Physical Time and Physical Space in General Relativity Richard J

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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 extension) 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 particular ﬁducial observer O is that function of position coordinates 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. If it were not so, not be large. They are finite but small enough that curvature this observer would not measure the local light speed c. effects are negligible over the region they span. When the Thus the notion that all fiducial observers measure the same ͓ ᐉ2ϭ␥ 1 2 spatial metric is diagonal to begin with d 11(dx ) speed c for light ͑Einstein’s postulate͒ is equivalent to the ϩ␥ 2 2ϩ␥ 3 2͔ 22(dx ) 33(dx ) , the local physical spatial coordi- notion that all fiducial observers use the physical time ˜t in nates are their local reference frames for the measurement of that 1ϭͱ␥ 1 ͑ ͒ speed. Also notice that, when we use local Cartesian coordi- dx˜ 11dx , 7a nates dx˜ i at a particular FO, the local line element ͑8͒ takes 2ϭͱ␥ 2 ͑ ͒ dx˜ 22dx , 7b the Minkowski form 3ϭͱ␥ 3 ͑ ͒ 2ϭϪ 2 ˜2ϩ͑ 1͒2ϩ͑ 2͒2ϩ͑ 3͒2 ͑ ͒ dx˜ 33dx . 7c ds c dt dx˜ dx˜ dx˜ . 10 ␮ But such differentials are not exact and cannot be integrated Equation ͑10͒ does not imply that the local frame dx˜ is to give global Cartesian coordinates ͑if they could, the space inertial. The observer O can be moving arbitrarily, and there would not be curved͒. can be a gravitational acceleration in this frame. ͓The local

215 Am. J. Phys., Vol. 72, No. 2, February 2004 Richard J. Cook 215 metric g␣␤(O)ϭ␩␣␤ is Minkowskian, but the derivatives of ␮ ͔.x are not necessarily zero at this point˜ץ/ ␤␣gץ the metric We use phrases such as ‘‘the local reference frame’’ or ‘‘immediate neighborhood,’’ because the differential of physical time in Eq. ͑9͒ may not be an exact differential. If dt˜ is not exact, Eq. ͑9͒ defines the physical time only in a small neighborhood about O. That is to say, unless they are very close to one another, different FOs use different physical times for their interpretation of nature. Only when the differential of physical time is exact does there exist a single global physical temporal coordinate ˜t (x0,xi) for all FOs. In this case, specific values of the physical time ͓˜t (x0,xi)ϭconstant͔ define global hypersurfaces of simultaneity in the reference frame under consideration. No- Fig. 2. On Einstein’s rotating disk, two firecrackers explode at points A and tice that, when dt˜ is exact, its integral ˜t (x0,xi) is a global B on the disk’s edge equidistant from observers O and P. Observer O sees temporal coordinate for which the line element ͑8͒, namely the two flashes at the same instant and concludes the explosions occurred simultaneously. Observer P, moving with the disk, sees the flash from B 2ϭϪ 2 ˜2ϩ␥ i j ͑ ͒ before that from A and concludes the explosions are not simultaneous. Thus, ds c dt ij dx dx , 11 the question of simultaneity is ambiguous in this rigid rotating reference ϭϪ ϭ frame. is in Gaussian normal form, that is, g00 1 and g0i gi0 ϭ0. This form of the metric implies that the FOs of such a reference frame are freely falling ͓xiϭconstant are geodesics of the metric ͑11͔͒, and the clocks of all the FOs run at the are simultaneous has no unambiguous answer. In this case, same rate and remain synchronized ͑no gravitational time there does not exist any temporal coordinate t(x0,xi) with dilation in this reference frame͒. These are the ‘‘comoving’’ the property that equal values of this coordinate for separated ͑or ‘‘synchronous’’͒ reference frames.4 Only in such frames events implies simultaneity of those events. We call such does a global physical time exist. Examples include the frames ‘‘asynchronous’’ frames. It is bad enough ͑or ‘‘good Robertson–Walker metrics of cosmology, the comoving co- enough’’ because it is true͒ that judgments of simultaneity ordinates used by Oppenheimer and Snyder in their early are different in different reference frames ͑as in special rela- studies of gravitational collapse,5 and the inertial frames of tivity͒, but for the concept of simultaneity of separated special relativity. In all other reference frames ͑other than events to become meaningless in a single reference frame comoving ones͒ the differential of physical time is not exact, ͑even a rigid one͒ is even more difficult to swallow ͑but and there does not exist a global physical time coordinate. equally true͒. That our notion of simultaneity at a distance Gravitational time dilation is a symptom of the lack of a loses meaning in an asynchronous reference frame is, in the global physical time ˜t (x0,xi). author’s view, one of the most counterintuitive ideas in all of When the physical time differential is not exact, it often is physics, and surprisingly little emphasis is given to it in possible to make it exact by means of an integrating factor many textbooks on general relativity. ͓dt¯ϭdt˜/R is exact for some function R(x0,xi)]. In this We can begin to understand how the concept of simultaneity becomes ambiguous by returning to Einstein’s rotating case, integration gives what we call a ‘‘synchronous tempo- disk. Consider an observer at the center O of the disk and ¯ 0 i ͑ ͒ ral coordinate’’ t (x ,x ), and the metric 8 is written in another at P on the edge of the rotating disk. Let two fire- terms of this time coordinate as crackers explode at points A and B on the disk’s edge equi- 2ϭϪR 2 2 ¯2ϩ␥ i j ͑ ͒ distant from observers O and P as in Fig. 2, and let the ds c dt ijdx dx , 12 explosions be timed so that observer O sees the flashes at the Rϭ ␶ ¯ where d 0 /dt is the rate of a fiducial clock on the time same instant and concludes, therefore, that the flashes occur scale ¯t . Clearly R describes gravitational time dilation. A simultaneously. Observer P, who is also ‘‘at rest’’ in the most important feature of the metric ͑12͒ is that equal values rotating frame, sees the light from B before that from A because ͑from the vantage point of inertial space͒ he is mov- of the temporal coordinate ¯t , say ¯t ϭ¯t for widely sepa- A B ing toward the light coming from B and away from the light rated events A and B, implies that these events occur simul- coming from A. Observer P concludes that the firecracker at taneously ͑according to the Einstein definition͒ in the given B exploded before the one at A because he observed it first reference frame. This result follows because the condition ͑ ˜ϭ and the points A and B where burn marks are left on the for neighboring events to be simultaneous (dt 0) can be disk͒ are at equal distance from him. In this way we see how written as dt¯ϭ0 and dt¯ is integrable. Hence, the locus ¯t two observers, both at rest in a rigid reference frame, can ϭconstant is a global hypersurface of simultaneity. Ex- disagree about the simultaneity of two events, and thus ren- amples of this case include the Schwarzschild metric in der the concept of simultaneity ambiguous. Schwarzschild coordinates and any other time-orthogonal The hallmark of an asynchronous reference frame is that ϭ metric with metric coefficients g0i 0. the metric components g0i not be zero, that is, that the metric The final and probably most interesting case occurs when tensor not be time-orthogonal. Examples of such metrics in- the physical time differential ͑9͒ is not exact and cannot be clude rotating reference frames, the Kerr metric6 in Boyer– made exact by applying an integrating factor. In such refer- Lindquist coordinates7 representing a rotating black hole, ence frames, the question of whether widely separated events and the Go¨del metric8 representing a model rotating uni-

