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Talk:Counterexamples to Relativity (view source)
Revision as of 04:00, August 4, 2010
, 04:00, August 4, 2010→Curl of the gravitaional field: Loooong discussion of the twin paradox, vectors, scalars, etc.
::: What about black holes, though? Surely their gravitational fields aren't conservative, since once an object passes the event horizon, you can't retrieve it. [[User:PhyllisS|PhyllisS]] 01:24, 3 August 2010 (EDT)
::: What about black holes, though? Surely their gravitational fields aren't conservative, since once an object passes the event horizon, you can't retrieve it. [[User:PhyllisS|PhyllisS]] 01:24, 3 August 2010 (EDT)
Phyllis:
You seem to be very curious about this topic. I'm going to try to give an intuitive, but nevertheless scientifically correct, explanation of what is going on with relativity, the "twin paradox", and vector fields, potential functions, and path integrals. This explanation will probably seem long and tedious, for which I apologize in advance. I also apologize if it seems that I am being too "folksy", or talking down to you. Please bear with me, and please pay close attention.
We have a parking lot, and two twins, who are fitness enthusiasts and always wear pedometers wherever they go. There are two spots, "X" and "Y", painted on the parking lot. Both people stand on spot "X", set their pedometers to zero, and start walking. Twin A simply walks directly to spot Y. Twin B, being more into fitness, walks all over the place, eventually arriving at B.
Now there are quite a number of things we can say. First, the temperatures vary all over the place. They are a ''scalar field''. That means that they are associated with ''location on the parking lot'', not with any particular observer. They are objective measurements that everyone agrees on, because they are aspects of space itself. Our fitness enthusiasts are also amateur meteorologists, and carry thermometers around with them.
::Twin A: "When I was at the green Toyota, I noticed that the temperature was 67 degrees Fahrenheit."
::Twin B: "By coincidence, I also wandered past the green Toyota, and got the same reading."
By the way, since temperature is a scalar field, it has a gradient, which is a vector field. That field is conservative, according to the theorem of mathematical physics that says that curl grad Φ = 0 always. This gradient is a ''vector field''. Like the scalar of temperature, it is a property of the ''space (parking lot) itself''. If the twins had been measuring this gradient (perhaps they carry around fancy "differential thermometers"), they would have gotten the same vector at the green Toyota.
There is also a theorem of mathematical physics, sort of the opposite of the theorem above, that says that, if a vector field V has a curl of zero:
*You can make a scalar field <math>\Phi\,</math> (a property of the space itself, not tied to any particular observer) that it is the gradient of. That scalar field is called the "potential" for the (conservative) vector field. (By the way, this is very closely related to "exact differential equations" that you wrote about! Do you see the connection?)
*If you integrate that vector field along any path between two points A and B (that is, you calculate
:::<math>\int_A^B \vec{V} \cdot dl</math>
for that path, where "dl" is the "line element" along the path), you will get <math>\Phi(B)-\Phi(A)\,</math>.
*Since <math>\Phi(B)-\Phi(A)\,</math> is a property of the scalar field itself (and the points A and B), it follows that that path integral is the same for all paths. And if the path ends on the same point it started on, the integral is zero.
::Twin A: "I was measuring the gradient of the temperature as I walked, and calculating its path integral as I went. I got an answer of 4 degrees."
::Twin B: "I was doing the same. My integral was much harder to calculate, because I was going all over the place. But I also got 4 degrees. Hey, wait a minute! The temperature at the start point was 68 degrees, and at the end point it was 72 degrees. That explains it."
Now someone at the edge of the parking lot was running a Van deGraff generator, so there were electric fields all over the place. The twins are also physics students, and carry electroscopes wherever they go. They measured the electric field, and calculated its path integrals. The electric field is conservative (in the absence of varying magnetic fields), so they got the same integral. That integral was 600 volts (it's only static electricity, so it isn't dangerous). Since the electric field is conservative, there is, by the previous theorem, a potential function. That function was 600 volts higher at point B than at point A.
Now here's the kicker:
::Twin A: "I walked directly from A to B. My pedometer says 150 feet."
::Twin B: "I took a long route all over the place. I walked half a mile."
