Relativity: from Special to General

1. Geraint F. Lewis Sydney Institute for Astronomy School of Physics University of Sydney Relativity: from Special to General

2. The goal is to understand the theory of relativity. At its heart, relativity is based upon an apparently simple question, namely “how are coordinate systems relate to one another”. Answering this question leads to our modern description of gravity. However, it iseasyto misunderstand the ideas of relativity. Here, we will focus upon a full and modern interpretation, allowing a straight-forward step from the special theory (without gravity) to the general theory.

3. It’s all about coordinates! Coordinate System: These are amongst the most important, yet most neglected, aspect of science. To undertake any aspect of science, we need to define a coordinate system, such that we can uniquely label any event. An example of an event is the popping of a cork. This takes place at a spatial location, given by (x,y,z) and at time, t. Note that in Newtonian Physics, the notions of space and time are completely separate. But while I see the popping of the cork at (x,y,z), how does someone moving with respect to me see this event?

4. Understanding relativity Understanding relativity begins not with Einstein, but with Galileo. He stated, in 1632, that a person on-board a smoothly sailing ship, sailing with constant velocity with no rocking and rolling, cannot detect their motion. What this means is that any physics experiment (and at the time, this meant mechanics) done onboard the moving ship will yield exactly the same result as an experiment done on the shore. Furthermore, this means we don’t need two sets of equations, one for those at rest with regards to absolute space, and another for moving observers.

5. Galilean relativity Galilean Transforms: These allow us to transform from one coordinate system (at rest) to one moving (in this case, at a constant velocity of v in the positive x-direction – this is the ‘primed frame’). Note that the individual observers will disagree on where the event happens, but they will agree on the time that it occurs.

6. Galilean relativity: An example Primed Frame Rest Frame V  If we are at rest with respect to the juggler (in the Rest Frame), we will see a ball move purely in the z-direction. If we are moving in the x-direction (the Primed Frame), the ball will have both z- and x-direction motions. However, we will agree on the times the ball is in the hand.

7. Galilean relativity The take-away message about Galilean transforms is that the laws of mechanics are invariant between observers moving at constant velocity with respect to each other. The laws of mechanics can be boiled down to one equation and transforming to the primed frame results in the same equation. While observers disagree on the distance between events, they all agree on the time that has elapsed between them.

8. Accelerating frames If our primed frame is accelerating, rather than moving with constant velocity, our transformation becomes more complicated; Substituting this into Newton’s second law results in This implies that we cannot use Newton’s second law in an accelerating frame. This means that even if an object has no force acting upon it, it will appear to accelerated when viewed from an accelerating reference frame.

9. Accelerating frames Think about camera recording a car crash, one inside the car and the other on the side of the road. Without a seatbelt, during the collision (as the car decelerates), the camera on the side of the road will record the driver continuing forward at constant velocity (we can make the seat slippery to ensure there are no forces on the driver). However, a camera in the car will record the driver accelerating forward (relative to the car). A physicist watching the recording will be able to deduce that this is an accelerating frame as there will be no force causing the apparent acceleration.

10. The road to relativity Einstein’s inspiration that led him on the road to special relativity was the work of James Clark Maxwell. Maxwell was able to unite electricity and magnetism into a single picture, electromagnetism. This insight, which proves the mathematical framework for all modern electronics, communication, power transmission etc.

11. The road to relativity Maxwell showed that light was a self-propagating electromagnetic wave, represented by a coupled wave equation. One number that appeared here was the wave velocity, c, which depends upon fundamental properties of the vacuum. Einstein’s question: What is this speed relative to?

12. The road to relativity It was suggested that the electromagnetic waves were carried in an aether, like ripples in a pond. This aether effectively defined an absolute reference frame (in which Maxwell’s equations hold). If we are moving with respect to the aether, we need to take our velocity into account when using Maxwell’s equations. This means that while we cannot detect absolute motion with mechanical experiments, we can with electromagnetic experiments.

13. The birth of relativity “On the electrodynamics of moving bodies” was published by Einstein in 1905. In this paper, Einstein proposed that, just like Newton’s second law, that Maxwell’s equations should hold in all reference frames (i.e. we don’t need different equations for “moving observers”). This automatically implies that all observers should measure the same speed of light. However, Maxwell’s equations are not invariant under Galilean transformations. What transformations do we need?

14. The light clock The light clock is a simple device which perpetually bounces a light pulse between two mirrors. Consider two observers, one stationary with respect to the clock, and the other moving at uniform velocity. The time between ticks as seen by the stationary observer is;

15. The light clock For the moving observer, the light ray covers a larger distance, but special relativity demands that we measure the speed of light to be the same, so the speed of light as measured by the moving observer is

16. The light clock Combining these, we can see that This means that the clocks of the observers tick at different rates.