216 Am. J. Phys., Vol. 72, No. 2, February 2004 Richard J. Cook 216 description of the motion of frame K from the point of view of the fiducial observers in K¯ . What then is it that makes the Lorentz transformation preferable to the Galilean transformation? The answer is that the Lorentz transformation is expressed in terms of the physical time and physical space coordinates in frame K instead of the coordinates (ct,x,y,z) used in the Galilean transformation, which are actually Fig. 3. Inertial reference frame labeled K ͑and Kញ ) with coordinates x, y, z physical times and physical distances in frame K¯ . This prop- ͑or xញ , yញ , zញ) moves out along the ¯x-axis of inertial frame K¯ with velocity erty of the Lorentz transformation is convenient ͑but not nec- vϭdx¯/dt¯. essary͒ for the calculation of physical quantities in the new frame. Let us show this explicitly. The metric ͑14͒, written in verse. There is, of course, a time coordinate t in these met- terms of the Galilean coordinates x␮ϭ(ct,x,y,z) reads rics, but we must understand that the equality of this time, ϭ v say tA tB for separated events A and B, does not mean that ds2ϭϪ͑1Ϫv2/c2͒c2 dt2ϩ2ͩ ͪ dx͑cdt͒ϩdx2ϩdy2 these events are simultaneous. ͑One wonders why such a c thing is called a ‘‘time coordinate’’ at all, because it does not ϩ 2 ͑ ͒ have the most fundamental features one associates with the dz , 15 ͒ ϭϪ word ‘‘time.’’ No doubt it is the conceptual problems sur- that is, the nonzero metric components are g00 (1 rounding the failure of simultaneity in such frames that mo- Ϫ 2 2 ϭ ϭ ϭ ϭ ϭ v /c ), g0x gx0 v/c, and gxx gyy gzz 1. Therefore tivates many authors to eliminate the metric terms containing the spatial metric, Eqs. ͑2͒ and ͑3͒,is g by transforming to a different reference frame in which oi 2 these terms are zero. But, if one wishes to work in a rotating dx dᐉ2ϭ ϩdy2ϩdz2. ͑16͒ frame ͑such as the frame rigidly attached to earth͒,orifone 1Ϫv2/c2 wishes to study certain ‘‘frame dragging’’ effects, these terms ͑ ͒ are necessarily present and give rise to such interesting ef- Equation 16 can be expressed in terms of the physical ញ ϭ ϭ ͱ Ϫ 2 2 ញ ϭ ϭ fects as Coriolis forces, the gravitomagnetic field, and the space differentials dx dx˜ dx/ 1 v /c , dy dy˜ dy, Sagnac effect.9,10 In fact, the experimental demonstration of and dzញϭdz˜ϭdz which, in this case, are exact differentials the Sagnac effect ͑the different light travel times for propa- that integrate to global physical space coordinates gation in opposite directions around a closed path in a rotat- ͒ 11 x ing frame using a ring-laser gyro or the global positioning xញ ϭ , ͑17a͒ system12 may be interpreted as a verification of the failure of ͱ1Ϫv2/c2 simultaneity in rotating reference frames, because such an ញ ϭ ͑ ͒ effect would be inconsistent with the invariant light speed c y y, 17b ͑ if a global physical time or even a global synchronous tem- zញϭz ͑17c͒ poral coordinate͒ existed. for frame Kញ (ϭK). We can also express the metric ͑15͒ in V. THE LORENTZ TRANSFORMATION frame Kញ in terms of the physical time differential in this ͑ ͒ Let us return now to the statement made in Sec. I that ‘‘In frame, Eq. 9 , which also is an exact differential that inte- general relativity the Galilean transformation is just as valid grates to the global physical time as the Lorentz transformation.’’ Specifically, consider the vx Galilean transformation ញtϭ˜t ϭͱ1Ϫv2/c2tϪ . ͑18͒ c2ͱ1Ϫv2/c2 xϭ¯xϪv¯t , ͑13a͒ If we substitute the transformations to physical variables ͑17͒ yϭ¯y, ͑13b͒ and ͑18͒ into the Galilean transformation ͑13͒, we obtain the Lorentz transformation zϭ¯z, ͑13c͒ ¯xϪv¯t tϭ¯t , ͑13d͒ xញ ϭ , ͑19a͒ ͱ1Ϫv2/c2 from an inertial frame K¯ with metric, yញ ϭ¯y, ͑19b͒ ds2ϭϪc2dt¯2ϩdx¯ 2ϩdy¯ 2ϩdz¯2, ͑14͒ zញϭ¯z, ͑19c͒ to another inertial frame K as depicted in Fig. 3. The time ¯t ¯Ϫ 2 ¯ t v¯x/c is the time on synchronized clocks at rest in frame K, and ញtϭ . ͑19d͒ frame K moves in the positive direction along the ¯x axis at ͱ1Ϫv2/c2 speed vϭdx¯/dt¯ as measure by the FOs in K¯ . Observe that the Galilean transformation, far from being We sometimes hear that Galileo’s transformation ͑13͒ is wrong, is a fully correct kinematic description of the motion ͑ ‘‘wrong’’ and Einstein’s transformation the Lorentz transfor- of frame Kញ (ϭK) in terms of the space and time variables of mation͒ is ‘‘right.’’ But surely this statement cannot be cor- ¯ rect when general relativity allows arbitrary space–time co- frame K, and as soon as we express the Galilean transforma- ordinate transformations, and Eq. ͑13͒ is a perfectly valid tion in terms of the physical time ញt and physical space coor-

217 Am. J. Phys., Vol. 72, No. 2, February 2004 Richard J. Cook 217 dinates xញ , yញ , zញ of frame Kញ , it becomes the Lorentz transfor- dx˜ϭrd␪, ͑23b͒ mation. dy˜ϭr sin ␪ d␾, ͑23c͒

VI. THE UNIVERSAL LORENTZ dz˜ϭdr/ͱ1Ϫr /r, ͑23d͒ TRANSFORMATION s where we have taken the ˜z-axis in the outward radial direc- The local physical coordinates in different reference tion, the ˜x-axis in the d␪-direction, and the ˜y-axis in the frames are related by a Lorentz transformation regardless of the motion of those frames. To see that this is so, consider d␾-direction. We think of cdt˜, dx˜, dy˜, dz˜ as finite but two arbitrary reference frames ͑two sets of fiducial observers small Minkowski coordinates measured from the given fidu- in relative motion͒ with space–time coordinates x␮ and y ␮ cial observer. Clearly the Schwarzschild metric ͑22͒ takes the connected by the coordinate transformation y ␮ϭy ␮(x␯)or form ds2ϭϪc2 dt˜2ϩdx˜ 2ϩdy˜ 2ϩdz˜2 in these local coordi- its inverse x␮ϭx␮(y ␯). At a particular fiducial observer X in nates. the x-frame, we construct physical coordinates dx˜ ␮ Now consider freely falling fiducial observers ͑a different reference frame͒ who fall radially inward from rest at r ϭ(cdt˜,dx˜ 1,dx˜ 2,dx˜ 3) for which the metric ͑8͒ at this point 2 ␣ ␤ ϭϱ. For this initial condition, the time equation of motion takes the Minkowski form ds ϭ␩␣␤ dx˜ dx˜ . We think of ␮ derived from the metric ͑22͒ reads dx˜ as finite but small coordinate values covering a limited neighborhood about observer X. At the same event in the rs dt y-frame we similarly construct local physical coordinates ͩ 1Ϫ ͪ ϭ1, ͑24͒ dy˜ ␮ measured from the fiducial observer Y of that frame r dt¯ who, at the instant under consideration, is at the same place ¯ as observer X. The metric at observer Y is also Minkowskian where dt is the differential of proper time at the falling observer. The radial equation of motion ͑once integrated͒ is in the local physical coordinates of this observer, ds2 ␣ ␤ ϭ␩␣␤ dy˜ dy˜ , and there may be a gravitational field in 2 dr rsc 2GM either or both of these frames. ϭϪͱ ϭϪͱ . ͑25͒ The overall transformation ͑from local physical coordi- dt¯ r r nates dx˜ ␮ to coordinates x␣, then to coordinates y ␯, and ␤ Fortuitously, this relativistic equation is the same as the cor- finally to local coordinates dy˜ ) takes the Minkowski tensor ͓ ␩ responding Newtonian equation. If readers are unfamiliar ␮␯ into the same Minkowski tensor and, therefore, can only ͑ ͒ ͑ ͒ ⌳␣ with the derivation of Eqs. 24 and 25 , they may consult be a Lorentz transformation ␤ : Ref. 2 where these results are simply derived as Eqs. ͑19͒ ␣ ␣ ␤ ͑ ͒ ͔ ͑ ͒ ͑ ͒ ͑ ͒ dy˜ ϭ⌳␤ dx˜ , ͑20͒ and 32 on pp. 3–22. If we use Eqs. 23a , 23d , and 24 in Eq. ͑25͒, we find that the physical velocity of the fall is with ␮ ␯ 2 ␩ ϭ␩ ⌳ ⌳ ͑ ͒ dz˜ rsc ␣␤ ␮␯ ␣ ␤ . 