Nov 12 2003 More on Gravity, Einstein's Special Relativity, Einstein's General Relativity Gravity: Newton's Law of Gravity, formulated in the late 1600s, is F=GMm/r**2 Using Newton's law of gravity, we can derive orbits of the planets around the Sun, we can send spacecraft to the planets, we can calculate the masses of stars in binary systems... When gravity is small, this law works just fine. So you get the right answer in almost all "everyday" things like sending rockets to Venus, or throwing a football down the field for a touchdown. But we know that Newton's law is incomplete. It has some holes in it. So in certain circumstances, basically as the gravitational force of one object on another gets stronger, or as the needed accuracy gets greater, we see these holes. 1) We don't get the right answer for Mercury's orbit. We can see errors in predictions of Mercury's orbit that were glaringly apparent more than 100 years ago (not apparent with naked eye, but easily apparent with good measuring equipment). 2) It implies that the speed of the force of gravity is infinite. 3) it implies that the force of gravity gets infinite as the distance of two objects from each other goes to zero. 4) If you use a GPS receiver to know where you are in the woods, or in your car, you get the wrong answer. So GPS receivers are useless if you use Newton's law of gravity to understand how radio signals propogate. We get really worried when we finid infinities in equations. Usually, rather than infinity being a property of Nature, it's a sign that our description of Nature has a hole in it. Special Relativity (1905): All that we can do is to give you a couple of examples of Relativity. We'll tell you a couple of the "nonintuitive" aspects of Relativity, since ultimately they matter for our real goal, which is explaining Black Holes. 1) Let me quote two nicely written paragraphs (Abell's "Exploration of the Universe,"4th edition, page 243): "The principle of relativity states that there is no physical experiment by which one can detect his state of uniform relative motion. What this means is that if two observers, moving uniformly with respect to each other, perform the identical experiment in their own moving environment they will obtain identical results, so neither can say, from anything the experiment told him, that he was or was not moving, or how fast he was moving. For example, two people standing in the aisle of an airliner going 1000 km/hr can play catch eaxctly as they would on the ground. On that same airplane you can drop a heavy and light object together, and they will hit the cabin floor at the same time, and they fall at the same rate as they would if you had dropped them from the ground (provided that the airplane is moving uniformly- in a straight line at a constant speed). You can play table tennis quite normally on a moving ship on a calm sea. You can swing pendulums in an automobile (so long as it is not turning or accelerating in some other way) and they will swing in the same way, with the same periods, obeying the same pendulum laws as do pendulums in the laboratory. When you are moving uniformly, you experience no physical sensation that will tell you that you are in motion. You can, of course, look out the window and see the ground moving by, but if you were stationary and the GROUND were moving you would feel and see the same thing. It is common to sit in a train in a station, and momentarily wonder whether it is your train or the one on the next track that starts to move. for that matter, none of us can feel the motion of the earth carrying us about the sun...so emphatically do we not feel it, that scarcely three centuries have elapsed since it has been generally accepted that, indeed, the earth DOES move." 2) The Speed of Light a) it's fast, so fast that human reactions cannot measure it. We're clever enough, however, to invent machines that can measure the speed of light. that said, the speed of light was first measured by comparing predictions of the times of eclipses of Jupiter's moons to the real times, and noting that the eclipses were "early" when Jupiter was closest to Earth, and "late" when Jupiter was farthest. If light takes some time to travel from one point to another, then we can understand these observations. b) Nothing can go faster than the speed of light. We have shown this in the lab, and we'll be justifying (not proving) this statement on the 18th. c) No matter how fast you are moving, and no matter how fast the person sending out the light beam is moving, when you detect the light beam, it's always going at the speed of light.When you think about cannon balls, that's not the case. 3) Maxwell's equations of electromagnetic waves We aren't about to go into Maxwell's equations. Suffice it to say that they are four equations that describe all of electricity and magnetism. One of the remarkable things about Maxwell's equations is that they predict that electromagnetic waves (think about jiggling electrons) propogate at the speed of light! That's remarkable, except that we now know that light is an electromagnetic wave. 4) the speed of light coupled with Maxwell's equations We pointed out above that light always travels at the speed of light. We pointed out that Mawell's equations say that electromagnetic waves always travel at the speed of light. The first statement probably seems weird to you, that light always passes you by at the speed of light, no matter how close to the speed of light you're travelling. So let's relax that first statement, and see what sort of trouble we get into. If you're driving down the highway at 81 miles an hour, and you come upon a car travelling 80, it seems like you and the other car are hardly