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. So far we have looked at the special theory of relativity, but today we will expand our understanding to incorporate gravity. This framework is Einstein’s theory of general relativity. A lot of the framework is very similar to special theory, but it does require several new concepts to be considered, especially the idea that space-time is curved.

3. Special Relativity: Reminder Remember, the basis of the special theory of relativity is that there are no mechanical or electromagnetic experiments that can reveal relative motion. This requires all observers who are not accelerating to measure the same speed of light. This requirement ensures a relation between the spatial components and time (remember, these were separate). Light rays travel along paths with ds=0, whereas the distance through space-time for a massive object is its proper time. Everyone disagrees on the individual components, but agrees on ds.

4. Why Not Gravity? While Einstein was guided by electromagnetism, he next asked himself how gravity fits in. Remember, Newton’s equation for the gravitational force between two objects is Einstein was puzzled – Special relativity says that the mass you would infer for an object depends on its velocity relative to you. Also, the distance between objects depends upon the relative velocity of the observer. No matter which way he tried to put this together, he could not get a covariant form for the inclusion of gravity, so different observers would claim different physical results.

5. Steps to General Relativity Einstein’s revelation began with the realization of equivalence of gravitational and inertial mass. Simply put, all masses fall at the same rate. What happens if I drop a laboratory? Clearly everything falls together. What Einstein realized is that effectively gravity has vanished.

6. Equivalence Principle Einstein’s “happiest thought” came from the realization he could take the equivalence principle further. Simply put, Einstein reasoned that; There is no experiment that can distinguish between uniform acceleration and a uniform gravitational field.

7. Time & Acceleration If we consider light signals being sent down the rocket, the relation between emission and receipt is Remembering the equivalence principle, then in a gravitational field we should expect This is gravitational time dilation.

8. Spacetime is “Curved” We could suppose that spacetime is flat and the gravitational potential somehow influence the running of the clock. This is a little like measuring distances on a flat map, but having to change the length of the ruler. While some aspects of relativity do adopt this view, it is “simpler, more economical and ultimately more powerful” to keep our measuring devices (rulers and clocks) fixed and assume the geometry in which they sit is curved. Newtonian gravity can be expressed in the Static Weak-Field Metric in which time and space are curved

9. The Equivalence Principle If we consider the path of light crossing an accelerating rocket, then an observer within the rocket would see the path of the beam as curved. Via the equivalence principle, an observer in a gravitational field should see the path of a light ray to be also curved. However, a freefalling observer in a gravitational field sees a straight path!

10. The Field Equations How do we get the geometry? This is a very straight-forward equation(s); Left-hand side: This is all geometry. The terms in there were discovered as part of non-Euclidean geometry in the 1800’s. Einstein was able to bring these terms together to describe the geometry of space-time. Right-hand side: This encodes the distribution of mass and energy in the stress energy tensor (with some fundamental constants). Einstein originally felt this equation(s) was unsolvable!

11. Geodesic Equations Given the geometry, how does something move? This is the geodesic equation. Basically, it says that we need to correct the acceleration of an object by adding a term which depends upon the geometry of the space time (called the Christoffel symbol) and the velocity of the object. Note, this is the velocity in space time, not just in space. This equation is not special to relativity! Think about how a airplane flies between cities.

12. Perihelion Shift of Mercury Mercury is the closest planet to the Sun, with a semi-major axis of 58x106 km and eccentricity of 0.21. The orbit of Mercury has been known to precess for quite a while. The vast majority of the precession is due to Newtonian effects. However, a residual precession of 42.98±0.04 “/century could not be explained. The prediction from Einstein’s analysis of the orbit in the weak-field limit predicts;

13. Gravitational Lensing Einstein’s first prediction was that light would be deflected as it passed by massive objects. He calculated that a light ray grazing the Sun would be deflected by Made in 1916, this prediction could not be tested until the end of WWI. Eddington organized two expeditions to observed an eclipse in 1919. With the Moon blocking out the Sun, the positions of stars could be measured, agreeing (roughly) with Einstein’s prediction. Now measured to an accuracy of ~1%.

14. Gravitational Redshift The final test proposed by Einstein in 1916 was the gravitational redshift. This was finally measured by the Pound-Rebka experiment in 1959 by firing gamma rays up and down a 22m tower at Harvard. Measuring a frequency change of 1 part in 1015, their measurement agreed with the Einstein prediction with an uncertainty of 10%. Five years later, the accuracy was improved to a 1% agreement and now measurements can accurately agree to less than a percent accuracy.

15. Shapiro Time Delay The Shapiro delay has also been measured using space probes, including Mariner in 1970 and Viking in 1976. The expected delay is of order 100s of microseconds over a total journey time of ~hrs, but atomic clocks are accurate to 1 part in 1012. A recent measurement using the Cassini space probe found the agreement to be (Bertotti et al 2003)

16. Schwarzschild Geometry When faced with the field equations, Einstein felt that it analytic solutions may be impossible. In 1916, Karl Schwarzschild derived the spherically symmetric vacuum solution, which describes the space-time outside of any spherical, stationary mass distribution; This is in Geometrized units, where G=c=1. Note, that the geometrized mass has units of length and so the curved terms in the invariant above are dimensionless.