17. Length contraction While we won’t derive it, the requirement that the speed of light remains constant also requires that we have length contraction in the direction of motion. So, the moving observer sees the mirrors separated by a smaller distance than the stationary observer.

18. Muons to the rescue! Muons are fundamental particles with a life-time of 2.2 microseconds. They are created in the upper atmosphere (at about 100km) from cosmic rays. They are traveling at almost the speed of light, and so the maximum distance they should travel is So, all muons should rapidly decay into other elementary particles (namely electrons and neutrinos). However, we readily detect muons reaching the Earth’s surface.

19. Muons to the rescue! Ground point of view: The muon is actually traveling close to the speed of light, at v=0.99998 c. With regards to an observer on the ground, the life time of the muon is time dilated to and the distance it can cover is So the muon can reach the ground.

20. Muons to the rescue! Muon point of view: To the muon, it’s life-time is unchanged, and so to it, the distance it covers is only 660m. However, this is the length contracted distance, and the corresponding distance in the ground (rest) frame is given by and, again, the muon can reach the ground.

21. Lorentz transformations Lorentz transformations: We need a set of transformations, akin to the Galilean transformation, that can relate the coordinates of an event between coordinate systems. In the following, we will assume that c=1. If we have a light ray traveling in the x-direction, then ; This implies that i.e. the two observers will see a different dt and dx, but in combination they get the same measurement of the speed of light. While this seems rather obvious, it actually represents one of the fundamental properties of space-time.

22. Lorentz transformations An event in one coordinate system will have coordinates (t,x,y,z). The requirement that the speed of light be the same means that we can calculate the location of the event in a reference frame moving at velocity v to be; Where Note that when the velocity is small, these become the Galilean transformations we saw earlier.

23. Speed of light: An aside In 1887, Michelson and Morley set up an experiment to measure the velocity of the Earth through the aether. Their null result surprised them, with many theoretical ideas to explain this, such as aether drag. Lorentz suggested that instruments physically shrank in the direction of motion through the aether, coming up with his transformations to account for this. Clearly Einstein’s ideas of the consistency of the speed of light explains this, although people are unsure if he knew of the M-M experiment.

24. How fast can you go? We can use the derivative of our Lorentz transformations to work out the addition of velocities; here, A see the projectile moving with a velocity of Note that if u’ = 1 (so it is equal to the speed of light), so will u. Otherwise, u is always less than the speed of light. So, if we define a “rest frame” for the projectile (with u’=0) then we can never see it move at the speed of light.

25. Space-time diagrams We’re going to be making use of space-time diagrams. These normally show t verses one or two spatial dimensions. Remember that light rays lie on paths with Here dsis the distance through space-time, and so light rays trace out null paths (i.e. zero distance through space-time). The light rays carve space-time into causally connected and disconnected regions.

26. Space-time diagrams Massive objects trace out a line through space-time (known as a world-line) with regards to a particular non-accelerating coordinate system. Because these objects cannot travel at (or above) the speed of light, they are constrained to lie within their future light cone. This means that all events that the massive object can interact with lie within their future light cones. Also, all events that can have influenced the massive object must be in the passed light cone. This defines causality!

27. Aside: The twin paradox The twin paradox has been a constant source of confusion for students of relativity. The basic premise is quite simple. A pair of twins start off on Earth. One rockets off and travels at constant velocity on a journey to a distant star, eventually returning home. The journey is apparently symmetric, the twin on Earth sees the other head out at vand back at –v. The traveling twin sees the the one on Earth heading off at –vand back at v. Shouldn’t the twins be the same age?

28. Aside: The twin paradox • However, relativity tells us that the twin who travels out and back is younger. • How can this be if the journey is symmetrical? Surely each twin calculates that the others clock running slowly in comparison to theirs? • The solution to the twin paradox requires the realization of a couple of things; • We need to calculate the distance through space-time for a massive object. • We can only compare clocks at events in space-time.

29. Massive particles Like light, massive particles trace out a path through space-time, and again we can define a space-time distance to be Again, different observers will disagree on the individual components, but will agree on ds. What will these components be for the massive particle? As, in its own rest frame, it is not moving, then dx=dy=dz=0, and τ is known as the proper time and corresponds to the time experienced by the particle.

30. Massive particles So, for a massive particle the distance through space-time corresponds to the time experienced on their journey. Equating these expressions, it is straight-forward to see that This recovers the time-dilation formula we derived earlier on.

31. Aside: The twin paradox Here is the standard twin paradox picture, with the blue representing the twin who stayed on Earth, while the red is the traveling twin. The coordinate system is at rest with regards to the twin who remains on Earth. The important points are the two events where the twins are in contact (i.e. at the start and the end of the journey). To calculate the time each experiences, we need to look at the distance they trace out through space time.

