseconds a day.  However:

• Each satellite is 20 200 km above the surface of the earth;

• Each satellite has an orbital speed of 3900 m s-1.

• As a result of special relativity, each clock ticks slower by about 7 microseconds a day when compared to a clock on the ground.

• Also due to curvature of space and time as predicted by Einstein' theory, the clocks gain about 45 microseconds a day.

Therefore in total the clocks gain 38 ms every day.  This corresponds to a distance of 11 km.  Accumulated uncertainties to this extent would lead to the system rapidly becoming useless, so the time shift is compensated for.

The Orbit of Mercury

Like all plants, Mercury's orbit is elliptical, in accordance with Kepler I.  Its point of closest approach to the Sun is called the perihelion.  It had been noted by astronomers that the perihelion of Mercury's orbit advanced slightly with each orbit.  The change in position is about 0.67 o per century.  The idea is shown below:

Image by Rainer Zenz - Wikimedia Commons.

The change in position is called a precession.  The orbit is anti-clockwise and the movement of the perihelion is anti-clockwise.  This happens because the orbit of Mercury is within the warp (curvature) made by the Sun in space-time.

Gravitational Red Shift

Light being emitted by stars has been observed as being slightly red-shifted as it leaves the gravitational depression in space-time caused by a star.  This was first identified in 1925 by the astronomer Walter Sidney Adams (1876 - 1956) on the star Sirius B.  There were difficulties in the measurement due to interference of light from Sirius A.  The first accurate measurements were carried out in 1954 on the white dwarf star 40 Eridani B, which indicated a red shift of 21 km s-1.  The red-shift from Sirius B was finally worked out as 89 km s-1 using the Hubble telescope.

Image by Vlad2i, and mapos, Wikimedia Commons

An experiment to demonstrate "red-shift" in gamma rays was carried out by the physicists R V Pound (1919 - 2010), and G A Rebka (1931 - 2015).  Gamma rays from an iron-57 source fixed to a loudspeaker were transmitted vertically up a 22.5 m tower from the basement.  Gamma rays from an identical source fixed to a second loudspeaker placed at the top of the tower were transmitted down the tower.  The results suggested that a red-shift with a uncertainty of about 1 % was detected.

Gravitational Lensing

After Einstein had made his predictions about space-time, the British physicist, Arthur Eddington (1883 - 1944) observed an eclipse of the Sun on the island of Principe (off the West coast of Africa) in the year 1919.  He predicted that the positions of several stars behind the Sun would change slightly as a result of the action of gravity on light.  And the predictions were supported by the results.

In 1936 Einstein developed his theory of space-time further.  He argued that light could follow the gravitational warp made by a sufficiently large object.  Therefore the light would follow a path rather like the path of light through a glass lens.  Therefore gravity could have a lensing effect on light.  In 1979, astronomers were observing a distant galaxy when they noticed multiple images of a quasar around the galaxy.  This was evidence to support Einstein's idea.

Gravitational lensing works like this:

Suppose we are observing a distant large black hole behind which was an even more distant star.  The black hole would completely obscure the star, of course.  However the warp caused by the gravity of the black hole in space-time would guide the light around the black hole, and bring it to a focus at the Earth.  We would see multiple images of the star, like this:

By measuring the amount by which the light has been deflected, astronomers can calculate the mass of the object that is causing the deflection.   However when the lensing caused by galaxies was analysed, the masses of the galaxies have been found to be much greater than the mass that could be accounted for using calculations from the visible light.  This suggested the presence of dark matter, named such for the simple reason it could not be seen.  Dark matter is considered in Astrophysics Tutorial 7.

World Lines

A world line is a unique line that is traced through space-time by an object.  The path can be plotted on a time - space-time graph like this:

The world lines are shown on a two-dimensional graph here, but there can be a second space-time axis to give a three-dimensional graph.  They can also be curved.  This graphs are widely used in string theory.  Some theoretical physicists have suggested that these lines might give the ability to travel forward and backward in time - every science fiction writer's dream.