Physical time and physical space in general relativity Richard J. Cook Department of Physics, U.S. Air Force Academy, Colorado Springs, Colorado 80840 ͑Received 13 February 2003; accepted 18 July 2003͒ This paper comments on the physical meaning of the line element in general relativity. We emphasize that, generally speaking, physical spatial and temporal coordinates ͑those with direct metrical significance͒ exist only in the immediate neighborhood of a given observer, and that the physical coordinates in different reference frames are related by Lorentz transformations ͑as in special relativity͒ even though those frames are accelerating or exist in strong gravitational fields. ͓DOI: 10.1119/1.1607338͔ ‘‘Now it came to me:...the independence of the this case, the distance between FOs changes with time even gravitational acceleration from the nature of the though each FO is ‘‘at rest’’ (r,␪,␾ϭconstant) in these co- falling substance, may be expressed as follows: ordinates. In a gravitational field (of small spatial exten- sion) things behave as they do in space free of III. PHYSICAL SPACE gravitation,... This happened in 1908. Why were another seven years required for the construction What is the distance dᐉ between neighboring FOs sepa- of the general theory of relativity? The main rea- rated by coordinate displacement dxi when the line element son lies in the fact that it is not so easy to free has the general form oneself from the idea that coordinates must have 2 ␣ ␤ an immediate metrical meaning.’’ 1 Albert Ein- ds ϭg␣␤ dx dx ? ͑1͒ stein Einstein’s answer is simple.3 Let a light ͑or radar͒ pulse be transmitted from the first FO ͑observer A) to the second FO I. INTRODUCTION ͑observer B) where it is reflected and returns to A. If the time between transmission and reception of the reflected I often start my lecture on the meaning of the line element ͑ ͒ ␶ in general relativity ͑GR͒ with the provocative statement, pulse as measured by A’s standard clock is d A , then the ‘‘In general relativity the Galilean transformation of classical Einstein distance between A and B is defined by the radar ᐉϭ ␶ ͑ ͒ mechanics is just as valid as the Lorentz transformation of formula d cd A/2. Using the general metric 1 for light special relativity, because all space–time coordinate transfor- (ds2ϭ0), we find in the Appendix that this distance dᐉ is mations are equally valid in general relativity’’ ͑the democ- given by the spatial metric racy of coordinate systems͒. This statement invariably sparks dᐉ2ϭ␥ dxi dxj, ͑2͒ a heated discussion about what is physically significant in ij GR and serves to introduce the following comments on the where meaning of the spacetime metric. I claim no new results in g g the following—only that this approach to understanding the ␥ ϭ Ϫ 0i 0 j ͑ ͒ ij gij . 3 metric has been useful for the author and may be of some g00 help to others who attempt the daunting task of teaching GR We call Eq. ͑2͒ the metric of physical space for the given to undergraduates. reference frame. It measures proper distance at the point of interest, that is, local radar distance is the same as proper II. REFERENCE FRAMES distance ͑the distance measured with a standard ruler͒. Notice that, when g Þ0, the spatial metric components For our purpose, a reference frame will be defined as a 0i ␥ are not simply the spatial components g of the full collection of fiducial observers distributed over space and ij ij moving in some prescribed manner. Each fiducial observer metric g␣␤ . The example of Einstein’s rotating disk with ͑ ͒ i ϭ measuring rods along the diameter and circumference, as de- FO is assigned space coordinates x (i 1,2,3) that do not ͑ ͒ ͑ ͒ ͓ iϭ picted in Fig. 1, nicely illustrates the use of Eqs. 2 and 3 . change. The FOs are ‘‘at rest’’ (x constant) in these coor- ␪ ͔ In inertial space with polar coordinates (r0 , 0) and origin dinates. Each FO carries a standard measuring rod and a 2 standard clock that measure proper length and proper time at at the center of the disk, the space–time metric reads ds ϭϪ 2 2ϩ 2ϩ 2 ␪2 his/her location. The basic data of GR are the results of local c dt dr0 r0 d 0. A transformation to the frame ro- ⍀ ϭ ␪ϭ␪ measurements made by the FOs. ͑These are the ‘‘10,000 lo- tating with the disk at angular velocity (r r0 , 0 cal witnesses’’ in the words of Taylor and Wheeler.2͒ ϩ⍀t) puts the metric into the form In special relativity, the FOs usually sit