17. Equations of Motion We can work out the equations of motion in the vicinity of an object defined by the Schwarzschild metric (outside of a spherical mass distribution). These are (where one dot is velocity, and two dots is acceleration); While these may appear daunting, they are easily solved on a computer.

18. Black Holes Black holes are masses that have completely collapsed, so that they are fully described by the Schwarzschild metric. Something weird happens when we fall towards the origin of the Schwarzschild metric; while the proper motion for the fall is finite, the coordinate time tends to r=2M as t->∞. Clearly, there is something weird about r=2M (the Schwarzschild radius) for massive particles. But what about light rays? Remembering that light paths are null so we can calculate structure of radial light paths.

19. Black Holes We can calculate the gradients of light rays from the metric Clearly the light cones are distorted, and within r=2M all massive particles are destined to hit the origin (the central singularity). In fact, once within this radius, a massive particle will not escape and is trapped (and doomed)! But how do we cross from inside to outside? Again, light curves tell us about the future of massive particles.

20. Eddington-Finkelstein We can make a coordinate transformation to Eddington-Finkelstein coordinates (think of changing from cartesian to polar). This straightens inward going light rays. While this makes the picture “nicer”, causality is preserved. Once you have crossed the event horizon, you are doomed, with your future at the central singularity.

21. Collapse to a Black Hole We can trace light rays in the Eddington-Finkelstein coordinates and consider what happens when a black hole is formed (or when a thing falls in). Outside of the event horizon, the rays are travel outwards, but at a steeper gradient. Once inside, the light rays move inwards. With this, it’s apparent that a distant observer can only see the infalling object for a finite amount of its proper time. So, while he can, in theory, see the faller for an infinite amount of time, he will see him freeze before hitting the event horizon. The radiation is also redshifted, meaning that the image will fade into the infra-red and then radio and then be unobservable.

22. White Holes Another coordinate transformation can straighten outgoing light rays. The result is a white hole and massive particle at r<2M are destined to be ejected and cannot return. Note that while this is still the Schwarzschild solution, this behaviour is not seen in the original solution. Is this a physical solution? We don’t think so, but so far there is no physical reason to think not.

23. Kruskal-Szekeres Coordinates The maximal coordinate system for the Schwarzschild solution is the Kruskal-Szekeres coordinates. These “expand out” the features of the Schwarzschild geometry, and so include a black hole (at the top) and the white hole (at the bottom). Remember, all of the physical predictions (i.e. what an observer experiences) are exactly the same. Interestingly, these coordinates reveal the presence of another, infinitely large universe (over on the left).

24. Special Relativity & Causality So, if there is a party at Alpha Centauri, and you leave too late, no matter how fast you travel, the best you can do is approach the slope of a light cone. Hence, you are destined to miss the party - or are you?

25. The Alcubierre Warp Drive All we have looked at so far has been in the flat space-time of special relativity. What if we consider general relativity where we can bend and flex space-time? Worried about missing parties in Alpha Centauri, in 1994 Alcubierre proposed a modification of flat space-time of the form (assuming c=1). Here, is the (inverse) slope of a path across our space-time diagram.

26. The Alcubierre Warp Drive So, looking at the interval, there is one remaining term to consider The function fis simply defined such that where and that f declines smoothly from unity to 0.

27. The Alcubierre Warp Drive With this, and looking at the interval, when rs>R this is just the same as the flat, special relativity space-time. Inside R the space-time is curved; to understand this, it is easiest of consider what happens to light cones (Δs2=0). The bubble of curved space-time has tiled the light-cones away from what you would expect in special relativity.

28. The Alcubierre Warp Drive From Hartle’s Gravity (Honours Textbook)

29. The Alcubierre Warp Drive We can adjustVs(t) so that we can get an arbitrary slope (velocity) over the space-time. If we look at the path of the massive particle, then we can happily follow this path and remain in the light cone. So, with the bubble, our rocketeer can cross our space-time diagram at any velocity they want (although they remain within their local light cone and so never actually exceed the local speed of light).

30. The Alcubierre Warp Drive From Wikipedia Basically, the warp drive works by (approx.) squeezing the space-time in front of it, while expanding the space-time behind it (although this is not how all warp drives work). So, apparently, you can travel faster than the speed of light, without ever going faster than the speed of light.

31. Warp Drive: Advantage The Alcubierre Warp drive has several major advantages: non-inertial: The warp drive does not accelerate. The rocketeer follows a free-fall path and so for the duration of the journey, they are weightless (just happily float around). no time dilation: unlike other relativistic travel, those in the Alcubierre bubble suffer no time dilation, so there are no problems about getting home and being the same age as your great-great-grandchildren. So, on the face of it, this is fantastic! It means we can happily explore as much of the universe as we want. Surely, there is a flaw somewhere?