32. Aside: The twin paradox For the twin on Earth: They are spatially at rest in this coordinate system, so the time they experience (their proper time) is For the traveling twin: In this coordinate system, the traveling twin heads out at a velocity v, and back again at the same velocity. The distance traveled through space-time is

33. Aside: The twin paradox Putting these together, we get that the experienced time for each of the twins is related by The traveling twin is younger. This is the “standard” relativistic analysis, but fails to address the apparent paradox. Surely, the traveling twin calculates that the twin on Earth’s clock should run slowly?

34. Aside: The twin paradox Let’s examine the twin paradox from a different point of view, namely from the reference frame attached to the outward going twin. In this coordinate system, the twin on Earth is traveling at a velocity v. Again, the important quantity that we need to look at is the distance traveled through space-time. However, things are a little trickier as the second segment of the traveling twins journey is at a different velocity. But we will come to that.

35. Aside: The twin paradox For the Earth-bound twin: We can calculate the distance as before and find that For the traveling twin: We need to break the journey into two segments. For the first part, in this coordinate system, the traveler is at rest, and so So, the traveler does calculate that the twin on Earth’s clock runs slowly.

36. Aside: The twin paradox For the second segment of the journey: What is the velocity of the return journey in these coordinates? We can use our addition of velocities to calculate this; Hence, the distance through space-time is The important thing to note is that for the traveling twin, the outward and inward journeys are the same in terms of experienced time.

37. Aside: The twin paradox So: We can not combine these to get and combining with the result for the twin on Earth gives; Which, while ugly, simplifies to give The traveling twin is younger!

38. Aside: The twin paradox The truth is that no matter how we analyze this situation, the traveling twin will come out being younger. The important thing is to work in “inertial” frames, which move with constant velocity with respect to each other. However, to use the rest frame of the outward and inward journey, we would need to jump between frames (i.e. accelerate) and would need the Lorentz transformations.

39. Aside: The twin paradox One of the key properties of relativistic motion is that we can connect two events in space-time with a myriad of different paths. Each will experience a different amount of proper time (governed by the length they trace out through space-time). However, the path that maximizes the proper time is a “free-fall” path (i.e. no rocket packs to accelerate the traveler). These free-fall paths are Geodesics.

40. World-line and null geodesics So, this is the description of space-time, with light cones defined by null-geodesics (light rays) and the world-lines of massive objects (time-like geodesics).

41. The journey through space-time For a massive object, the distance it traces out through space-time is its proper time (i.e. the time it experiences). The coordinates it traces out in someone other reference frame, however, will depend upon the inertial motion of the observer. However, we can use this to define a velocity through space-time.

42. The journey through space-time The velocity of a line is a vector which is tangent to the path describing the motion. Now, we are in 4-dimensional space-time and so we can define a 4-velocity to be; But remember, we have a relationship between the coordinates (t,x,y,z) and the proper-time τ which implies We can take this to mean the length of the 4-velocity is a constant!

43. The journey through space-time If we remember that and we we multiply by the mass of the object, this expression becomes Let’s call the first term in this expression E (it’s actually the total energy of the particle) and the second p (it’s the relativistic momentum) this becomes

44. Energy If the velocity is small (much less than the speed of light) then this expression approximates to The first term in this expression is classical kinetic energy, but what is the m term doing in there? It is saying that even a body that is not moving possesses energy, and for non-relativistic motion, this energy (associated with the rest mass) dominates. Typically (with the c’s put back in) this energy is

45. Energy: No faster than c? So, why can’t I travel at, or faster, than the speed of light. We can see this by looking at the total energy; The total energy depends upon the gamma factor and the rest mass. If we start off at rest, the gamma factor is unity and so we have only the rest mass. As we add energy (e.g. by firing a rocket) the total energy increases. But this can only change the gamma factor. So, adding a finite amount of energy results in a finite gamma factor, and hence a velocity less than the speed of light. To get up to the speed of light, we would need to add an infinite amount of energy!

46. Electricity and magnetism One other interesting consequence of the theory of relativity is what happens to electric and magnetic fields. For a bar magnet, a stationary observer would say that there is only a magnetic field. However, a moving observer would see both a magnetic and electric field. The physical predictions would be the same.

47. Seeing relativity http://www.anu.edu.au/Physics/Searle/Obsolete/Download.html

48. Seeing relativity: Aberation http://www.anu.edu.au/Physics/Searle/Obsolete/Download.html

49. Seeing: No relativity http://www.anu.edu.au/Physics/Searle/Obsolete/Download.html

50. Seeing: Aberration http://www.anu.edu.au/Physics/Searle/Obsolete/Download.html