Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!mhuxt!houxm!whuxl!whuxlm!akgua!gatech!seismo!rochester!bullwinkle!batcomputer!cpf From: cpf@batcomputer.UUCP Newsgroups: net.physics Subject: Coordinate frames in GR Message-ID: <438@batcomputer.TN.CORNELL.EDU> Date: Mon, 24-Mar-86 17:29:07 EST Article-I.D.: batcompu.438 Posted: Mon Mar 24 17:29:07 1986 Date-Received: Sat, 29-Mar-86 06:20:42 EST Reply-To: cpf@batcomputer.UUCP (Courtenay Footman) Organization: LNS, Cornell University, Ithaca NY Lines: 70 After several very bad articles on this subject, there have recently been some very good ones, especially the one that noted that the Earth does rotate, and that this is unambiguous because the geometry near the Earth is a Kerr geometry. However, I feel that the lack of significance of coordinate frames should be stated forcefully. Coordinates are NOT fundamental in general relativity. What is fundamental the geometry of spacetime that is described by the metric ds**2 = gAB(p)dxAdxB, where g, the metric, is 10 functions of the point p in space time. All the physics is here. We can now make any (almost) coordinate transformation, xA = xA(xAold), and get exactly the same physics. (Although it is nice if the transformation is invertible and sufficiently differentiable.) An example of a legal transformation is: x0 -> x0, x1 -> x1, x2 -> x2 , x3 -> x3 - v * x0, where v is a constant. Pick v = 10. We now have a coordinate system in which every "velocity" is faster than light! What does that mean? It means that a dx3/dT for a time-like particle is greater than 1 (c = 1). In fact, in this coordinate system, the velocity of light is greater than the "velocity of light"! So what? A light-like world line still has interval zero (ds**2 = 0). In this coordinate frame, the metric is somewhat strange looking, but the physics is normal. If I want to make a coordinate system in which the earth is motionless (i.e dx{1,2,3}/dT = 0 for points on the earth's surface, I can do that (in an infinite number of ways). If I pick a simple way of doing that, Alpha Centauri's velocity will have some strange looking numbers in it, but no physics will be changed by my changing coordinate systems. To restate this, coordinate systems have no reality -- we are free to choose any one we want to. Normally we choose on the basis of simplicity, and this conditions us to expect certain things of a coordinate system, but nothing stops us from choosing a perverse one. They all give the same physics. I can do anything to my coordinates that I want to: x0 -> x0**2, x1 -> exp (x2) - x3, x2 -> gamma(x1**2), x3 -> arctanh(x3). I don't even want to thing about what my metric would look like (even for flat space), but there is nothing that stops me from doing it. As a final example of the meaninglessness of coordinates, consider two point masses orbiting each other. This problem is not exactly soluble, but I can make an approximate solution: Newtonian gravity. I can now make a coordinate transformation so that both particles are "stationary"; that is x1, x2 and x3 are constant for all x0. The metric will look odd, with trigonometric functions of x0 in strange places, but it will be a correct description of the physics (up to the Newtonian approximation). It is not a particularly useful description, but is a correct one. To sum up, the evolution of the coordinates of a particle are completely meaningless until one looks at the metric (that is, the geometry) and figures out what the particle is actually doing! [P.S. For the semi-expert. If you don't understand what I am talking about in the next paragraph, don't worry about it.] An example of a coordinate system with some perverse features is ordinary Schwarzschild geometry. It seems like no particle will every reach 2M, since it takes infinite (coordinate) time to reach 2M. Actually, of course, a particle does reach 2M in finite proper time, and in due course achieves it demise at r = 0. One can remove this singularity by going to another coordinate system, e.g Kruskal-Szekeres. (This coordinate singularity is, in its way, more disturbing than the "faster than light" velocities of a rotating coordinate system, and has produced considerable confusion among science fiction writers.) -- -------------------------------------------------------------------------------- Courtenay Footman ARPA: cpf@lnsvax.tn.cornell.edu Lab. of Nuclear Studies Usenet: {decvax,ihnp4,vax135}!cornell!lnsvax!cpf Cornell University Bitnet: cpf%lnsvax.tn.cornell.edu@WISCVM.BITNET