What sort of things happen in accelerated frames? Why use them? Next Topics: Is curvature necessary? Conventionalism. Gravitational waves space is real Singularities Global properties of GR cosmology Topology, time travel, and other oddities Term paper deadlines: Next Week: Consultations. Topic Outline/draft fragments. Complete draft Final revised draft Early versions are encouraged at each stage. Accelerating Clocks
We saw that clock rates must appear different to an accelerated observer. Let's investigate further. Here's a pair of our simple two-mirror clocks viewed over a brief interval during which they accelerate toward our left, at rest in the middle of the interval. a v a a The light leaves the middle, starting the clock tick on both sides. The light reaches the left side, which was moving toward it, before reaching the right side, which moved away. v
The left-side tick finishes, as the middle side moves toward that light. The right-side tick doesn't, as the middle moves away from it. An observer can conclude objectively that the left clock is running faster than the right clock. (e.g. by film exposed on only the left side). The clock the acceleration is toward runs fast, the clock the acceleration is from runs slow. Time and Position We could make a whole stack of these clocks, applying the same argument to each. The farther the clocks are away in the direction the acceleration is toward, the faster they run. The farther they are away in the opposite direction, the slower they run. Everybody agrees on this, even if they can't agree on which clock is "right". (unlike disagreement between inertial frames, in which they can't even agree on who's faster.) Notice that this effect is just what we needed in the twin "paradox". As Alice accelerated toward Beth, (while turning back) she concluded that Beth's clock's were running fast.
How Big an Effect? Lets calculate the effect of the acceleration is on clock rates: B is a massive object, A is less massive, held in orbit by a string. B ~doesn't accelerate. B says A's clock runs slow by a factor of: What does A say about B? If A weren't accelerating, she would see B as slowed by a factor of 1/g. . But since A circles B repeatedly, they can compare times on each orbit, with no change in signal transmission times. So A and B must agree about who is faster and by what factor. Thus A also thinks that B's time runs faster by a factor of g. . The net effect of A's acceleration in speeding the rate at which 1 1
a R g. 1 c a R A sees B's clocks run is thus a factor of: 1 1 v c c This agrees with our argument from stacked light clocks: the clock rate increases with the distance in the direction of the acceleration, decreases the other way. This is also just the amount needed to get Beth and Alice to agree. (We won't try to make the argument accurate beyond the linear term in the acceleration- we would have to worry about who measures "a" and "R".) 2
2 2 2 2 The Accelerating Observer sees something very strange. In her reference frame, identically constructed clocks run at different rates, depending on where they're located!
It would seem that a reasonable law of physics should be that identically constructed clocks run at the same rate. Thus this sort of gross effect should tell A that she is accelerated. In an accelerated frame, you cannot go through the exercise of building a lattice of identical synchronized clocks to define the coordinate system: the clocks don't stay synchronized. It certainly looks like accelerated frames are a curiosity, since in one we would have to abandon some simple laws of physics. At one historical point, the only reason to insist on looking at the laws that apply within accelerated frames was Einstein's Machian prejudice that the laws of physics should depend only on the relations among objects, not on absolute motion in any sense. Defining Coordinates Einstein insisted on the (Machean) idea that each observer must construct his own view of space and time by some actual observable operations: we entirely shun the vague word space, of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by motion relative to a practically rigid body of reference. With no absolute space or time to rely on, we need an operational
definition of position and time measurements. So lets replace Newtonian space & time by an ensemble of meter sticks and clocks Distribute a set of clocks on a lattice. Clock synchronization is done by sending a round trip signal between two clocks. The signal is assumed to arrive at the second clock halfway between emission and reception by the first. The position of each clock is measured similarly. Half of the round trip time, divided by c, gives the distance. How do we test if our frame is inertial? Do Inertial Frames Exist?