21 vϭ ϭϪͱ . ͑26͒ Hence, the differential Lorentz transformation ͑20͒ is not dt˜ r limited to inertial frames, but applies to arbitrary spacetime ͑ ͒ ͑ Equation 26 is the velocity that enters the differential Lor- coordinate transformations. It applies to accelerating frames entz transformation and frames in arbitrarily strong gravitational fields.͒ It follows that all of the tensors of special relativity in Minkowski dt˜Ϫv dz˜/c2 coordinates, such as the electromagnetic field tensor F␮␯ dtនϭ , ͑27a͒ ͱ Ϫ 2 2 (F01ϭE1, F02ϭE2, F03ϭE3, F12ϭB3, F23ϭB1, F31ϭB2, 1 v /c ␣␤ϭϪ ␤␣ ␮ϭ ␮ ␶ F F ), the four-momentum P mdx˜ /d , the dxន ϭdx˜, ͑27b͒ stress-energy tensor T␮␯, transform as in special relativity between local physical reference frames. dyន ϭdy˜, ͑27c͒ A. Example: Transformation to a falling reference dz˜Ϫv dt˜ frame in Schwarzschild space dzនϭ , ͑27d͒ ͱ Ϫ 2 2 Start with the static Schwarzschild metric, 1 v /c dr2 to the local physical coordinates (ctន, dxន , dyន , dzន) of the 2ϭϪ͑ Ϫ ͒ 2 2ϩ ds 1 rs /r c dt Ϫ falling fiducial observer. Notice that the physical fall velocity 1 rs /r ͑26͒ approaches c as r approaches the Schwarzschild radius ϩr2͑d␪2ϩsin2 ␪ d␾2͒, ͑22͒ Ͻ ͑ ͒ rs , and, for r rs , the Lorentz transformation 27 fails be- describing a nonrotating black hole of Schwarzschild radius cause there can be no fiducial observers at rest in Schwarzs- ϭ 2 child coordinates (rϭconstant) at these values of r. rs 2GM/c in the Schwarzschild reference frame with coordinates (ct,r,␪,␾), and transform to a frame that falls ra- As an example we note that, exactly as in special relativ- dially inward from rest at infinity. For a fiducial observer at ity, the differential Lorentz transformation leads to the veloc- rest in Schwarzschild coordinates (r,␪,␾ϭconstants), the ity transformation local physical coordinate differentials are ˜uxͱ1Ϫv2/c2 ន xϭ ͑ ͒ ˜ϭͱ Ϫ ͑ ͒ u Ϫ z 2 , 28a dt 1 rs /rdt, 23a 1 v˜u /c

218 Am. J. Phys., Vol. 72, No. 2, February 2004 Richard J. Cook 218 ˜uyͱ1Ϫv2/c2 ds2ϭg ͑dx0͒2ϩ2g dxi dx0,ϩg dxi dxj ͑A1͒ ន yϭ ͑ ͒ 00 0i ij u z 2 , 28b 1Ϫv˜u /c as follows. ͑This derivation is an abbreviation of a proof ͒ ˜uzϪv found in Ref. 4, pp. 233–236. Light propagating from A to uន zϭ , ͑28c͒ B does so in coordinate time dx0 determined by the null 1Ϫv˜uz/c2 out condition ds2ϭ0: iϵ i ˜ from the Schwarzschild physical components ˜u dx˜ /dt to ͑ 0 ͒2ϩ i 0 ϩ i jϭ ͑ ͒ g00 dxout 2g0i dx dxout gij dx dx 0. A2 the physical velocity components uន iϵdxន i/dtន in the freely falling frame. The solution of this quadratic equation is Similarly, the physical components of the electric and Ϫg dxiϪͱ͑g g Ϫg g ͒dxi dxj magnetic fields, charge, and current densities, and the com- 0 ϭ 0i 0i 0 j 00 ij ͑ ͒ dxout . A3 ponents of the stress-energy tensor all transform under the g00 Lorentz transformation ͑27͒ as they do in special relativity, On the return path light travels the displacement Ϫdxi and so long as we use local physical coordinates in the two takes coordinate time frames of interest. g dxiϪͱ͑g g Ϫg g ͒dxi dxj 0 ϭ 0i 0i 0 j 00 ij ͑ ͒ VII. CONCLUSION dxback . A4 g00 The above arguments attempt to make it clear that an es- The total coordinate time out and back is dx0ϭdx0 sential difference between the special and general theories of out ϩ 0 relativity is that, in the former, there exist global physical dxback , and the proper time evolved on the clock at A in coordinates ͑the Minkowski coordinates͒ but, in the latter this time is physical coordinates ͑coordinates with direct metrical sig- ͱϪg dx0 2 g g nificance and a Minkowski metric͒ exist only in the imme- ␶ ϭ 00 ϭ ͱͩ Ϫ 0i ojͪ i j ͑ ͒ d A gij dx dx . A5 diate neighborhood of each fiducial observer. But aside from c c g00 this essential difference, the Lorentz transformation still ap- Therefore, the