32. Metric-Mechanics Remember, Alcubierre defined the structure of space-time, although this is really only one side of the story. The Einstein Field Equations relates geometry (lhs) to the distribution of matter and energy (rhs). We would like to specify the rhs and then calculate the lhs, but this is very difficult. But, once we specify the geometry, we can calculate the mass/energy distribution. So, what do we get for the Alcubierre warp dive?

33. The need for negative energy The result is that to make the Alcubierre drive work, we need material with negative energy density. This sounds bizarre, but represents a tension, rather than a pressure. While we don’t know of such material in the lab, the dominant energy in the universe, dark energy, appears to have similar properties. http://www.orbitalvector.com/

34. The need for negative energy Other than negative energy, the Alcubierre warp drive has other rather worrying properties. The curvature of space-time at the edge of the bubble means that the tidal gravitational forces are huge, and if the bubble runs into you, it will be bad (although the driver may not see it coming!) http://www.orbitalvector.com/

35. The awful truth There are other geometries which allow weird (faster than light) through the twisting and bending of space-time (such as wormholes), although these rely on defining geometry and then calculating mass/energy distribution.

36. Cosmology: Isaac Newton First mathematical laws of gravity and motion. Thought that an isolated group of stars would collapse in on itself. An infinite universe of stars should collapse into isolated islands of mass. A finely tuned universe could be balanced and static.

37. Cosmology: Albert Einstein General Relativity: viewing gravity as curved space time (1915). “Cosmological considerations on the general theory of relativity” (1917). Einstein thought the universe was static and unchanging, although his equations were dynamic. Added a cosmological constant term which acts as an repulsive force, balancing gravity.

38. Cosmic Dynamics If we assume that matter and energy in the universe are isotropic and homogeneous, then the geometry of space-time is given by the Friedmann-Robsertson-Walker metric; The part in brackets is the space part, and the geometry of space is defined by k. From this, we can define some dynamical equations; Where a = R/Ro is the scale factor. The Λ-term is special fluid whose energy density is constant as the universe expands.

39. Cosmology: Alexander Friedmann Friedmann wrote “On the curvature of space” in 1922. He came to the conclusion that Einstein’s cosmological equations predicted that the universe evolved with time, either expanding or collapsing. Einstein wrote that Friedmann had made a mathematical error and his results were invalid. In 1923, Einstein retracted his objection and agreed relativistic universe was dynamic.

40. Einstein’s Biggest Blunder? After Friedmann’s work, Einstein threw away his Cosmological Constant, calling it his biggest blunder. There is a persistent myth that Einstein fudged the equations of relativity, adding anti-gravity to make a static universe. However, this is not correct. The addition of a cosmological constant term was a completely legitimate mathematical exercise. Einstein’s blunder was choosing a specific value for the cosmological constant to balance gravity, not its addition. It was not discarded, just set to zero.

41. Edwin Hubble In the 1920s, Hubble measured the speeds of nearby galaxies. He found nearly all were rushing away from us, with their velocity increasing with distance, exactly as predicted in the relativistic model of the universe.

42. Modern Measurement The search for Hubble’s Constant, the rate of the expansion of the universe, has dominated astronomy since Hubble’s day. Velocities are easy to measure, distances are hard. The issue was only resolved in the last decade with use of the Hubble Space Telescope.

43. Understanding Expansion A good way to understand expansion is with a “conformal diagram”. It simply has us and all other objects in the universe as a series of straight lines.

44. Understanding Expansion Friedmann’s equations give us the “Scale Factor” and the distances as a function of time are;

45. Understanding Expansion • Notice how as we go back in time, R(t) goes to zero. This means the distance between any two objects also goes to zero. This is the location of the “Big Bang”.

46. Which Scale Factor? The shape of the scale factor depends upon the mix of energies (matter, radiation, other stuff) in the universe. Universes only containing matter slow down over time, while other universes slow and then accelerate. Which is our universe? www.astro.ucla.edu/~wright/intro.html

47. Which Scale Factor? On conformal diagrams, light rays travel at 45 degrees and it’s simple to see that light we receive now set out from distant objects long ago.

48. Which Scale Factor? The velocity of an object (its redshift) tells us the scale factor at the time the light set out, while the brightness of an object tells us how far the light has travelled.

49. Cosmological Supernovae Supernovae are exploding stars whose true brightness is well known. Using the Keck and Hubble Space Telescope, ten years ago we were able to do this experiment.

50. Which Universe? The supernovae appeared fainter than expected, showing that the universe does not contain only matter. A third of the cosmos is matter, the most of which is dark (does not radiate, but we can feel its gravitational pull). Heavy elements (that’s us!) make up 0.03% of the universe. Some mysterious substance, dark energy, make up 70% of the universe. www.lsst.org