arXiv:1803.08346v2 [gr-qc] 1 Apr 2018 enpbihdi e.2. Ref. in published been detail. re- in without derivations 1 and Ref. definitions of their equations peating and Schutz. intro- notations GR of of using one with level that as proficient minimum such is to the books, at ductory easy reader techniques is the and It that concepts provided middle. re- the is, depiction the by that ally in mistaken sheet profoundly sun rubber how the a demonstrate like of weight dipped space surrounding and massive a stretched as in is be, or that possibly intrinsically, can they non-turning are as sun ‘straight’ i.e., our around geodesics, planets of actually orbits bound that suggests limit. non-relativistic the in even the relativity, for of misleading alone conceptually essence space and curved incorrect factually in is geodesics commonly of a depiction that mathe- popularized GR of and understanding to physically precise, return nonetheless matically possess to already but wish who introductory, immense students I an an remind modestly, and over more poses concepts Far cosmos basic GR the scales. that of of perspective range understanding grand this our in any for equations with those nor be or not constant paper, shall that the We with in equations. concerned curvature field to Einstein contribute cosmologi- corresponding A further may radiation. constant and cal matter stress-energy four-dimensional from a derived with pseudo- a tensor a connection as in in curved manifold fol- is geodesics Riemannian which test-particles space-time null point-like relativistic or (4D) ideal time-like that low is (GR) ativity xetfrafwmnrcags hspprhsjust has paper this changes, minor few a for Except depiction misleading such of version simplistic most A rel- general of theory Einstein’s in element central A ewrs eea hoyo eaiiy rvtto,Schw Gravitation, relativity, of Geodesics. theory curvature, General middle Keywords: the relativi in of numbers: essence dipped PACS the sheet for rubber misleading and a mistaken as is represented resu body, as sun, planets the of orbits by geodesic of depiction popularized eaiitclmt hsdmntae tabsclvltec the level basic a at demonstrates This limit. relativistic time erca n ie oriaetm orsodt nunphys prope an a to mater for correspond equations coordinate-time for given orbit gravity any geodesic at and contrary, metric the mechanics On Newtonian limit. of sections conic edscobteutosi h cwrshl emtyo g of geometry Schwarzschild the in equations orbit Geodesic .INTRODUCTION I. n t niaeped-imnincneto ihspace. with connection pseudo-Riemannian intimate its and pc n pc-iegoeisi cwrshl geometry Schwarzschild in geodesics Space-Time and Space eateto hsc,TeCtoi nvriyo America, of University Catholic The Physics, of Department 1 hsIwl show will I Thus Dtd pi ,2018) 3, April (Dated: oez Resca Lorenzo aee with labeled 1.6 nRf .Teoeadol eta parameter central gravity ( only Eq. Eq. introduces and metric, at and Schwarzschild one in curvature arriving The GR Schutz, characterizes 1. that by Ref. discussed in (10.36) and derived are enn htalmti esrcomponents, tensor metric all black-hole that or meaning star mass non-rotating of spherical singularity a 1. outside Ref. uum of subsection corresponding in the outlined in procedure and the 8.1 be following Table can in by comparisons and units performed conversions general easily unit retain but may paper, this we transparency, greater of and ain,i sotncnein oepeste nge- in them express to setting convenient by often units ometrized is it lations, 3, Ref. in provided is ground- example. 1916 two for in Schwarzschild’s published papers of breaking centenary the cel- information ebrating technical and hystorical interesting Most ean nhne ytime-reversal, by unchanged coordinate-time, remains the of dependent ds h cwrshl coordinates, Schwarzschild The cwrshl ercdsrbssaetm ntevac- the in space-time describes metric Schwarzschild nodrt ipiyo hre Reutoso re- or equations GR shorten or simplify to order In 2 rshl erc pc-iecraue Space curvature, Space-time metric, arzschild I CWRSHL EDSC AND GEODESICS SCHWARZSCHILD II. G tn rmtecraueo pc produced space of curvature the from lting = = ∗ g r y vni h o-eaiitclimit. non-relativistic the in even ty, o etnsgaiainlcntn.Frsake For constant. gravitational