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1 The Equivalence Principle and Foundations of General Relativity K. Arthur Croker Department of Physics and Astronomy University of Hawaii at Manoa Abstract These are the notes I used personally for an hour-long whiteboard presentation to about people at Chaminade University on Februrary 15th, No power-point, no slides, none of that fancy stuff. Apparently things went very well, as people began to spontaneously videorecord the presentation. I really think its important to make a dynamic and engaging presentation, and if I could cite it, I recall a NASA article that found power-point to be less efficient at making information stick. For animations and wild images, computer is great. But for bread and butter, good theatrics and pedagogy cannot be beat. Not all of the information here was presented, and it was not presented exactly as written down: these are notes. The flavor is intact though, especially there was no mention of Black Holes! If you have any questions or criticisms, please do not hesitate to contact me. Much of what was discussed is based on Weinberg Chapter 3, along with various things from d Inverno s Introducting Einstein s Relativity and Wheeler s Spacetime Physics. For video segments, please see my webpage. 1 Intro [give everyone rulers] All of you know who Einstein is, and know that he was incredibly smart. You probably haven t gotten a chance yet to learn about just what he did (he did an awful lot!), so today we re going to give a solid teaser (we want to keep you in the Physics department). Perhaps you think that he is famous because the ideas he mastered were so intricate that only someone of his geniuscouldgrapplewiththem. Wewillseetodaythatthisisnotthecase, thereisnothingintricate about the essence of his ideas, only their implications! We re going to develop an understanding of what Einstein described as the happiest thought of his life, something called the Equivalence Principle. But first, we need to know what this [hold a ruler] actually is. 2 Guest Visitor i remember this guy? It s going to make a guest appearance. What about square roots? 3 Rulers What is this? [Ask] Okay. What makes a ruler different than this [Hold up an unmarked stick]. It has markings. What is special about the markings? [Ask]. They are equally spaced. What about this side? What s different? [Ask: spacings different]. Here s a line [draw a line]. By s, I will mean a measurement using one side of the ruler, by s I will mean a measurement using the other side of the ruler. For example, this line has s = 10 inches and s = 25.4 cm. Of course s and s are describing the same object, just using a different ruler. 1

2 4 Curved? Zoom in for Flat [Draw xy plane, A, B] Here is a road connecting A and B. How far do we have to drive to get from A to B? Pythagoras Law. If we place one ruler along x and another ruler along y, then s = x 2 +y 2 (1) would be how far we need to drive. What about this? [Draw mountain, Honolulu, Kaneohe] Now how far do we have to drive to get from Honolulu to Kaneohe? Can we still use Pythagoras? No. Unless there is a tunnel straight through the mountain, Pythagoras does not give us what we are looking for. Why does Pythagoras Law not work? We have to stay on the road, the road goes over the Pali Pass, and the mountain is not flat! We are now going to assign to the word flat a very specific meaning, and I m going to leave this written on the board for the rest of the lecture, so don t forget about it! When I say flat I mean that Pythagoras Law gives us the right idea of how far it is to drive on a road between to places. We are also going to assign a word to the phrase the right idea of how far it is to drive between two places staying on the road. This is the distance. Pretty obvious, right? Soforabasketballcourt, Pythagoras Lawgivesustherightideafordistance, sowecandescribe a basketball court as flat. For highways on Oahu, Pythagoras Law gets it wrong, so Oahu is not flat. What if, in Kansas, I tell you the distance between two places is r = x y 2 (2) is Kansas flat? [Ask] Well, what does our distance equation actually tell us? It says that moving along the ŷ direction takes you further than moving along the ˆx direction, right? r = x 2 +(2.54y) 2 (3) Lets draw what I mean [Draw the two diagrams] What did I do? I used centimeters along x and inches along y, I used different rulers. (They do silly things like this in Kansas) Does using different rulers make something stop being flat? No! All I ve done is stretch my ruler in the y direction, and stretching rulers does not change the object! So it doesn t change whether it is flat or not. Notice that the number in front just switches between the two units: 1cm = 2.54inches. Now that we know what it means to be flat in two dimensions, lets add a dimension. What does it mean to be flat in three dimensions? Remember a dimension is just the number of variables we have to play with. So above we had two dimensions: two variables, x and y. What would s be for three variables x, y, z if our three dimensional space is flat? We can just use Pythagoras twice on two different triangles, like this. Look what happens: r = x 2 +y 2 +z 2 (4) Pythagoras just turns into add up the squares of each variable, then take a square root of the whole thing. Flat always means Pythagoras Law still holds! Back to two dimensions. So, is this flat? [Draw horizontal line] Put your single ruler up against it. Yes. What about this? [Draw sloped line] Again put your single ruler up against it. Yes. 2

