Special Relativity David Berman Queen Mary College University of London Symmetries in physics The key to understanding the laws of nature is to determine what things can depend on. For example, the force of attraction between opposite charges will depend on how far apart they are. Yet to describe how far apart they are I need to use some coordinates to describe where the charges are. The coordinate system I use CANNOT matter. Symmetries in physics Let’s use Cartesian coordinates. Object 1 is distance x1 along the x-axis and distance y1 along the y-axis Object 2 is distance x2 along the x-axis and distance y2 along the y-axis The distance squared between the two objects will be given by: d ( x1 x2) ( y1 y 2) 2 2 2 Symmetries in physics The force is inversely proportional to the distance squared. What transformations can we do that will leave the distance unchanged? Translation: x1 x1 a y1 y1 b x2 x2 a y2 y2 b Symmetries in physics Rotations: x1 cos( ) x1 sin( ) y1 y1 sin( ) x1 cos( ) y1 x 2 cos( ) x 2 sin( ) y 2 y 2 sin( ) x 2 cos( ) y 2 Symmetries in physics We can carry out the transformations described of translations and rotations and yet the physical quantity which is the distance between the two charges remains the same. That is a symmetry. We carry out a transformation and yet the object upon which the transformation takes place remains the same or is left invariant. Symmetries in physics The important quantities in physics are those that are invariants. That is the things that don’t transform. Other things will change under transformations and so will depend typically on our choice of description, for example which way we are facing. Space and Time We live in both space and time. There are the usual three dimensions of space we are used to and also one more of time. We perceive time as being very different to space though. How different is it really? To arrange a meeting I need to specify a time and a place. I can describe the place by using some coordinates and the time by specifying the hour of the day (that’s just a time coordinate). Space and Time Distances in space can given as we have shown. x x 2 x1 y y 2 y1 Distances in time would also be given by the difference of the two times that is: t t 2 t1 Space and Time How do we add up distances in different directions? We’ve already seen that it is NOT just the sum of the distances in the different directions rather the total distance is given by: d x y 2 2 2 Space and Time How do we find the distance in space-time. That is given the distance in space and the distance in time how can we combine them to give the total distance in space-time? Wrong guess: s t d 2 2 2 Space and Time Einstein had a better idea. He combined space and time found the right way to describe distances in spacetime. Einstein In 1905, while working as a patent office clerk in Bern, published his work on special relativity. His insights in that paper were essentially that space and time should be combined in one thing, spacetime. He also realised the right way to construct invariant distances in spacetime. The same year he also published two other key papers in other areas of physics. It really was an enormous break through year for Einstein. Spacetime The distance in spacetime is given by: s d t 2 2 2 Spacetime When we measure distances we use the same units for x and y. If we didn’t then we could convert between units in the distance formula like so: d x w y 2 2 2 2 With w the ratio of the two different units. Instead we pick w=1 and use the same units for our x and y distances. Spacetime For spacetime, what is the choice of units of time that will set w=1 and give us the equivalent unit for time as for space? If we measure space in meters then we should measure time in light meters. (More about this later). Spacetime Given that the distance in spacetime is given by: s d t 2 2 2 What are the transformations that leave this distance invariant? What is the symmetry? That is how can we transform space and time so that the distance in spacetime remains the same. Spacetime Lorentz realised that there was a symmetry in nature where you could transform space and time distances in the following way. Lorentz Transformations x b( x vt ) t b(t vx) 1 b 1 v 2 Lorentz Transformations Spatial distances can shorten Time distances can also shorten The spacetime distance is the same that is it is invariant under these transformations. v is a velocity Units are chosen such that time is measured in light meters. Lorentz Transformations Distances in space will depend on the velocity of the observer. Distances in time will depend on the velocity of the observer. This is just like saying that spatial distance in one direction depends on which way you are facing. The equivalent to the angle you are facing is velocity you are moving at. Experiments Thousands of experiments have been done checking the Lorentz transformations and the altering of time and space depending on velocity. Experiments Lifetime of elementary particles Orbiting atomic clocks Collider physics Michaelson Morley experiment: Speed of light is constant no matter what your velocity Experiments In the experiment carried out by Michaelson and Morley an attempt was made to measure speed of light parallel to the motion of the earth and at right angles to the motion of the earth. According to our usual notions of how velocities add there should have been a difference. They found the speed of light was the same whether it was directed alongs the earth’s motion or not. This agrees with relativity, the speed of light is the same no matter how fast you are going! Consequences How big is a light meter? Speed of light is about 300000000m/s One light meter is about 0.0000000033333 s To convert to velocities measured in m/s we need to divide by c- the speed of light a big number. Most velocities in every day are much much less than the speed of light which is why we don’t notice the Lorentz transformations in ordinary life. Consequences Notice that v/c can’t be 1 or the Lorentz transformation become infinite and time and space become infinitely transformed. We can’t travel faster than the speed of light. Consequences Just as space and time rotate into each other so do other physical quantities. What matter is the invariant quantity. Energy and Momentum also transform into each other under Lorentz transformations. The invariant quantity is: m E p 2 2 2 Consequences Putting back in c, the speed of light so that energy and momentum would be measured in SI units this equation becomes: m c E p c 2 4 2 2 2 If p is zero we get the celebrated equation: E mc 2 Conclusions Space and time should be combined to spacetime a single entity. The invariant measure of distance on spacetime is s d t 2 2 2 With the unit the light meter. Lorentz transformations leave this distance invariant.