Newton’s for 1 = − nrlt n rtclrl of role critical and entrality HI O-EAIITCLIMIT NON-RELATIVISTIC THEIR 2 µν pta umnfl fSchwarzschild of submanifold spatial r clgravitational ical (  dθ dx nrlrltvt euet ordinary to reduce relativity eneral a atce ntenon-relativistic the in particles ial 1 ) µ − 2 µ ytewihn fta massive that of weighing the by dx + 0 = c G orsodnl,acommonly a Correspondingly, ahntn C20064 DC Washington, ν 2 r 2 2 , M M r sin 1 ,  2 2 tteoii.Ta erci static, is metric That origin. the at r , S θ ( ,admti ieelement line metric and 3, cdt ( repulsion ≡ dφ ) c c G ) 2 2 2 1 + o h pe flight of speed the for 1 = , 2 ,i cwrshl radius, Schwarzschild is ), M.  ntenon- the in 1 relativistic x − t µ n h geometry the and , t c G ( = − → 2 2 t ,θ φ θ, r, ct, M r t .  g µν − 1 r in- are , ( dr ,also ), ) 2 (1) (2) + 2

For a material point-like test-particle of vanishing mass µ m > 0, we thus introduce the four-momentum p = 2 µ 4 ˜2 dx dr r 2 2 2 2M L m dτ , whose pseudo-norm c E˜ c + G . (9) dφ  ≃ L˜2  − r − r2  µ µ ν 2 2 pµp = gµν p p = m c (3) − In the non-relativistic Newtonian limit we must also consider low energies relative to mc2, implying that E˜2 √ 2 defines the invariant proper-time interval cdτ = ds . 1 << 1. We may thus rescale energy as | − Such dτ > 0 represents the ticking of an ideal clock− at- | tached to the material test-particle. 1 m(E˜2 1)c2, (10) The geodesic equation for momentum covariant com- E ≡ 2 − ponents, and assume that << mc2 in the non-relativistic |E| dp 1 ∂g limit. We then arrive to the gravitational Newtonian m β = να pν pα, (4) dτ 2 ∂xβ  orbit equation dr 2 2mr4 Mm L2 is derived by Schutz quite generally, arriving to Eq. + G . (11) (7.29) in Ref. 1. For the static Schwarzschild metric, dφ ≃ L2 E r − 2mr2  the time-component (β = 0) immediately provides con- This coincides with Eq. (3.12) in Ref. 4, for example. servation of energy, The Newtonian orbit Eq. (11) is readily integrated to yield r = r(φ), which represents (arcs of) ellipses for 0 rS dt p0 E/c mcE˜ = g00p = 1 mc = const. < 0, parabolae for = 0, and one branch of hyperbolae ≡− ≡− − − r  dτ Efor > 0, having theE center of force at r = 0 located at (5) theE focus inside that branch: see Fig. 5.2 in Ref. 4, for The spherical symmetry of the Schwarzschild metric example. leads to planar geodesic orbits, which we may thus as- sume to be equatorial, maintaining constant the polar an- π gle θ = 2 = const. Spherical symmetry and the geodesic III. GEODESIC ORBIT EQUATIONS IN SPACE Eq. (4) for the azimuthal angle φ, or β = 3 component, SUBMANIFOLDS OF SCHWARZSCHILD implies further conservation of angular momentum value METRIC as

φ 2 dφ A most natural way to consider proper space by itself pφ L mL˜ = gφφp = r m = const. (6) ≡ ≡ dτ in the Schwarzschild metric is to regard it as a three- dimensional (3D) submanifold at any given coordinate- Introducing these conserved quantities into the con- time, t, exploiting the fact that the full space-time metric servation of four-momentum pseudo-norm, Eq. (3), we is static. This leads to a submetric line element for xi = arrive at the geodesic equation of motion for radial dis- (r, θ, φ) spatial coordinates, also labeled with i = 1, 2, 3, tance, given by 2 ˜2 ˜2 dS2 = g dxidxj dr 2 ˜2 2 2M L G 2M L ij = c E c + G + , (7) −  dτ  − r − r2 c2 r r2 r 1 = 1 S (dr)2 + r2(dθ)2 + r2 sin2 θ(dφ)2. r as demonstrated by Schutz in Eq. (11.11) and illustrated  −  in Fig. 11.1 of Ref. 1. (12) We may further derive a geodesic orbit equation in Since spatial coordinates are in fact space-like only out- terms of the azimuthal angle, φ, rather than the proper side the Schwarzschild radius, we are only interested here 2 time, τ, by dividing both sides of Eq. (7) by (dφ/dτ) , in that region within our event horizon, having r > rS. as provided in Eq. (6). That yields In that region, dS2 > 0 represents the line element of a 3D positive-definite Riemannian submetric. dr 2 r4 2M L˜2 G 2M L˜2 = c2E˜2 c2+G + . (8) We can parametrize curves in that 3D spatial subman- dφ L˜2  − r − r2 c2 r r2  ifold with an affine parameter, λ, such that tangent vec- i tors V i = dx have a positive-definite norm The last term in Eq. (8) is responsible for the famous dλ i i j 2 GR correction to the perihelion precession of Mercury’s ViV = gij V V = C > 0. (13) orbit: cf. Ref. 1, pp. 287-291. The non-relativistic limit of the GR geodesic orbit Geodesic curves in the 3D spatial submanifold then obey equation is obtained by assuming that r >>rS and that the equation ˜ r >> L /c for non-relativistic velocities. Thus we may dV 1 ∂g k = ij V iV j . (14) correspondingly drop the last term in Eq. (8), yielding dλ 2 ∂xk 