3 The distance between two points on these objects (lines) is given correctly by just reading off what the ruler tells us, which happens to be the same thing that Pythagoras law would tell us if we considered just x for the first line, or the combination of x and y for the second. What did I do? I just rotated my rulers, I did not change the object. What about this? [Draw diagram as in notes. near A has small curvature, near C has large curvature, D is not regular] Is this object flat? NO. Put your ruler between two points anywhere on the curve, and we have the mountain situation all over again. Pythagoras Law does not give you the correct distance. Remember, distance is defined to be how far we would have to drive to get between two points if we have to stay on the road. But what if we zoom in really close to A? [draw draw draw] Now its looking pretty flat. What about C? [draw draw draw] Its taking alot longer to look flat, we have to zoom alot more, but again, it eventually looks flat. This is puzzle piece number one: if you have curved object and you zoom in really close to it, it looks FLAT. 5 Inertial motion happens in flat 4D spacetime Remember Newton s first law of motion: an object in motion will continue along in a straight line at constant speed unless acted on by a force.. So if the constant speed is zero, then an object at rest will stay at rest. But what if you pass the HPD on the highway? You are speeding, but you only see the officer drift behind you slowly. Suppose you couldn t see anything else on the road, no other cars, no road, no buildings: only the officer. Do you even know you are moving? The point I am trying to make is the idea of constant speed depends on how you are moving too, not just how something else is moving. Takes two to tango. A little before 1905, there was alot of experimental evidence that the speed of light, c = 10 cm second 3 10 was constant for EVERYBODY ALWAYS. Why is this weird? Imagine the police officer is a light wave. No matter how fast you drive, and you can tell you are speeding up because you can feel the acceleration, but the police officer always looks like he is moving at the same speed. All the smartest people around 1905 had a really hard time with this, something was very wrong. Here s how Einstein started winning. Say I asked you to use your ruler to measure this book. Easy, line it up with the book and read off the numbers. Now say you are in the car with the ruler and, as you drive by the book, I ask you to use your ruler to measure the book. Hard! Its really tricky to see where and when the edges of the book line up with markings on the ruler! Why would you ever even try to do that? To measure distance, the obvious way to do it is to be at rest along with what you are trying to measure. Einstein s first brilliant idea was if its stupid to measure length with a moving ruler, its just as stupid to measure time with a moving clock. But why would you ever be thinking about rulers and clocks together? 2.54cm = 1inch cm = cseconds [make them examine their ruler!] If the speed of light is constant, you can use it like a conversion factor between time and space, just like converting between inches and centimeters. Lets exit the H1, we re on the offramp and approaching a stop sign. [DRAW IT OUT] 3 (5)

4 This is what nature sees, and she just uses two different rulers. Einstein then showed how two different people s rulers and clocks get mixed together if they are in uniform motion. All this mixing stuff is called Special Relativity. But most importantly, Einstein showed something else: there is one quantity that never gets mixed up: s = x 2 +y 2 +x 2 (ct) 2 (6) But this looks really familiar though. How would I measure the distance s in a flat four dimensional space? Remember what flat means! s = x 2 +y 2 +x 2 +w 2 (7) But we now know Nature uses a different ruler for time: w = ct. But if we plug this in above, we don t get the minus sign! What s wrong? Nature is trickier than that!: remember imaginary numbers? 1 = i? Nature uses an imaginary ruler! s = x 2 +y 2 +x 2 +w 2 = x 2 +y 2 +x 2 +(ict) 2 = x 2 +y 2 +x 2 (ct) 2 (8) Remember if I rotated my head in two dimensions, I would get different x and y for the same object? The mixing of space and time is like rotating your head, except how fast you are moving corresponds to the angle your head is tilted. Remember how the distance did not change when we rotated our head? If you are in inertial motion, constant speed compared to something else, your rulers and clocks can t be trusted separately to agree with someone else s rulers and clocks, you have to look at the distance formed from x 2 +y 2 +z 2 t 2. Then everyone else in inertial motion agrees on that distance, and its just Pythagoras law, with a really weird ruler for time. And we know what Pythagoras Law means... This is puzzle piece number two: inertial motion is the same thing as living in a FLAT four dimensional world 6 Gravity? Now we are ready. Everything above was at constant speed, inertial motion. No forces. What if we consider gravity? Remember what Newton said about gravity between two objects F = GBb r 2 (9) G is just some fixed number, r is the distance between the center s of the objects. Now, what are B and b? B is some property of one of the objects (the larger one) that determines its contribution to the strength of gravity between them. Likewise, b is some property of the other object (the smaller one) that determines its contribution to the strength of the gravity. Maybe you think you know what b and B are? Lets line up a bunch of objects. [planet, moon, asteroid, pokemon right in front of, you]. Do these all fall toward the Earth at the same rate? Lets look at their accelerations. 4