local radar distance dᐉϭcd␶ /2 is the radical plies in general relativity for transformations between local A in Eq. ͑A5͒, and the spatial metric reads physical coordinate frames in arbitrary relative motion and in ᐉ2ϭ␥ i j ͑ ͒ arbitrary gravitational fields. The differential Lorentz trans- d ij dx dx , A6 formation is not limited to local inertial reference frames, and by using local physical coordinates, students transfer es- with spatial metric tensor sentially all they have learned in special relativity of the g g ␥ ϭ Ϫ 0i oj ͑ ͒ transformation properties of particles and fields to the ij gij . A7 broader context of general relativity ͑and they are not fooled g00 into thinking that the more general coordinate markers al- lowed in general relativity in any way change the relations 1The quote is taken from C. W. Misner, K. S. Thorn, and J. A. Wheeler, between physical quantities expressed in the Lorentz trans- Gravitation ͑W. H. Freeman and Company, San Francisco, CA, 1973͒,p. formation͒. 5. It was pieced together from ‘‘Einstein’s Autobiography,’’ in Albert Ein- stein Philosopher-Scientist, edited by P. A. Schilpp ͑Library of Living Recently the physics community has witnessed the publi- Philosophers, Evanston, IL, 1949͒, pp. 65–67. cation of a truly outstanding undergraduate level textbook on 2E. F. Taylor and J. A. Wheeler, Spacetime Physics ͑W. H. Freeman and general relativity by James B. Hartle. The local physical co- Company, New York, 1992͒, Chap. 2, pp. 39–40. ordinates discussed in this paper are components on a local 3A. Einstein, The Meaning of Relativity, 5th ed. ͑Princeton University ‘‘orthonormal bases’’ in Hartle’s more elegant notation.13 Press, Princeton, NJ, 1955͒, pp. 27–28. 4 ͑ The utility of working with local physical coordinates ͑the L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields Addison- ͒ Wesley, Reading, MA, 1971͒, Chap. 11, pp. 290–295. subject of this paper is emphasized, in Hartle’s words, by 5J. R. Oppenheimer and H. Snyder, ‘‘On continued gravitational contrac- the recommendation that we should ‘‘calculate in coordinate tion,’’ Phys. Rev. 56, 455–459 ͑1939͒. bases and interpret the result in orthonormal bases,’’ and our 6R. P. Kerr, ‘‘Gravitational field of a spinning mass as an example of alge- observation that transformations between local physical co- braically special metrics,’’ Phys. Rev. Lett. 11, 237–238 ͑1963͒. ordinate frames are Lorentz transformations is equivalent to 7R. H. Boyer and R. W. Lindquist, ‘‘Maximal analytic extension of the Kerr ͑ ͒ Hartle’s statement that transformations between orthonormal metric,’’ J. Math. Phys. 8, 265–281 1967 . 8K. Go¨del, ‘‘An example of a new type of cosmological solutions of Ein- bases are Lorentz transformations. Finally, it is worth noting stein’s field equations of gravitation,’’ Rev. Mod. Phys. 21,447–450 that components on an orthonormal basis have traditionally ͑1949͒. been called ‘‘physical components.’’ 9I. Ciufolini and J. A. Wheeler, Gravitation and Inertia ͑Princeton Univer- sity Press, Princeton, NJ, 1995͒, Chap. 6, pp. 315–374. APPENDIX: DERIVATION OF THE SPATIAL 10E. J. Post, ‘‘Sagnac effect,’’ Rev. Mod. Phys. 39, 475–494 ͑1967͒. 11F. Aronowitz, in Laser Applications, edited by M. Ross ͑Academic, New METRIC York, 1971͒, Vol. 1, pp. 134–189. ᐉ 12D. W. Allan, M. A. Weiss, and N. Ashby, ‘‘Around-the-world relativistic A formula for the Einstein length d of the coordinate ͑ ͒ i Sagnac experiment,’’ Science 228, 69–70 1985 . displacement dx between fiducial observers A and B is eas- 13J. B. Hartle, Gravity: An Introduction to Einstein’s General Relativity ily derived from the general line element, ͑Addison-Wesley, San Francisco, CA, 2003͒, Chap. 7, pp. 152–158.

219 Am. J. Phys., Vol. 72, No. 2, February 2004 Richard J. Cook 219