Gravity as Geometry:
A Point-Form Summary.
This section of the course notes, and the associated PowerPoint presentation, makes the following critical points:
Newton thought of gravity as a force. Following Einstein, we now consider it more correct to say that a lump of matter affects the surrounding space and time by introducing a distortion or curvature. This has an effect on our measurements of physical phenomena, speeds of moving particles, the trajectories they follow, and the passage of time
these distortions also affect the motion of light through the curved space-time. One helpful visualisation is to imagine a trampoline distorted by (say) a cannonball at rest on it, and then to imagine particles and light moving along paths that follow the curvature of the surface
the fact that light follows curved paths has been demonstrated in various ways. It was first shown, in a famous 'eclipse' experiment in 1919, that the trajectory of starlight bends very slightly as it passes by the edge of the sun
the distortion of space-time near the sun also causes an anomalous precession of the orbit of Mercury. For a time, it was thought that a new planet interior to Mercury might be responsible, but Einstein's theory explains it perfectly (and no planet has been found!)
the bending of light following various trajectories through space also results in gravitational lensing, resulting in the formation of multiple or distorted images of remote objects as the light passes near or through regions of high density
gravity has yet another effect on light: as a photon moves away from a massive object, climbing away from its gravitational influence, it loses some energy. It does not (and can not!) slow down, but it is red-shifted to longer wavelengths (lower energies). These effects are very small on Earth, but can be measured; the effect is real
gravity is weak in the Solar System. To create a very strong gravitational field, it is necessary to collect a huge amount of material together and/or to compress material to very high density
Laplace imagined a star so massive that light could not escape it. He visualised light rising up like a stone thrown into the air, coming to a halt, and falling back. Light does not do this: a local observer hovering there would measure it travelling at 'c', the speed of light. But the deformations in space-time can force the light to follow a curved trajectory which cannot escape to the outside world. The light is trapped -- and so also would be all material particles
some people resist the notion of an object so dense that nothing can escape it, and hope or suggest that black holes may never happen. But massive stars seem unlikely to be able to shed enough material to wind up as neutron stars, and immensely massive accumulations of material seem even more destined to collapse to supermassive black holes. There is no imaginable 'special physics' to stave off the collapse
we can calculate the "Schwarzschild" radius which is the size to which a massive object would have to be compressed before gravity would inevitably and irreversibly take over. For the Earth, that is about 1 cm; for the sun, it is about 3 km
when the black hole forms, nothing can come out from inside it. But it does not magically 'vacuum up' all the surrounding material, and its gravitational influence at a large distance remains unchanged. The only differences are that it vanishes from sight, and that you can get very close to the edge of the hole itself, where the tidal forces can be dangerous
Associated Readings from the Text.Please look at: Section S3, pages 463-469. Chapter 18, pages 583-586.A Fresh Look at Gravity.What I have said about Einsteinian relativity is true, demonstrable, and amply confirmed by any number of physics experiments. But I have not yet said anything about gravity. My descriptions would be correct even in completely empty space, trillions of miles from any matter. What difference, if any, does gravity make to the behaviour of light and time? The difference becomes apparent when we recognize that gravity is no longer thought of in quite the way Newton formuated it. Newton thought in terms of forces - the Earth curves in its path because the sun pulls on it with a certain force. Following Einstein, we now describe gravity in geometrical terms: a lump of matter distorts space in a way which means that particles moving nearby will follow curved trajectories. I gave a simple analogy in class: imagine a big rubber sheet stretched out flat (something like a trampoline). If you roll a marble across the sheet, it will follow a straight path. But now put a heavy bowling ball gently onto the rubber sheet, so that it sits at rest in the very center. You can appreciate that the sheet will now have a pronounced `dip' in it, and the same marble as before will now move in a curved path when it rolls by. This is not because the bowling ball itself applies a force to the marble, but rather a result of the distorted geometry of the rubber sheet. In like fashion, big lumps of matter deform space itself. (You have to imagine such deformations in three dimensions, rather than the two-dimensional sheet we have been discussing. Such visualisations are not easy!) Objects passing by the sun or indeed any lump of matter move along paths which are determined by these geometrical distortions. So matter distorts space -- and time, too! Consequently, the local strength of gravity, which depends on the presence of lumps of matter, has an influence on how we measure intervals of time and distance. Where there are large concentrations of matter (``strong gravitational fields'') the distortion is very pronounced; far from matter, the