3

Spherical symmetry leads again to planar geodesic We may consider a weak-field limit of the GR geodesic curves, which can be thus assumed to be equatorial, hav- orbit Eq. (18) in the spatial submanifold by assuming π ing θ = 2 = const. Spherical symmetry further leads asymptotically large distances. Thus, ignoring the last from Eq. (14) for k = 3 to a conservation law of the form term in Eq. (18), we obtain

2 4 2 dr r 2 2 G 2M L φ 2 dφ 2 C C 2 2 . (19) Vφ L = gφφV = r = const. (15) dφ  ≃ L  − c r − r  ≡ dλ We should now consider whether to attribute to the L If we try to compare that orbit Eq. (19) with the New- constant in Eq. (15) some angular momentum interpre- tonian orbit Eq. (11), we have to associate the positive 2 norm constant C with 2m s > 0, yielding tation, as we properly did in Eq. (6). Angular momen- E tum depends on velocity, which depends on time. In our 2 4 2 spatial submanifold, however, coordinate time is fixed, dr 2mr 2 s Mm L s E G . (20) and geodesics can be parameterized only in terms of an dφ ≃ L2 E − mc2  r − 2mr2  affine parameter, λ, ultimately proportional to their arc lengths. Nevertheless, we can always reparameterize λ By comparing signs relative to that of the familiar ef- by multiplication with any arbitrary constant, including fective centrifugal repulsive potential, we may then con- one that gives to λ dimensions of time over mass, thus clude that the GR curvature of a Schwarzschild spatial making Eq. (15) at least dimensionally consistent with submanifold at any given time, t, corresponds to a weakly Eq. (6). We may further regard angular momentum as a repulsive gravitational potential, generator of spatial rotations, which coincide in both the space-time metric and its spatial sub-metric, since both 2 s Mm + E G > 0, (21) preserve the same spherical symmetry. From these per- mc2  r spectives, it is permissible to maintain a nomenclature or interpretation of the L constant in Eq. (15) as at least in a weak-field limit. Corresponding orbits can thus only proportional to some angular momentum value, which be arcs of hyperbolae. Furthermore, assuming a vanish- 2Es may be positive, negative, or zero. In fact, we are about ing prefactor, 0 < mc2 << 1, in the non-relativistic limit to derive a spatial geodesic orbit Eq. (18) that will elim- of the repulsive gravitational potential, those orbits be- inate any explicit appearance of the affine parameter λ, come virtually Euclidean straight lines, as expected for just as we did for the space-time geodesic orbit Eq. (8). an essentially flat spatial submanifold of Schwarzschild Ultimately, we will be able to obtain a spatial geodesic metric in the non-relativistic limit. That is incompatible, orbit Eq. (29) that depends on only two geometrical pa- for example, with bound orbits ( < 0) for the attractive E rameters, the orbit periastron, rp, and the Schwarzschild and much stronger gravitational potential, radius, rS, thus eliminating any need or appearance of Mm angular momentum, L, and energy, s, independently. G < 0, (22) Combining Eq. (15) with conservationE of norm, − r Eq. (13), we obtain of the physically correct Newtonian mechanics in the non- − relativistic limit of space-time GR, i.e., Eq. (11). r 1 dr 2 L2 i r φ S 2 In any case, regardless of whether one may or may not ViV = VrV + VφV = 1 + 2 = C .  − r   dλ r wish to consider weak-field and/or non-relativistic limits, (16) it is clear from direct comparison of the exact space-time This provides a geodesic equation for the radial coordi- geodesic orbit Eq. (8) and the exact spatial-submanifold nate in a spatial submanifold at any given time, t, namely geodesic orbit Eq. (18) that the former equation demands gravitational attraction exclusively, whereas the latter dr 2 r L2 r L2 = C2 C2 S + S . (17) equation invariably contains one term, namely its sec- dλ  − r − r2 r r2 ond, which corresponds to gravitational repulsion!