The pedometer readings ''are not a scalar field''. They are not a property of the space itself. They are properties of the observers. Even though they, in some sense, measure an aspect of the parking lot (how many molecules of asphalt one passes), they are artifacts of the twins' actions.
The twins could have been integrating their motion vectors; that's sort of what pedometers do. But those vectors are not a vector field on the space itself. It makes no sense to ask whether that "vector field" is conservative, because it isn't a vector field. A vector (or scalar, or tensor) field has to be a property ''of the space itself''. These "pedometer vectors" are just things that the twins make up as they walk.
Now for the "twin paradox". The parking lot is replaced by "Minkowski space", also called "4-dimensional space-time". "Points" in this space are now "events", complete with a time. Events A and B are now the act of the twins saying goodbye as one of them got into the rocket, and the act of them re-uniting after B returns. Twin A took a direct route (called her "world line") from A to B. She used a coordinate system in which the spatial coordinates of A and B were the same (Cape Canaveral, latitude yada yada, etc.) and the time coordinate differed by 30 years (2010 to 2040.) B went to Alpha Centauri and back. When she returned, they were both using the same coordinate system (location is Cape Canaveral, latitude yada yada, time is 2040.) But she looks at her watch, and only 5 years have elapsed! What the watch shows is ''not a scalar field on spacetime''. It was ''not the path integral of a vector field on spacetime''. What she integrated was the ticking of her watch, nothing more.
The path that A took is called a geodesic. It is the Minkowski-space equivalent of a "straight line". But, because of the peculiarities of relativity, it shows the ''longest'' elapsed time (30 years) of all paths, rather than the shortest. By going to Alpha Centauri, twin B took a shorter path, in terms of the way path length is measured in Minkowski space.
Very interesting fact: The path length in Minkowski space, that is, the sum of the tiny distances as measured by the Lorentz/Minkowski metric, ''is the same as the local time''. That is (assuming you are using the "spacelike convention"), everyone's wristwatch measures path length along their own world line. The twins simply followed paths of different lengths. That's all there is to the "twin paradox". (That is, that's all there is to it, if you analyze it correctly, as I have described above. Most introductory treatments of relativity don't do it this way. They just throw the Lorentz transform at you.)
Now there are a few points about the "twin paradox" that people find confusing.
First, aren't the laws of physics supposed to be the same for everyone? What made twin B's watch run slower? Well, she ''knew'' she was traveling at high speed. She brought an accelerometer with her in the rocket. Just as twin B in the parking lot knew she was walking all over the place, turning around and such, twin B in space knew that her world-line was turning, and therefore wasn't a straight line (geodesic). How did she know? It takes force to make you deviate from a geodesic (this is really pretty much the same as Newton's laws of motion), and she felt the force.
Second, how can we analyze the curvature of B's world line? Here's where general relativity has to come in. As soon as world lines start to curve, you have to measure their curvature, that is "geodesic curvature". You get into complicated issues of curved coordinate systems (you're in one now; it's what you perceive as "gravity"!), and curved spacetime, and so on. And you get into the <math>\Gamma\,</math> symbols, which measure the deviation from a geodesic, and hence the "fictitious forces" that you feel. This is why general relativity is related to the "twin paradox", in that the space ship followed a curved trajectory and experienced acceleration. But, to analyze the plain facts "twin paradox", all you really need to know is that twin B followed a crooked line. Place your ruler on a diagonal on the graph of Minkowski space, draw the line out to Alpha Centauri. Turn the ruler, draw the returning line. Ignore the impossibly sharp corners. Use special relativity to analyze the Lorentz transform for each section of B's world line.
Oh, and to try to answer some of your specific questions, the gravitational field, under either Newtonian or relativistic mechanics, is a conservative field. Its curl is zero. If it weren't, conservation of energy would be violated, and we could make a perpetual motion machine by having a planet run around in circles picking up energy. The "curl=0" aspect of gravity under general relativity is more complicated, because true vector fields have to be on Minkowski space, but it still conserves energy. When Mercury orbits the Sun, its perihelion precesses because of relativistic effects, but its energy is conserved.
I'll try to think some more about your black hole question and get back to you.
[[User:Simeon|Simeon]] 00:00, 4 August 2010 (EDT)