distortions are less.Light Itself.Einstein also realized that light itself should move through space in a way which follows the distortions introduced by lumps of matter. In other words, the trajectory of light `curves' when it passes a lump of matter. (This is true for lumps of any mass, but the effects are naturally more pronounced the bigger the mass is.) This may surprise you. If I throw a piece of chalk across the room, it follows a path which is clearly curved, but beams of light (from a flashlight, say) seem to follow a straight path. The reason the beam of light looks straight is due to the enormous speed of light - after all, if you throw the chalk much faster, the curve it follows is less steep - but in fact there is a real change of direction in the path followed by the light. As the light crosses the room, it `falls' a little towards the floor. The problem is that the drop is very very small! Over the last eighty years, Einstein's Theory of General Relativity has been tested in a number of ways and found to be correct in every context. We now consider some of these tests.The Bending of Starlight by the Sun.Knowing that light will curve as it passes a big lump of matter, Einstein predicted that stars would appear to shift positions very slightly, but measureably, if their light skims past the sun. It is worth emphasising that this bending will be very small, because even the sun distorts space only a small amount. The effects can best be seen as a difference. Here is how to accomplish the task: In January (say), take a picture of a field of stars in the midnight sky. Your telescope is pointed directly away from the sun. Six months later, take another photograph with the telescope again pointed in the direction of that field of stars. The difference is that the Earth has now moved around in its orbit and is on the other side of the sun, so the sun lies directly between you and the stars! This sounds pointless, because the sun will fog the picture and the daytime sky will be too bright to allow us to see the background stars. But there is a clever solution: we can do the second part of the exercise at the moment of a total solar eclipse, so that the sun is masked out! Then the prediction is that the pattern of remote stars should look somewhat distorted geometrically because of the bending of the various rays of light as they pass the sun on the way to us. This famous experiment was first done in 1919. Einstein's prediction was confirmed and he was made famous. There is an interesting socio-political sidelight to this: Einstein was German, and the eclipse expedition was coordinated and run by the British astronomer Eddington in the immediate post-World-War-I years. This had a symbolic significance about the way in which pure scientific endeavour could transcend national boundaries and differences.The Anomalous Precession of Mercury's Orbit.If the solar system contained only the Sun and one planet, and if Newton were right about gravity, that planet would move in an elliptical path which retraced its tracks forever, orbit after orbit. As it happens, the presence of other planets slightly messes things up, but in predictable ways. The orbit of the Earth, for instance, is not the same at all times: it shifts around a bit because the Earth is influenced not just by the gravity of the sun but also by the gravitational tugs of the other planets, principally Jupiter (the most massive planet). A figure on page 467 of your text makes the point (in much exaggerated fashion) that the effect of this is to cause the orientation of a planet's orbit to precess (slowly change in orientation). These planetary perturbations are complex, but can be dealt with. (The Earth's precession is fully understood in terms of the influence of the other planets.) A century or more ago, however, astronomers were puzzled by the particular behaviour of Mercury. Even after the influence of all other planets had been accounted for, the orbit of Mercury was precessing somewhat more than it should. But why? One early explanation which was considered was the possibility that there might be undetected planets even closer to the sun than Mercury itself is. Searches were mounted, and some detections were even claimed (with the putative planet being named Vulcan, by the way), but eventually it was realised that there is no planet there. Einstein solved this long-standing problem. He showed that the anomalous precession is a consequence of the way gravity distorts space and controls the motions of planets when they get especially close to massive bodies, where the curvature of space is most pronounced. Newton's formuation is not quite correct in these close confines, but the General Relativistic formulation exactly explains the observed behaviour.Look at All Those Fish! Gravitational Lensing.I have a tank of fish at home, and when I stand in exactly the right position I can see several images of the same fish. The reason is that light from the fish can reach my eye along several different paths. Light changes direction abruptly when it passes from the water through the glass to the air, and careful positioning can reveal multiple images of the same fish. Light can behave similarly as it travels through intergalactic space. This is shown beautifully in the images shown on pages 468-469 and on pages 686-687 of your textbook. Of course, in these cases the light is not passing through different regions of water, glass, and air! Instead, it changes direction because it passes large lumps of matter - whole galaxies and clusters of galaxies. In this way, two beams of light from a source may start out in slightly different directions but be brought