2 Dividing both sides of Eq. (17) by dφ , as provided  dλ  IV. NULL GEODESICS IN SPACE-TIME in Eq. (15), we eliminate any explicit appearance of affine parameter and obtain once again a geodesic orbit equa- The pseudo-Riemannian Schwarzschild metric in tion, expressed in terms of the azimuthal angle, φ. Since space-time also admits null geodesics, having ds2 = 0 L2 C2 has square-length units in Eq. (17), we have consis- in Eq. (1). Null geodesics are travelled exclusively by ex- tently actly massless (m 0) point-like test-particles, having a ≡ µ four-momentum pµ = dx with null pseudo-norm 2 4 2 2 dλ dr r 2 2 G 2M L G 2M L = C C + . (18) µ µ ν dφ L2  − c2 r − r2 c2 r r2  pµp = gµν p p =0. (23) 4

π Conservation of energy and angular momentum in θ = 2 = const. The corresponding line element in their geodesic equation lead to a radial component equa- Eq. (12) thus reduces to tion −1 rS dr 2 E2 L2 G 2M L2 dS2 = 1 (dr)2 + r2(dφ)2 (26) = + . (24)  − r  dλ  c2 − r2 c2 r r2 for the (r, φ) coordinates. Let us then embed the corre- The spherical symmetry of the Schwarzschild metric has sponding 2D submanifold in the ordinary 3D Euclidean led again to planar equatorial geodesics, maintaining θ = space, by associating r2 with (X2 + Y 2) and by defining, π = const. In general units, Eq. (24) coincides with 2 for r rS , Eq. (11.12) in Ref. 1. The corresponding geodesic orbit ≥ equation is 2 2 r Z =4rS 1 . (27) 2 r −  dr r4 E2 L2 G 2M L2 S = 2 2 2 + 2 2 . (25) dφ  L  c − r c r r  This is known as Flamm’s paraboloid of revolution about the Z axis.7 It derives from straightforward integration The last term in Eq. (25) is responsible for the GR de- after setting− dS2 = (dZ)2 + (dr)2 + r2(dφ)2 equal to dS2 flection of star-light grazing the sun, first famously con- in Eq. (26). Pictures and discussions related to Flamm’s firmed by observations in the eclipse of 1919: cf. Ref. 1, paraboloid are ubiquitously provided, including classic pp. 293-295. textbooks such as: Ref. 8, p. 837, Fig. 31.5; Ref. 9, pp. Estimating qualitatively the linear momentum of the r dr E 136-142; Ref. 10, p. 155, Fig. 6.10; Ref. 5, pp. 70-73, Fig. massless test-particle, or ‘photon,’ as p = dλ c , and 2 2 37; Ref. 1, p. 257, Fig. 10.1, even sketched on the cover E L∼ E its angular momentum as L r c , we have r2 c2 . of that book’s first Edition. Ironically, it is precisely a G ∼ ∼ For r >> c2 M, the last term in Eq. (24) or in Eq. (25) misguided interpretation of such correct pictures that is becomes asymptotically smaller than each of its two pre- mainly responsible for the confusion of popular rubber- ceding terms. One may neglect that last term in the sheet analogies of curved space. weak-field limit, which thus reduces to flat space-time. For L = 0, it is easy to show that the geodesic G Thus, for r >> c2 M, the massless (m 0) photon is no Eq. (17) leads to corresponding parabolic curves as given longer subject to any gravitational potential.≡ The pho- in Eq. (27). These are geodesic orbits for which the test- ton essentially moves along a straight Euclidean line as particle is radially directed, thus maintaining φ = const. in flat space-time. This contrasts with the much older For L = 0, it is possible to prove that the geodesic prediction of some star-light deflection by grazing the orbit Eq.6 (18) admits no circular orbit solutions with sun made by Cavendish (1784) and Soldner (1801) based r = r0 = const > rS. In fact, the only bound solution on a purely Newtonian description of light particles: cf. is the minimal circle with r = rS , which is the unsta- Ref. 5, Sec. 5.4, pp. 85-88, and Ref. 6. ble geodesic orbit that encircles the ‘throat’ of Flamm’s However one may regard or disregard qualitative weak- paraboloid at Z = 0. Other than that, all other circles field estimates, it is evident that the exact geodesic or- commonly drawn at various r = const > rS do not repre- bit Eq. (25) for the massless (m 0) photon is pre- sent geodesic orbits on Flamm’s 2D spatial submanifold ≡ cisely missing the attractive Newtonian gravitational po- for any arc length. The geometry of Flamm’s paraboloid tential that is instead prominent in the exact geodesic or- is in fact hyperbolic-like, with a negative intrinsic Gaus- G M bit Eq. (8) for a material point-like test-particle of mass sian curvature K = 2 3 that quickly vanishes for − c r m > 0, independently of how small or ‘vanishing’ that r >> rS , rapidly reaching the asymptotic limit of flat mass may be. 2D space.9 Therein, geodesic orbits become virtually Eu- If we consider instead the spatial submanifold of clidean straight lines, while still asymptotically bending Schwarzschild metric at any given coordinate-time, t, its away from the ‘throat’ of Flamm’s paraboloid. line element dS2 > 0, as given in Eq. (12), corresponds The geodesic orbit Eq. (18) admits only a single turn- to a positive-definite Riemannian metric within the event ing point, obtained by equating Eq. (18) to its minimum horizon, where r > rS . Therein, null geodesics cannot zero value. One can then express the orbit periastron as exist on the spatial submanifold, by definition. 2 2 2 L L rp = 2 , (28) C ≡ 2m s V. FLAMM’S PARABOLOID E for any rp > rS . One may then re-express the orbit There is in fact a rigorous procedure to depict the met- Eq. (18) exclusively in terms of rp and rS as ric of space alone as a submanifold of Schwarzschild met- ric at any given time, t, which can be summarized as fol- 2 4 3 lows. First of all, let us consider only a two-dimensional dr r rS r 2 = 2 2 r + rSr. (29) (2D) space to represent equatorial planes, maintaining dφ  rp − rp − 5

