Lecture 17

Relativity

In Einstein's words, he used to imagine how the world would appear to him if he was to move at the speed of light : in a "gedanken experiment" (a thought experiment) he described how looking at the time shown by a clock at rest, and moving away from it at the speed of light, he would always see the clock marking the same time, since he would always be moving together with the light photons emitted by the clock at the time he went by it. Time, as shown by the clock at rest, would then appear to have stopped, while instead a watch carried in his hand would still tick away: time does depend on the relative state of motion between clock and observer !!

As a more detailed example we can consider the "light clock" described in the book : from the point of view of an observer at rest, when the clock is in motion light has to travel longer distances to go to and from the mirror. Assuming that the speed of light is always the same, the separation between ticks for the moving clock is larger: time is passing more slowly for objects in motion...

As counter-intuitive as such a result might be, it has been confirmed in experiments performed with real, very precise, atomic clocks, and is verified continuously in the behaviour of unstable elementary particles generated by cosmic rays or particle accelerators. The lifetime of unstable particles moving at extremely high velocities is stretched consistently with the predictions of Einstein's relativity.

If we believe in the theory, and there is no reason not to believe in it, then we can assume that similar effects will apply also to living organisms: an astronaut leaving earth and travelling at extremely high velocities, will age more slowly than people left on the earth. More precisely, his clocks, both mechanical and biological, will tick more slowly than a clock on earth. If he was away for 1 year, as measured on his clock, many years would instead have elapsed on earth...

Einstein's theory allows to give precise formulae for time dilation. We will skip the derivation on pages 442-445, and just give the final result: if t_m is one unit of time (i.e. the time between ticks) for the clock moving with velocity v, and t_s is the time between ticks for the stationary clock, one has:

$t_{m} = t_{s}/\sqrt{1-v^{2}/c^{2}}$

t_m, i.e. the time between ticks, is larger than t_s, since t_m is given by t_s divided by a number less than 1. If we were to reach the point where v=c, then t_m would become infinitely large: if you move at the speed of light, time comes to a stop !!

It is easy to estimate that, as long as we are dealing with velocities typical of our daily lives, the time dilation effect is practically un-measurable. Example: how much slower is the time flow if we move on a super-sonic plane at about 1000 km/hour?

Once you get started on a path, you have to follow it to the end. In rigorous derivations, the original assumptions led to another surprising result: lengths are also depending on the motion of the objects, so that if an object has a length l_s when stationary, when in motion it will have a length given by

$l_{m} = l_{s}\times\sqrt{1-v^{2}/c^{2}}$

i.e. an object in motion contracts according to a factor depending upon its velocity. The theory tells us that the contraction occurs only along the direction of motion, the other two dimensions remain unchanged.

The next step is to see how we can reconcile the galileian transformation of velocities between moving frames with the requirement that the velocity of light is frame-independent. Relativity tells us that, if an object is moving with velocity v with respect to a frame which is itself moving with velocity u, then the velocity V of the object as measured by an observer at rest is

V = (u+v)/(1+uv/c²)

Such a formula implies that

1.

the velocity of an object cannot be made to exceed c just by going from one frame to another

2.

if an object has velocity c in one frame, it will have the same velocity c in any other frame.

From the relativity formulae, one can see that unphysical things happen (i.e. infinities are encountered) when an object was to increases its velocity up to the speed of light. On the other side, Newtonian physics tells us that, if we apply a constant force to a body, its velocity will keep increasing and eventually reach and exceed the speed of light. How can we resolve this inconsistency? Again the correct application of the relativity formulae would give us the answer: another necessary consequence of the starting assumptions is that the mass of an object depends on its velocity, according to:

$m(v) = m_{0}/\sqrt{1-v^{2}/c^{2}}$

As the velocity of the object increases, so does its mass. Given that F = ma, in the relativistic world a constant force will not cause a constant, but instead an ever decreasing acceleration, so that it will never be possible to accelerate an object beyond the speed of light. Still, applying a constant force will result in a steady increase of the body's energy. In Einstein's theory, the correct expression for the energy of a body is

E = mc², i.e. $E = m_{0}c^{2}/\sqrt{1-v^{2}/c^{2}}$

The consequence is that any massive object has an intrinsic energy, even when at rest, given by E_(v=0)=m₀c². As we have learnt, this surprising relation between mass and energy has been verified to the greatest extent in the nuclear and elementary particle domain.

General Relativity

The relativity rules we have discussed so far represent the "easy" part of Einstein's theory, since they only handle the case of uniform motions (i.e. motions with constant velocity); since it refers to a special case, the theory goes under the name of Special Relativity. Einstein's next endeavour was to work out the most general case, i.e. the case of motions with arbitrary accelerations. The more complete theory is known as General Relativity, and it is equally revolutionary, since it introduces a complete new way to look at the universe. Like before, we will only present the introductory arguments that form the base for the theory, and then mention its most remarkable consequences.

The starting point is the realization that forces and accelerations are indistinguishable. Newton's law, F=ma, had already recognized a correlation between force and acceleration, but Einstein's reasoning went a step further. The standard argument is the following :

suppose that, while standing in a room you drop an object: if you are on earth, the object will fall with constant acceleration g. You interpret this by saying that there is a force (the gravitational force in this case) acting on the object and causing it to accelerate towards the floor. Suppose now that the room you are in is somewhere in outer space, inside some vessel moving with constant acceleration g: if you drop the object, the floor of the room will move towards it with acceleration g, and the effect will be the same as if it had fallen because of the gravity.

Another way to picture the effect of accelerations is to think of what happens when you are in an elevator. When the elevator is at rest, you feel an attraction force towards the floor proportional to your weight, but when the elevator starts moving (i.e. it has an acceleration) you feel yourself pushed against the floor by a stronger force, i.e. it is as if gravity had become stronger.

The consequence of this type of resoning, as Einstein correctly pointed out, is that, in principle, we have no way to state whether certain effects are caused by a force or an acceleration: if the room where we drop the object has no windows, and we do not know where we are, we would have no way to tell whether the object is falling to the floor because of gravity, or whether the floor is coming towards it because it is accelerating...Given then that we have no way of telling whether forces are at play, we might as well get rid of the concept of force !!

Still, it is a fact that massive objects influence each other via what Newton had called the gravitational force, so how is this effect explained in General Relativity? Simple replies Einstein, a massive object affects the motion of other objects not because it exerts a force upon them, but because massive objects cause a deformation in the space around them. The earth is circling around the sun not because it feels the sun's attraction, but because that is the expected trajectory in the warped space-time around the sun.

As revolutionary as these concepts were, they made a very strong prediction that could be verified experimentally: if the space around a massive object is warped, then its effect should be felt by any body travelling by it, including a massless entity like light. In Einstein's view, light should be affected by gravitation, and this is clearly in contrast with Newton's law, which predicts that gravitation is only felt by objects with a mass.

The experimental test was done in 1919 (there were plans for earlier tests, but World War I had caused an interruption in scientific research). The idea was to compare the apparent position of stars when their light did or didn't go near the sun. To do the measurement, one had to wait for a total solar eclipse, so that stars would be visible even when their position was "behind the sun". The measurement was done, and it fully confirmed Einstein's theory.

More recently, other accurate verifications of the predictions of General Relativity were performed. If light is affected by gravitation, then light moving away from the earth should lose energy. But, when losing energy, light cannot change its velocity, since it always travels at speed c : light loses energy by changing its frequency (remember E=h$\nu$), so that light moving away from the earth will be slightly red-shifted. Again this effect was verified experimentally, to confirm the validity of General Relativity.