Einstein's operational recipe (make a bunch of identical rods and clocks, build a lattice of the rods to mount the clocks on, check the clock synchrony with light rays) also allows you to check if Special Relativity is correct. It assumes that once the clocks are synchronized by this procedure, they will stay synchronized. (It also makes other assumptions, but we needn't think about them.) So while you are getting used to the ideas that: Aristotle and instinct are wrong, there is no absolute rest frame. Newton and Galileo are wrong, there is no absolute time or distance. Einstein has at least opened the possibility that his Special Relativity also is wrong, there may be no inertial frames. Einsteins recipe for building a coordinate frame from rigid sticks already has a problem: Can we follow Einsteins Recipe? Rigid bodies cannot exist if Special Relativity is correct. Otherwise you could wiggle one end of a stick and transmit a signal to the other end infinitely fast.
Each atom only feels forces from its neighbors after a delay- the forces are transmitted at the speed of light. So if a force acts on any part, it always distorts the object at least temporarily. We also don't know if the clocks will stay synchronized. So the existence of inertial frames is not simply given. Oddities of Accelerating Frames Other strange things happen in uniformly accelerated frames. A light ray travelling at right angles to the acceleration seems to bend, as if it were falling in the direction opposite to the acceleration. Straight lines are supposed to be the shortest paths between two points. Now the quicker path here is the path of the uninterrupted light ray (we can see that easily in the lab frame) but in the accelerated frame the spatial path length of that light ray seems longer than the path of the ray that
makes a bounce off the wall. Isn't the speed of light supposed to be constant? If we tried to make our ordinary laws of physics work in such a frame, we couldn't identify light rays' paths with straight lines. How would we define straight lines? (stay tuned) Non-uniform Acceleration As long as we're looking at yucky reference frames, let's see what would happen if you used a nonuniformly accelerating reference frame, e.g. a merry-go round. Before the merry go-round accelerates, you get a bunch of little meter sticks, all stacked together and the same length. If you measure the MGR circumference C and radius R by counting out meter sticks, you find C = 2pR, whether you do this on the
MGR or the ground. Now set the MGR spinning. It will stretch, etc., but you tighten down any bolts needed to make its circumference still fall exactly above the previous circumference, traced out on the ground. If you measure on the ground, you get the same old C and R. Due to the Lorentz contraction, the rulers measuring the circumference on the MGR have shrunk, but not the ones used for the radius. Therefore, in the MGR measurement, C > 2pR. Not only that, the ratio C/R depends on the radius of the circle. (It gets bigger for bigger R.) This is not Euclid's plane geometry, but rather resembles the sorts of geometries you get if you try to confine measurements to curved surfaces. No wonder it's hard to find straight lines with familiar properties! Curved Geometries Here are 2-d surfaces in 3-d in which C/2R Sphere: Saddle:
"triangle" "triangle" If we want to accept non-uniformly accelerated frames as equally valid, we must accept such weird geometry as being the correct description of our 3-d space. In other words, we should be able to distinguish whether we are using a non-uniformly accelerated frame by whether measurements confirm Euclid's axioms. Don't worry about whether the space looks like a piece of some hypothetical Euclidean higher dimensional space, which would have absolutely no physical significance. Our Choice of Frames We saw that the outside observers in a rotating frame think that the clocks at the middle run fast. So if we try to use a rotating frame, clock rates depend on position. We have a choice: Reject accelerating reference frames, because they require clock rates to depend on position,
violate Euclidean geometry (with reasonable definitions of length), generally make a mess out of familiar laws of nature. Accept accelerating frames, and make new laws of nature that have all those weird effects. Anyone in his or her right mind would choose (1). Why then do physicists choose (2)? Why choose accelerating frames? Our previous argument suggests that physicists do so because of some mental defect. To confirm that: Where do you feel gravity? The principle of equivalence The gravitational mass of a body is equal to its inertial mass. Remember: inertial mass is the m that appears in p = mv or E = mc2 , or (approximately) F= ma.