The expression of the geodesic orbit in Eq. (29) thus static metric. However, this simultaneity condition ex- depends on a single initial condition, specified by the pe- cludes any bona fide conservation of energy for geodesics riastron, rp. That originates from the fact that we can constrained to that spatial submanifold. Thus, the pos- reparameterize the affine parameter λ in Eq. (15) by mul- itive norm, C2, of tangent vectors to space-constrained tiplication with an arbitrary constant. Thus, only the ra- geodesics can only be formally associated with a ficti- 2 L 2 2 tio 2 in r ultimately matters, rather than the L and tious ‘space-invariant’ energy, 2m s > 0, resulting in the C p E C2 constants independently. This is already implicit in non-relativistic limit of Eq. (20). Only that allows to Eq. (18). Still, it is typically more instructive or conve- interpret the spatial geodesic orbit as a gravitational or- nient to study or solve spatial geodesic orbit equations in bit, but with the critical difference of having a weakly the form of Eq. (29), whether analytically or numerically. repulsive potential, as given in Eq. (21). It is possible to integrate the orbit Eq. (29) for L =0 From yet a third perspective, notice that the derivation following a standard procedure, involving a separation6 of geodesic equations in space-time, Eq. (7) and Eq. (8), of variables, followed by a functional inversion, which even in the non-relativistic limit, Eq. (9), requires consid- yields r = r(φ): see, for instance, p. 13 in Ref. 4. If we eration of both time-like, gtt, and space-like, grr, metric drop the last and most relativistic term in Eq. (29), we tensor components on an equal footing. On the contrary, can directly derive the analytic solution, representing a the derivation of geodesic equations in the spatial sub- branch of hyperbola. The convex angle comprised by its manifold, Eq. (17), Eq. (18), and Eq. (19), completely asymptotes varies from a maximum of π, corresponding excludes consideration of gtt, having required dt = 0 at to a straight line, to a minimum of 126.87 degrees. the outset. Relativistically however, if rp becomes of the order of All these related perspectives indicate that what is crit- rS , while still maintaining rp > rS , spatial geodesic orbits ically missing from the space-only descrition of GR curva- become much more complicated, especially at short dis- ture is the fundamental concept of relativity of time and tances. In that situation, the second and fourth terms in simultaneity, and its pseudo-Riemannian connection to Eq. (29), representing gravitational repulsion and attrac- the relativity of space. Of course, the Newtonian account tion, respectively, produce comparable and competing ef- of gravity disregards absolutely that very concept. Ac- fects. It is always possible, however, to obtain an analytic cording to Newton, time and simultaneity are presumed solution in terms of elliptic integrals. In fact, we have to be absolute, while gravity is supposed to act instantly generated and studied many such orbits numerically and at all distances. Of course we currently know that it analytically. Detailed results and discussions are beyond takes minutes or hours for the sun to influence gravi- the scope of this paper and will be reported elsewhere.11 tationally its planets while they move closer or further Remarkably, whether encircling the ‘throat’ of Flamm’s around it. Thus, in retrospect of course, one might won- paraboloid once or multiple times or not at all, the an- der why Newtonian mechanics and gravity, absolutely gle comprised between the orbit asymptotes turns out to defying such fundamental relativistic principle of space- be always concave, varying from a minimum of π, corre- time connection, could have worked so well from the be- sponding to a straight line in the non-relativistic limit, ginning. to a maximum of 2π when rp is appropriately close to rS. Our derivations and equations may help to figure that out more precisely. First of all, by deriving orbit equa- tions, we have avoided the relativity of time evolution VI. INTERPRETATION OF RESULTS to appear explicitly. Secondly, by considering weak- gravity regions, for example in the solar sytem, where G We wish of course to understand more precisely the r >> 2 M 1.5 km, we have limited ourselves to a c ∼ physical and mathematical origins of the discrepancy be- nearly flat space-time. In the geodesic orbit Eq. (11) for tween GR geodesic orbits in space-time and those in that space-time manifold, all three terms in curly brack- space alone, which strikingly persists even in the non- ets are comparably small and of the order of << mc2 |E| relativistic limit of GR: cf. Eq. (11) and Eq. (20), along typically bound planetary orbits. By comparison to Eq. (21) and Eq. (22). Clearly, the central element is their planetary velocities, the speed of light is practically that geodesics of material test-particles are time-like in infinite. We may thus better realize why GR principles GR space-time, as shown by the negative pseudo-norm of of space-time relativity and curvature are not altogether their tangent vectors, Eq. (3), whereas geodesics in the incompatible with Newtonian absolute principles in the spatial submanifold of Schwarzschild metric at any given appropriate non-relativitic limit. coordinate-time, t, are space-like, as shown by the posi- On the other hand, why the dt = 0 denial