to the same point - your eye, or your telescope - at the end of a long journey. There are many examples of this kind of gravitational lensing in the heavens. The effect is most readily detected when the path followed by the light is very long, since that allows the tiny bending effects to accumulate to a significant amount and increases the chance that a beam of light will pass close by some very massive lump of material. Sometimes the images produced are `out of focus', and yield rings and arcs. It is a real pity that Einstein, who died in 1955, did not ever get to see these impressive examples of the way in which light is subject to the dictates of gravity!Gravitational Redshift.What happens if you throw a stone into the air, so that it is moving up and away from the Earth's surface? Well, it starts off moving quickly but loses kinetic energy (speed) as it goes because it is climbing out of the local `pit' -- which we call a gravitational potential well -- formed in space by the concentration of matter there. If the stone is not thrown quickly enough, it does not escape, but even if it does it loses some of its speed (kinetic energy) on the way. (Our `stretched rubber sheet' analogy works here as well. Think again about the cannonball on the trampoline. If you roll a smaller ball away from the cannonball, it loses some of its speed as it climbs up out of the depression. Indeed, if it is rolled too slowly to begin with, it will come to a momentary halt and then roll back down.) Now imagine yourself in a spacecraft hovering high above the surface of a planet from which a beam of light is being emitted in your direction. As the photons climb up out of the gravitational potential well, they do not lose speed: you will see the photons arrive and move past you at the speed of light, which is a universal constant. But they do lose energy, and will appear to be at longer wavelengths (or equivalently redder colours ) when they reach you. This is what we call the gravitational redshift. Here on the Earth, the gravitational field is very weak (or, in modern parlance, the total mass of the Earth is not large and consequently the distortion in the space-time continuum is small). Thus a photon shot straight upwards from the ground loses only a tiny bit of its energy in climbing up and away. If you turn on a yellow lamp on the ground, it will not look red when seen from the top parapet of a skyscraper (although it could and would if the Earth's gravitational field were fantastically stronger!). But the photons really are less energetic, and the effect, too tiny to be noticed by the eye, can be and has been measured with precision instruments. Photons really do lose energy in climbing out of a 'gravitational potential well'!Making a Strong Gravitational Field.As we have seen, there is now lots of experimental support in favour of Einstein's Theory of General Relativity, complete with its implication that light is affected by gravity. One consequence of this is that sufficiently strong gravitational fields should bend the trajectories of light quite a lot, and cause a pronounced gravitational redshift on photons climbing away from the concentration of matter. But where are we likely to find a particularly strong gravitational field? Most people would instantly say that what you need is a lot of matter to make a strong cumulative gravitational pull. This helps, of course, but it is as important to squash all the matter together into a region of very high density. An object which is very extended `dilutes' its potential gravitational pull. For instance, here in Kingston we feel the pull of the limestone rocks just beneath our feet much more strongly than equivalent lumps of material in Australia or Indonesia, on the other side of the world. Indeed, as Newton showed, the spherical form of the Earth means that the net effect of all the matter is really determined by our distance from the centre of the distribution (about 4000 miles). Suppose we squashed the Earth down to the size of a basketball, and found ourselves floating next to it in space. There is just as much matter as ever before, but we are now only a metre or so away from every single bit of it. The total gravitational influence would be enormous. (But remember that a person floating 4000 miles away would feel exactly the same total gravitational influence as before, and would fall towards the Earth no more rapidly than she would if she stepped off a cliff on the present Earth, at least to start with. The difference, of course, is that she would not promptly hit the surface, but would continue to fall and speed up a lot more as the fall continued towards the now-dense central lump!) In short, intense gravitational fields are found where matter is packed very densely. The more matter there is, the better, but the critical factor is really the compression. This is exactly what is provided by a black hole formed by the inward collapse of a star too massive for neutron degeneracy to provide a sustaining pressure.Black Holes Foreseen.Interestingly, this notion was foreseen by the French scientist Laplace a couple of centuries ago, although we now recognize that his reasoning was not correct. He imagined a body so massive that light could not escape, but supposed that the body could be internally supported in some way. (In short, it would be a large but invisible `star.') He did not realise that all material bodies would collapse inward to essentially zero volume under such extremes of gravity. Laplace also had the disadvantage of lacking our modern understanding of light and the way it is propagated. He though light was corpuscular - that is, he supposed light to be rather