Gravitational mass is the m that appears in Newtons law of gravity: F = GMm/r2. It tells us the strength of the gravitational force between two masses. Empirically, the two kinds of mass are the same to within a part per trillion (10-12). Different types of mass all show the same acceleration (from a given starting velocity) in a gravitational field In an objects calculated gravitational acceleration the object's own mass, m, drops out: a = F/m= GM/ r2 So gravity makes every different type of object accelerate together: the effect of gravity is completely describable classically by an acceleration field. Just as Galileo said. That means that you don't feel gravity in the same way that you feel other forces. Since all your parts are accelerating together, gravity creates no strains, tickles no nerves However, as Einstein put it, he was the first to interpret this fact. The principle of equivalence
In the famous elevator gedanken experiment we are somewhere in intergalactic space, with no planets or other junk nearby. Fred is resting at ease in his un-accelerated reference frame. Barney, on the other hand is inside a box and cant see out. Suppose there is a rope attached to the box, and some external agent pulls on the rope, accelerating the box at exactly 9.8 m/s2. Fred sees: a Barney sees: g g =a Fred says: The box (and Barney) are accelerating. So what? Barney says, I am not accelerating. I am in an elevator which is hanging from its cable
in a gravitational field. It's the same field that's making Fred fall, because no cable supports him. Who is correct? Einstein insists that in the absence of a reason for preferring one point of view, one must accept both. Einstein proposed another modest generalization No measurement of any sort can detect a uniform gravitational field. And, by the way: No local measurement can detect any gravitational field. Since any field is nearly uniform on a small enough patch If we accept Einstein's generalization we cannot distinguish an accelerated frame from a gravitational field. We said that no sane person would voluntarily accept accelerated frames, because they lead to all sorts of crazy effects. If gravity were completely uniform, you could get rid of it by transforming to another reference frame. Gravity can always be eliminated that way in a small region (e.g., inside Barneys elevator), but not over a large spacetime domain, because there is an
uneven distribution of matter. So no sane person can reject a universe with gravity: you can't get rid of gravity without getting rid of everything. The gravity isn't uniform, so our actual reference frames are like the ones with non-uniform accelerations. If Einstein is right, then a world with gravity has all those bizarre effects we found for accelerating frames, whether you like them or not. Observational Consequences But if you can't tell whether balls fall to the floor because of g or because you have chosen an accelerated reference frame, then g must make those weird effects that we found for accelerated reference frames. Otherwise you could use them to test whether you were accelerating. So g should give the strange clock effects found in accelerating frames. E.g. the Gravitational red shift: clocks at the bottom run slower than ones at the top. One can use the frequency of light as a clock (count the crests as they pass by. The most accurate clocks work this way). So, do an experiment drop light from the top of a tower:
Light becomes redder as it goes up and bluer as it goes down. (This can also be understood in terms of the energy of "photons".) The effect can be seen in starlight and also in buildings. (Pound & Rebka, 1960) Space must be non-Euclidean in a non-uniform g, just as with non-uniform a. On the merry-go-round, where things would fly outward, C > 2pR. Near a star, where things fly inward, there must be the opposite effect, C < 2pR. (Here each length is measured with a meter-stick with some uniform construction, say same number of nickel atoms.) The result is a curvature of the paths of anything, including light, beyond that given by treating gravity as a force. (Doubles the curvature of light.) CONFIRMED, 1919 solar eclipse Gravitational time-warp Classically, energy goes up as something falls. Well see that corresponds to frequency.
Clocks run faster away from the star, so synchronized processes like the wavefronts passing seem faster, measured by local clocks, below. The (locally measured) frequency of the wave is higher near the star, so the wavefronts are closer near the star. The wave must bend. This time-warp term is essentially just the classical force term, to a good approximation. Gravitational space-warp On the merry-go-round, there was missing space inside the circle. Near a star, there should be the opposite effect. A wave passing by the star has to traverse more distance on the part near the star than on the part farther away. The wavefronts farther away get ahead, which turns the wave as if it were being pulled toward the star. That's the extra curvature of light observed in 1919. It looks like the wavefronts are closer together near the star,
but that's only because I've forced the picture onto this nearly Euclidean page. Really there's extra space there and all the fronts are equally spaced, as measured locally. In a standard coordinate system (Schwarzschild) its the radial lengths that are larger than youd think from the circumferential ones, giving extra curvature on the way toward and away from the star, not on closest approach. Note the necessity for an operational definition of distance (and time).
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