of the tive norm of their tangent vectors, Eq. (13), for r > rS . relativity of time and assertion of simultaneity dooms From another perspective, conservation of energy, from the outset a spatial submanifold consideration of the Eq. (5), is associated with invariance under time- Schwarzschild metric as a possible explanation of New- translations, which strictly applies only to geodesics in tonian gravity in the non-relativistic limit? Evidently, Schwarzschild space-time geometry. A proper spatial the order in which certain limits are taken does matter. submanifold of that can be most sensibly considered by One thing is to derive a correct geodesic orbit equation in assuming simultaneity, i.e., dt = 0, in Schwarzschild’s GR space-time, Eq. (8), and then take its non-relativistic 6 limit, Eq. (9). An altogether different matter is to con- Let us consider alternatively a fictitious 4D pseudo- sider a spatial submanifold of Schwarzschild metric which Riemannian manifold with metric eliminates from the outset the fundamental principle of relativity of space-time, simultaneity, and curvature as 2 µ ν their pseudo-Riemannian connection. In dealing with ds =gµν dx dx − ‘nearly flat’ space-time, as weak as curvature and gravity r 1 may be, limits must be taken in the appropriate order, = (cdt)2 + 1 S (dr)2+ beyond the zero-order of ‘absolutely flat’ space-time. −  − r  r2(dθ)2 + r2 sin2 θ(dφ)2. (33)

VII. ALTERNATIVE APPROACHES TO This differs from the Schwarzschild metric in that the SPATIAL CURVATURE time-like metric tensor component is assumed to be the same as it is in special relativity, i.e., gtt = 1, whereas − Of course over time there have been many different the 3D spatial submanifold at any given coordinate-time, accounts of spatial curvature, as it may relate to gravity t, maintains the same curvature as in Schwarzschild met- separately from full space-time curvature. Reasonably ric. Following the same procedures that we adopted ear- simple approaches have been discussed in Refs. 12–14, for lier produces now the following geodesic orbit equation: example. It may thus be useful to recast at least parts of such accounts in terms of the covariant geodesic orbit dr 2 2mr4 2 Mm L2 G M L2 formulation, based on Eq. (4), that I have consistently = E G + . developed throughout this paper. dφ L2 E−mc2  r −2mr2 c2 r mr2  Let us first consider an approximate 4D pseudo- (34) Riemannian manifold with metric Whether coincidentally or not, this geodesic or- bit Eq. (34) formally coincides with Eq. (18) that I previously obtained for the spatial submanifold of 2 µ ν ds =gµν dx dx Schwarzschild metric at any given coordinate-time, t. There is an important distinction, however. My previ- rS = 1 (cdt)2 + (dr)2+ ous geodesic orbit Eq. (18) had a space-like origin. Thus −  − r  I was bound to associate the positive constant C2 with 2 2 2 2 2 r (dθ) + r sin θ(dφ) . (30) a positive energy term 2m s > 0. The geodesic orbit Eq. (34) has a time-like origin.E Hence, its energy may This differs from the physically correct Schwarzschild also be negative, thus yielding a weakly attractiveEgrav- metric in that the 3D spatial submanifold at any given itational potential. However, unbound orbits, allowing coordinate-time, t, has been devoided of any curvature in r , require > 0, which brings us back to the prob- Eq. (30). Following the same procedures that I adopted lem→ of ∞ a weakly repulsiveE gravitational potential. Further earlier produces here the geodesic orbit equation: analysis shows that this attractive/repulsive switching of the potential can occur only for L = 0. For L = 0, by 6 2 4 −1 ˜2 equating the orbit Eq. (34) to its minimum zero value, dr r 2 ˜2 G 2M 2 L one can show that there is again only a single turning- = c E 1 2 c 2 . (31) dφ  L˜2   − c r  − − r  point, which is a periastron that satisfies

G 2 In the weak-field limit, r >> 2 M, the geodesic orbit L c r2 = (35) Eq. (31) reduces to p 2m E for any r > r . Then the corresponding energy must 2 p S dr 2mr4 Mm 2 Mm L2 again be positive, yielding a weakly repulsive gravita-E +G + E G , dφ  ≃ L2 E r mc2  r − 2mr2  tional potential corresponding to that of Eq. (21). (32) It is also possible to determine null geodesic orbits for where we have relabeled the energy according to Eq. (10). the ‘splittable space-time’ metric given in Eq. (33). I For << mc2, this geodesic orbit Eq. (32) coincides obtain with Schwarzschild|E| result in the non-relativistic New- tonian limit, Eq. (11), including the correct attractive dr 2 r4 E2 E2 G 2M L2 G 2M L2 gravitational potential, Eq. (22). This provided a major = + . breakthrough from both physical and historical perspec- dφ L2  c2 −  c2 c2 r − r2 c2 r r2  tives. It confirmed to Einstein that Newtonian gravity (36) basically derives from the equivalence principle and its Whether coincidentally or not, also this null geodesic association with the gravitational redshift, even without orbit Eq. (36) for the ‘splittable space-time’ metric full knowledge of Einstein field equations: cf. Chap. 18 structurally coincides with my space-like geodesic orbit of Ref. 15, for example. Eq. (18) for the Schwarzschild spatial submanifold, if we 7 let the positive constant C2 in Eq. (18) correspond to the gard: cf. Ref. 9, Sec. 7.4, pp. 114-117. What Riemann E2 did not know around 1854 was of course the theory of spe- positive constant c2 in Eq. (36) in this case. The null geodesic orbit Eq. (36) for ‘splittable space- cial relativity and the formulation of Minkowski’s space- time’ critically