like small bullets. (He did not know about quantum mechanics and `packets' of energy; he was almost literally thinking of small bullet-like particles.) Laplace then argued that these corpuscles - I really should not call them photons, given his imperfect understanding - would lose speed as they rise away from a massive object, just as a bullet or stone itself slows and comes momentarily to rest before falling back. In this way, he hypothesised that there might be large objects in space whose mass and radius are such that even light is moving too slowly to reach `escape velocity.' Notice one point in particular. In Laplace's version, light rises quickly from the surface, but is slowed by gravity and comes to a halt before falling back. This is not correct. Light cannot be brought `to a halt.' If you hover near a black hole, you will not see photons making abortive efforts to get out but failing, coming to rest, and falling back.Are Black Holes Really Inevitable?The mathematics of black holes implies that the material within them rushes inward under the intense influence of gravity and (once the neutron degeneracy limit is exceeded) continues to contract rapidly until the volume dwindles to zero and the density at the very centre becomes infinite. (Such a region is called a singularity. ) The precise upper limit is not well known, but it seems likely that stars much more massive than about three solar masses face this prospect. There are many such stars in the galaxy. Some people, including a few astronomers, find the notion of material falling into `zero volume and infinite density' distasteful, and hope that stars can somehow avoid this messy end. There are a couple of possibilities, but each has problems. If you have a star which is, say, 15 times the mass of the sun, it is much too massive to be supported by neutron degeneracy. But perhaps the star will throw off enough material (at least 12 solar masses) during its late life, or even during the supernova event itself, to conveniently leave behind a sufficiently small remnant. The problem with this is to understand why the star would necessarily do so! There is no way that the star could `know' that it faces a fate which we find unpleasant and take steps to avoid it. And if even one star failed in this process, what would be the consequences? Perhaps some `brand new physics' arises when the star is collapsing to new regimes of high density. Maybe, for instance, the star passes through the stage at which neutron degeneracy could stop the collapse but, at some slightly higher density, some as-yet-unknown bizarre physical behaviour intrudes and does stop the collapse. There are two problems with this. The first is that we already understand the behaviour of matter at neutron star density pretty well, and something like a neutron star does not have to be compressed very much more to become a black hole - the density needs to be increased only a little. The second objection is worse. Consider something like a great cluster of stars, or the crowded central parts of a galaxy where the stars are very close together. Suppose their mutual gravitational attraction causes them to move in towards a common center. By the time you have a billion solar masses clumped into a volume 20 astronomical units in radius (comparable to the volume of the Solar System), the gravitational field is so strong that nothing can escape and further collapse to a singularity is inevitable. But at this stage the average density of the material is still very much less than that of water! So it is not possible to imagine any `new physics' coming to our rescue: any particular atom is surrounded by other atoms in not-particularly-uncomfortable proximity. How would the atom `know' that it should now behave in some new way to prevent further collapse? In short, even if some new physics could prevent the formation of stellar black holes (those formed by the collapse of a single star), there is absolutely no way that we can imagine Nature putting a stop to the formation of very massive black holes. In the end, of course, all this speculation is idle! The acid test will be to look out into space to see if there are black holes or not! (You may think that discovering them would be impossible, but I will later explain to you why this is not the case.)Light Near a Black Hole.In the figure below, you will see a representation of the way light moves near a black hole. Look in particular at the `blow-up' in the bottom part of the figure, which shows how photons move after being emitted from a source fairly close to the `edge' of a black hole. (I will define what we mean by the `edge,' which has the technical name of event horizon, in the following section.) You can see that the photons follow strongly curved trajectories, thanks to the strong gravity (or, in Einsteinian terms, thanks to the distortions in space-time). It is important to recognize that the photons do not slow down and fall back into the hole in the way Laplace imagined! At every stage in their trajectory, the photons are travelling at the speed of light. But they follow a path which carries them back into the hole, and once inside the event horizon the photons can only move inwards, toward the singularity at the very center. If the source is outside the event horizon, light emitted directly outward can escape the vicinity of the black hole, but a remote observer will see the light as very strongly red-shifted, since it will have lost so much energy in battling its way out against the pull of gravity. (It will still be seen to arrive at the speed of light.) Once the source falls within the event horizon, however, there is no photon