5 Remember Newton s second law tells us how to figure out something s acceleration if we 1) know its mass, 2) can tally up all the forces acting on it m i a = F = GBb r 2 (10) Does this look familiar? What s different? Here s the catch: Newton s second law works for all forces, for all objects, any kind! B and b are just numbers that tell you how strong gravity, one specific force, is. Just like electric charge Q and q. But what you usually see is this: m i a = GMm i r 2 (11) But why should the number appearing in the special law for a single type of force, the gravity force, be the same number that shows up when figuring out how any object responds to any type of force? This is really weird, Newton s second law has nothing more to do with gravity than it does with any other force. Its its own thing: give it any forces, and it tells you how to move. Does not care about the origin of the force. I m hammering on this hard: there is no reason why they should be the same, but Nature consistently shows us that they are the same in every experiment where we have tried to see if they are different. I have called them different things because they don t need to be the same, yet they seemingly are: b for gravitational mass, and m i for inertial mass. Its called inertial because if the forces are zero, Newton s second law turns into Newton s first law, which is the inertial law. That they seem to be the same is an experimental result, the assertion that they are exactly the same is called The Principle of Equivalence of Gravitational and Inertial Mass. As we will now see, this principle forms the foundation of Einstein s approach to the Universe: General Relativity. If we assume the Equivalence Principle is correct, that is m i = b (12) for all masses, regardless of composition: same for moons, same for asteroids, same for pokemon, same for you. Now if we try to figure out our accelerations due to gravity, the equations simplify right from the start: m whatever i m whatever i M Earth r 2 m whatever i M Earth r 2 a whatever = Gmwhatever i a whatever = G a whatever = GMEarth r 2 But absolute accelerations are not something you can measure if you are part of the action, just like absolute speed didn t make any sense when talking about the police officer on the highway. What we should look at is a relative acceleration between ourselves and something else. The simplest way is to look at the ratio of accelerations because if the ratio is 1, the separation between us and other never changes because even if we both speed up, we speed up together in the same way. A = aus = a( other ) GM Earth ( rus 2 ) GM earth r other 2 = r2 other r 2 us 5 (13) (14)

6 Now remember that those r s in Netwon s gravity equation are distances from the center of the Earth. Lets zoom in towards ourselves so we can t see the Earth anymore. Can we still feel its gravity? What do I mean? I mean can we still see things drifting apart from each other: if we can t see things drifting apart, it looks like there are no forces, so it looks like there is no gravity. So now that we ve zoomed in, we can see the moon, the asteroid, our pokemon and us. We will still be able to see the moon falling away from us because we are far from the moon. What if we zoom closer and put a box around just the asteroid, our pokemon and us? We can still see the asteroid falling away from us, but it takes a lot longer. What if we zoom very close so all we can see is our pokemon? He s right in front of us. 1 meter relative distance versus a billion meters (not centimeters) away from the center of the Earth? Lets calculate A A = (109 1) 2 (10 9 ) 2 = = Since acceleration is velocity per time, after 1 second, our pokemon is moving times faster than us. If the cop on the highway was moving only 1/100th more quickly than we were, he would probably still look like he was just matching speeds with us. How long would it take for our pokemon to be moving 1/100 more quickly than us? Five million seconds, which is about two months. So after two months, we still couldn t tell that he was falling away from us! What kind of motion is this: at rest and staying at rest? Inertial. So even if we are sitting in gravity, if we zoom in really close, the motion looks inertial. This is the final puzzle piece: if we zoom in really close in a space with gravity, the motion of objects looks inertial. 7 Together Lets put the three pieces together: 1. if you have curved object and you zoom in really close to it, it looks FLAT 2. intertial motion is the same thing as living in a FLAT four dimensional world 3. if we zoom in really close in a space with gravity, the motion of objects looks inertial Do you see it? Do you see Einstein s second brilliant idea? If we know inertial Physics is flat, and if zooming in to a spacetime with gravity gives inertial -FLAT- behaviour, then zooming out, we must conclude that spacetime with gravity is curved. 8 Acknowledgements I would like to thank all those who took time to listen to rough drafts of the presentation and offered pedagogical advice. Especially to John Learned for the composition of Pythagoras in 3D and Sandip Pakvasa for suggesting that gravitational charge be initially written with something other than M g,m g so as to aid in hammering home the oddity of equivalence. (15) 6

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