differs from the exact null geodesic or- time in particular. Einstein figured all that and how to bit Eq. (25) of Schwarzschild space-time metric, which put it together to formulate a general relativity theory for rules out the second term within the curly brackets of a pseudo-Riemannian space-time that accounts for grav- 3 Eq. (36). For light grazing the sun, our numerical in- ity as a manifestation of its curvature and connection. tegrations of the equivalent Eq. (18) and Eq. (36) indi- Now just about any intelligent person can at least in cate a ‘spatial bending’ of about half the total inward principle understand these ideas, among the grandest, if light deflection of 1.75 arc-seconds, which we recover for not the grandest, of all times. However, it still takes ma- the exact null geodesic orbit Eq. (25) of Schwarzschild jor training in both physics and mathematics to get to space-time metric.11 Our numerical integrations of null that point. Straightforward applications that I worked geodesics for the fictitious metric of Eq. (30) also indi- out in this article may have been evident to experts for cate a ‘gravitational red-shift bending’ of about half the more than a century. Still, this should remind us that total inward light deflection of 1.75 arc-seconds.11 Other ‘subtle is the Lord’ and that one should ‘make a theory authors may have reached similar conclusions by different as simple as nature allows, but not simpler.’ Accordingly, methods.12,15 ‘perceptual visualization’ of space alone as a rubber sheet It may seem curious that both time-like and null deformed by the weight of the sun in the middle is deceiv- geodesic orbits for the ‘splittable space-time’ metric es- ing and should not be used to suggest that planets merely sentially coincide with space-like geodesic orbits for the follow geodesic orbits in a curved space. What is needed proper spatial Schwarzschild sub-metric, for example. In is a much deeper physical and mathematical appreciation fact, it is possible to understand precisely all such matters of the relativistic connection between time and space, by keeping track explicitly of all gtt and grr factors and all which is ultimately a consequence of the constancy of the norm, pseudo-norm or null terms in the exact derivation speed of light in local freely-falling Lorentzian frames. of geodesic orbits for all metrics considered. A critical A computer-generated depiction of the popular rubber feature is that the product of gtt and grr is constant only sheet pinched and pulled down at its center has been fea- for the exact Schwarzschild space-time metric, but not for tured in a NOVA program on ‘Black Hole Apocalypse,’ the fictitious metrics of Eq. (30) and Eq. (33). Having broadcasted on January 10, 2018, on PBS.16 A commen- gttgrr = 1, as in Minkowski’s space-time, tells us that tary describing that depiction includes the following ex- − time and space bend inversely, relatively to each other, cerpts: ‘According to Einstein, the apple and the Space in Schwarzschild space-time. That reflects a central re- Station and the astronauts are all falling freely along a quirement of the equivalence principle, namely, that the curved path in space. And what makes that path curved? speed of light must remain a universal constant in any lo- The mass of the earth ... So, according to Einstein, the cal freely-falling Lorentzian frame, in curved space-time, mass of every object causes the space around it to curve just as it is in flat space-time. ... All objects in motion follow the curves in space. So, how does the earth move the apple without touching it? The earth curves space and the apple falls freely along VIII. SOME HISTORICAL REMARKS AND those curves. That, according to Einstein’s general the- CONCLUSIONS ory of relativity, is gravity: curved space. And that un- derstanding of gravity, that an object causes the space It was clear to Euclid, if not before, that the space around it to curve, leads directly to black holes.’ around us may or may not be absolutely flat. Thus Euclid Popular commentaries such as this may contradict not did not assert that he could mathematically demonstrate only Einstein’s genius, but also common sense: see pp. his fifth postulate on the basis of his other geometrical 59-61 and Fig 26 in Ref. 5, for example. Once space is postulates or elements. In subsequent centuries, many statically curved, any object that is released from any mathematicians and natural philosophers tried hard to given point in any given direction should follow the same either prove mathematically or demonstrate practically geodesic curve in such a-temporally curved space. Like- that space could or could not be perfectly flat or ‘Eu- wise, a particle constrained to move on the surface of a clidean.’ Gauss’s Theorema Egregium and his famous sphere should follow a geodesic great circle, regardless geodetic experiments with light rays led the way to math- of its encircling rate. Instead, we see that the object ematically rigorous theories of non-Euclidean geometries: follows vastly different trajectories, depending critically cf. Ref. 5, p. 61, pp. 160-163. The formulation of on the speed with which the object is initially released. Riemannian manifolds and geometry represents a crown- In fact, at the surface of the earth, the curvature of a- ing achievement and a momentous breakthrough in that temporal space is minuscule (K 1.7 x 10−27 cm−2) quest. Remarkably, Riemann himself attempted to apply and practically undetectable, as demonstrated≃ − by obser- his geometry to configurational spaces in Lagrangian me- vations from Euclid to Gauss, and up to the most ad- chanics including the influence of external gravitational vanced current technologies. Indeed, at the surface of the fields, but his efforts were doomed to failure in that re- earth, a particle approaching or traveling at the speed of 8 light practically follows a Euclidean straight line. The discussions, numerical results and technical support. I vastly different curvatures in trajectories of objects or acknowledge critical contributions of anonymous review- projectiles thrown at different speeds on or around the ers of the European Journal of Physics that I have in- surface of the earth, which are part of our every-day ex- cluded in my paper and further summarized in the Ap- perience, are consequence of space-time curvature, not pendix. of a-temporal space curvature. Except for this last but most important qualification, that central point is made repeatedly in the NOVA program on ‘Black Hole Apoc- alypse,’ which is outstanding.16 In any event, recall that even for absolutely flat or Eu- IX. APPENDIX clidean space the fictitious 4D pseudo-Riemannian met- ric of Eq. (30) for curved time correctly reduces to the non-relativistic Newtonian limit, thus correctly predict- In the formalism of modern differential geometry of ing all parabolic orbits that we ordinarily observe for Lorentzian manifolds, i.e., manifolds that are equipped projectiles thrown at non-relativistic speeds at the earth with a metric that is locally Minkowskian, static space- surface. On the contrary, the alternative fictitious 4D times, like Schwarzschild’s, are warped products of a 3D pseudo-Riemannian metric for exclusively curved space, Riemannian manifold as the base, modeling space, and Eq. (33), does not predict such correct parabolic orbits the real line as the fiber, modeling time. It is a prop- in the non-relativistic limit. erty of static space-times that geodesics, i.e., trajecto- Popular rubber-sheet funnel depictions of gravity and ries of freely falling test particles, do not typically cor- corresponding commentaries are often flawed from a rig- respond, i.e., they are not projected onto, geodesics of orous scientific perspective. I showed that by reviewing ‘fixed’ space. See, for instance, Chapter 7, pp. 204-209, how the Schwarzschild metric in relativistic space-time and Chapters 12-13, pp. 360-371, of Ref. 17. The main yields geodesic orbit equations in space for r = r(φ) that purpose of this paper has been to demonstrate this gen- reproduce all the correct conic sections for a Newtonian eral result by relatively simple means, showing by explicit attractive gravitational potential, G Mm , in the non- − r calculations for particular but important examples that relativistic limit. On the contrary, considering the curva- orbits of test particles moving along space-time geodesics ture of space exclusively, obtained as a proper submani- are different from orbits of points moving along geodesics fold of the Schwarzschild metric at any given coordinate- in the ‘a-temporal’ space of Schwarzschild’s geometry. time, t, yields geodesic orbit equations for r = r(φ) that correspond to a fictitious anti-gravitational poten- Further relevant references of greater or deeper histori- cal or technical interest include the following. Concerning 2Es Mm tial, + 2 G , weakly repulsive for material test- Riemann, an English translation of his famous inaugu-  mc  r particles in the non-relativistic limit. ral lecture can be found in Ref. 18. Extensive historical In the following Appendix, I briefly refer to a more ad- accounts on the developments of space, time and grav- vanced and general formulation of the differential geome- itation theory, beginning with Gauss’s, Riemann’s and try of Lorentzian manifolds that underlies explicit results Clifford’s ideas, are provided in Ref. 19, pp. 92-178, and obtained in this paper for the Schwarzschild geometry. I Ref. 20, Chapter 5. Clifford’s seminal contributions are also briefly refer to further literature of broader historical included in Ref. 21, pp. 546-569, and more extensively and general interest. in Ref. 22. Some conclusions similar to those of this paper have been reached recently from a different approach.23 In par- Acknowledgments ticular, it has been noted that geodesics with zero energy in 3D space coincide with null geodesics in 4D space-time. I am grateful to Drs. Rafael T. Eufrasio, Nicholas A. That is also the case in the limit of vanishing energies in Mecholsky, and Prof. Francesco Sorrentino for critical Eq. (18) and Eq. (25).

∗ Electronic address: [email protected]; solutions, Am. J. Phys. 84(7), 537-541, July 2016. URL: http://physics.cua.edu/people/faculty/homepage.cfm4 ;Fetter, A. L., Walecka, J. D., Theoretical Mechanics of Corresponding Author. Particles and Continua, McGraw Hill, 1980. 1 Schutz, B. F., A First Course in General Relativity, 2nd 5 Berry, M., Principles of Cosmology and Gravitation, Cam- Ed., Cambridge University Press, 2009. bridge University Press, 1976. 2 Lorenzo Resca, 2018, Eur. J. Phys. 39, 035602 (14pp), 6 Will, C. M., Henry Cavendish, Johann von Soldner, and Spacetime and spatial geodesic orbits in Schwarzschild ge- the deflection of light, Am. J. Phys. 56(5), 413-415, May ometry, https://doi.org/10.1088/1361-6404/aab12f . 1988. 3 Grøn, Ø., Celebrating the centenary of the Schwarzschild 7 Flamm, L., Physik Z. 17, 448, 1916. 9

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