trajectory which will carry the light to the outer region. Photons from even the most incandescently hot material imaginable cannot escape!The Schwarzschild Radius.As we have seen, the mathematics of the exercise tells us that gravity pulls matter inward ever more strongly until it shrinks to zero volume and infinite density. But we lose sight of it - no photons can escape - once the material reaches the event horizon, the boundary which I earlier described as the `edge' of the black hole. The name event horizon originates for technical reasons I do not want to go into, so let me tell you instead that it has another name: the Schwarzschild radius, named after the German astrophysicist Karl Schwarzschild. (The name is appropriate in that ``schwartz'' is the German word for ``black.'') The Schwarzschild radius is the defining point of a black hole in the following simple sense: once an object shrinks down to this size, for whatever reason, gravity is sure to dominate and the continued collapse to a black hole is inescapable. The Schwarzschild radius, which I will denote R(S), can be very simply calculated. It is given by the formula "R(S) = (2 G M) / (c**2)" where M is the mass of the object, G is the gravitational constant, and c is the speed of light. Let us consider a few examples: The Earth has a mass of about six trillion trillion kilograms. If we insert this into the formula, we will find that the Schwarzschild radius for such a mass is about one centimeter. In other words, if some agency could squeeze the Earth down to this size - about the size of a large grape - gravity would finish the job for us, pull all the atoms irretrievably into the vanishing singularity, and prevent any light from escaping. Of course, there is no chance that the Earth ever will get into such a pickle: its internal rigidity and the strength of its constituent materials sustain it. But you can imagine yourself, with god-like powers, crushing it down far enough between mighty hands to make it vanish in this way. The sun is about 300,000 times as massive as the Earth, so its Schwarzschild radius is about 300,000 centimeters - about 3 kilometers. Once again, I emphasize that the sun will not become a black hole spontaneously as it ages and evolves, but with magical powers you could force it to become one by squeezing it down to this size. A star three times the mass of the sun has a Schwarzschild radius of about nine kilometers. This is not very different in size from a neutron star, which emphasises the point I made earlier about the inevitability of black holes. If we accept that a massive star can turn into a compact neutron star, as the existence of pulsars seems to prove, then it is clear that there are processes which come perilously close to squashing material down to the very verge of becoming black holes. Surely in at least some cases, for the more massive stars in particular, black holes must come into existence. An agglomeration of stars - a big cluster at the core of a galaxy, for example - with the mass of a billion suns would have a Schwarzschild radius of about three billion kilometers. This is about the size of the Solar System out as far as the planet Uranus, but if all the stars in the cluster were to settle into a volume of this size, the average density of the material would be considerably less than that of water. This is the point I made earlier: there is no reason to expect `surprising new physics' to spring up to prevent the further collapse of material of this kind. By this stage, howver, the total collapse to the singularity is already inevitable.Not Really Vanished.It is important to reemphasise a point I made much earlier. When a black hole forms, it does not magically start to suck all surrounding material into it (but neither does its gravitational influence vanish). If the sun were to become a black hole, the Earth would feel exactly the same gravity as it always did, and would now merely orbit the invisible central spot where the sun was (and still is, in some sense). This is true of all the planets, including Mercury. The difference is that you could now travel very close to the spot marking the very centre of the sun. (At present, you can't do this because you will run into the hot gases of the sun's outer parts about a million kilometers out!) But travelling close to a black hole is dangerous, because that is where the gravitational distortions are strong, the tidal forces will rip you to shreds, and so on. In the next lecture, we will discuss the fate of a brave astronaut who decides to plunge into the hole. Previous chapter:Next chapter0: Physics 016: The Course Notes, spring 2005. 1: The Properties of the Sun: 2: What Is The Sun Doing? 3: An Introduction to Thermonuclear Fusion. 4: Probing the Deep Interior of the Sun. 5: The Sun in More Detail. 6: An Introduction to the Stars. 7: Stars and Their Distances: 8: The HR Diagram: 9: Questions Arising from the HR Diagram: 10: The Importance of Binary Stars: 11: Implications from Stellar Masses: 12: Late in the Life of the Sun: 13: The Importance of Star Clusters in Understanding Stellar Evolution: 14: The Chandrasekhar Limit: 15: Supernovae: The Deaths of Massive Stars, 16: Pulsars: 17: Novae: 18: An Introduction to Black Holes: 19: Gravity as Geometry: 20: Finishing Off Black Holes: 21: Star Formation: 22: Dust in the Interstellar Medium: 23: Gas in the ISM: 24: The Size and Shape of Our Galaxy: 25: The Discovery of External Galaxies: 26: Galaxies of All Kinds: 27: The Expanding Universe: 28: Quasars and Active Galaxies: 29: The Hot Big Bang: 30: The Geometry of the Universe: 31: Closing Thoughts: Part 1:Part 2:Part 3: |
(Wednesday, 22 April, 2026.)