on a rigid lattice of ⍀r 2 ⍀ Cartesian coordinates in inertial space; there is a different set ds2ϭϪͫ1Ϫͩ ͪ ͬc2 dt2ϩ2ͩ ͪ r2͑cdt͒d␪ϩdr2 of FOs in a different inertial reference frame. In Schwarzs- c c child space, the FOs reside at constant values of the ϩr2 d␪2, ͑4͒ Schwarzschild space coordinates (r,␪,␾), and, in an ex- ϭϪ Ϫ ⍀ 2 panding universe, the FOs sit at constant values of the co- with nonzero metric components g00 ͓1 ( r/c) ͔, ␪ ␾ ϭ ϭϪ⍀ 2 ϭ ϭ 2 moving Robertson–Walker space coordinates (r, , ). In g0␪ g␪0 r /c, grr 1, and g␪␪ r on coordinates 214 Am. J. Phys. 72 ͑2͒, February 2004 http://aapt.org/ajp 214 IV. PHYSICAL TIME Physical time ˜t in the immediate neighborhood of a par- ticular fiducial observer O is that function of position coor- dinates xi and coordinate time x0ϭct that, when used to measure speed dᐉ/dt˜, places the speed of light ͑the one-way speed͒ at the invariant value c in all directions. Now the general line element ͑1͒ can be written in terms of the physical space metric ͑2͒ as ds2ϭϪc2 dt˜2ϩdᐉ2, ͑8͒ where dt˜ is defined by i g0i dx Fig. 1. Einstein disk rotating about its center O with angular velocity ⍀ dt˜ϵͱϪg dtϪ . ͑9͒ 00 ͱϪ ͑viewed from inertial space͒. Standard measuring rods are laid end-to-end c g00 along the diameter and circumference of the disk and move with the disk. The proof of Eq. ͑8͒ is by direct expansion of this equation using Eqs. ͑2͒, ͑3͒, and the definition ͑9͒ of dt˜. We have used the notation dt˜ because for light (ds2ϭ0), Eq. ͑8͒ x␮ϭ(ct,r,␪). A glance at the metric ͑4͒ might miss that the shows that dt˜ is the time differential that gives the speed of spatial geometry in this reference frame is non-Euclidean. But application of Eqs. ͑2͒ and ͑3͒ gives the spatial metric light dᐉ/dt˜ the value c in all directions, that is, dt˜ is indeed the differential of physical time. r2 d␪2 ͓ iϭ ͑ ͔͒ ᐉ2ϭ 2ϩ ͑ ͒ At observer O at dx 0 in Eq. 9 , the physical time d dr Ϫ͑⍀ ͒2 , 5 ␶ 1 r/c increases at the same rate as the proper time 0 at that loca- ˜ϭͱϪ ϭ ␶ which clearly shows no change of radial distance (dr tion (dt g00dt d 0), and, at the coordinate displace- i ϭdr ) and an increase in measured circumferential distance ment dx from O, the physical time is synchronized with the 0 3 ͓from rd␪ to rd␪/(1Ϫv2/c2)1/2] due to the Lorentz con- clock at O by the Einstein synchronization procedure. By traction of measuring rods placed end-to-end on the circum- the Einstein synchronization procedure, we mean the process ˜ ference and hence moving in the direction of their lengths in which, when the clock at O reads time t 0 and this time is with speed vϭ⍀r. transmitted over the coordinate displacement dxi of length When three-space is flat, we can transform to Cartesian dᐉ at the speed of light, the physical time at the end of this space coordinates for which the spatial metric takes the Eu- journey has the value ˜t ϭ˜t ϩdl/c, that is, it contains the ᐉ2ϭ␦ i j i 0 clidean form d ij dx dx , and the coordinates x mea- retardation correction dᐉ/c required to account for the finite sure distance directly. We shall refer to such coordinates as and invariant propagation speed of the time signal. ͓That the ‘‘physical’’ space coordinates. But, when three-space is ͑ ͒ ˜ curved, global physical coordinates do not exist. We can, definition 9 of the physical time differential dt is consistent however, always transform to local Cartesian coordinates with the Einstein synchronization procedure follows from the dx˜ 1, dx˜ 2, dx˜ 3 in the immediate neighborhood of any chosen fact that dt˜ is the time differential that gives the speed of FO, in which case the local space metric takes the form light the value c in all directions.͔ Another way of saying this is that, in the immediate neighborhood of the given fiducial dᐉ2ϭ͑dx˜ 1͒2ϩ͑dx˜ 2͒2ϩ͑dx˜ 3͒2. ͑6͒ observer O, the condition dt˜ϭ0 ͑or ˜t ϭconstant) defines a ͑Here and in the following we use a tilde to denote differen- hypersurface of simultaneity. tials that may not be exact and, therefore, may not be inte- The all important point of this discussion is that, as a 1 2 grable to global functions.͒ The proper distances dx˜ , dx˜ , temporal coordinate, the physical time ˜t is a very special dx˜ 3 are local physical coordinates measured from the given one. It is the time actually used by the fiducial observer for FO, and are written as differentials to suggest that they can- measurements in his local reference frame.