moving. And in fact, you approach and separate at 1 mile an hour! You can look inside the other car, see what station is playing on the radio, no problem. If this were true for light, you could approach a light beam and what you'd see is that the light beam has stopped. It's still an electromagnetic wave, and it's jiggling up and down. But it's not travelling at the speed of light, which is forbidden by MAxwell's equations. Well, either Maxwell's equations are wrong, or the analogy to the car story is wrong. Light is weird in the sense that it doesn't obey what us slowly moving folks call common sense. But that's not light's problem, it's ours. 5) An example of why the speed of light can't depend on the motion of the source. Consider a binary star orbiting edge-on, with circular velocity V. Sometimes one member of the binary is moving towards us, sometimes it's moving away from us (as it orbits the other object). So, if the speed of light WERE NOT invariant, sometimes it would head towards earth at velocity C+V, where c is again the speed of light. On the other side of the orbit it would head towards earth at velocityi c-V. Since the star is far away, these two snapshots of the binary will arrive here at very different times. We can set things up so that you'd see multiple copies of the system at once, so you wouldn't just see two stars, sometimes you'd see three or four or more. (Think about a student who arrives in class promptly at 2, and another who comes in at 2:30. If you take a picture at 2:45, you see them both) After having examined thousands and thousands of binary stars, we've never seen a double star split into three for a while! 6) Another example In nuclear physics labs, you can accelerate a particle to almost the speed of light. Some particles will decay, emitting a photon. the lab will measure that photon to be going at the speed of light, not c+V or C-V or anything else. 7) More aspects to the Theory of Relativity. Remember, this theory (like all scientific theories) can be checked or tested. We have never made an experiment that differed in its results from the predictions of relativity. And this theory is about to celebrate its 100th birthday! a) Moving clocks run slowly This is real, you can change the decay time of atomic particles by changing their speed. the particle "thinks" it's decaying in the normal time. But an outside observer sees otherwise. So, when one observer looks at the other, and the second observer looks at herself, they see different things. b) Distances shrink for moving observers A moving observer will see external rulers shrink, too. So we'd see a fast rocket go to the stars in a certain time, having travelled a certain distance. The observer on that rocket will see that it took a different time to get to that star. His clock was running slowly. He'll measure he travelled a different distance, too. c) A fast moving object weighs more than a slowly moving one. we can measure this in the lab! This is the reason it's impossible to go faster than the speed of light. As you go faster, you weigh more. So to go faster takes more energy. To get to the speed of light takes infinite energy. If you travel quickly, you age slowly. So you can go very long distances before "your 80 years are up," before you die. Of course, your friends left back on Earth would have long ago died while you were aging slowly. So there is a time machine! But it doesn't play by all of the rules of a "science fiction time machine." If you could travel at almost the speed of light, you could live for almost forever as seen from earth. We'll just mention here that if it were possible to travel faster than the speed of light, all of these experimental tests of relativity would have given different answers! 8) Last comment about Special Relativity We now know people will make different (but correct) measurements of each other. And we understand how they're expected to disagree, depending on how fast they're moving. Most motions, like a baseball falling back to Earth after someone hits a monumental pop fly, are not uniform motions, but are accelerated motions. the ball falls faster and faster, not at one speed. Einstein solved that problem in 1916 with the General Theory of Relativity. GR, as it's known, changes how we look at gravity. 9) First comment on GR We'll rediscuss this idea on the 18th. a) Ignoring air friction, two parachutists freely falling to Earth can play catch with a baseball, and to them the ball moves back and forth in a straight line. b) On Earth, balls don't travel in straight lines because gravity is pulling them downward. c)Way out in space, far from anything, balls would travel in straight lines because nothing is pulling on them. So how can a and c be giving the same answer? Before we answer that, let's think about watching the astronauts in the Space Station. You let go of a hot dog and it doesn't fall to the ground, it just sits there. That's because you and the hot dog are falling at the same speed. We can see movies of this! Check out the "vomit comet", a special airplane to train astronauts. Here's one link http://www.spie.org/web/oer/december/dec97/vomit.html Why are we "weightless" in a spacecraft orbiting earth, or onboard the vomit comet? Well, something is cancelling gravity! So let's go back to the parachutists falling down towards earth, and the astronauts far from any sources of gravity. Put both sets of people, along with their baseball, in a box. You can't tell which pair is falling and which pair is far from gravity. So, here's the [General Relativity] PRINCIPLE OF EQUIVALENCE An accelerating object can mimic gravity, and you can't tell the difference. We'll give you a better definition on the 18th