Deriving the Schwarzschild solution from a local Newtonian limit
DDeriving the Schwarzschild solution from a local Newtonian limit
Markus P¨ossel ∗ Haus der Astronomie and Max Planck Institute for Astronomy, K¨onigstuhl 17, 69124 Heidelberg, Germany (Dated: September 9, 2020)The Schwarzschild metric is derived in a manner that does not require familiarity with the for-malism of differential geometry beyond the ability to interpret a general spacetime metric. As such,the derivation is suitable for an undergraduate course on general relativity. The derivation uses in-falling coordinates that are particularly well adapted to the situation, as well as Einstein’s equationin the simple form introduced by Baez and Bunn. That version of the vacuum Einstein equationscorresponds to requiring a particular local Newtonian limit: that, to first order, the deformation of a“test ball” of freely falling, initially-at-rest test particles is governed by the tidal forces of Newtoniangravity.
I. INTRODUCTION
The Schwarzschild solution plays a key role in teachingabout general relativity: It describes the simplest versionof a black hole. By Birkhoff’s theorem, it more gener-ally describes the gravitational field around any spheri-cal mass distribution, such as the Sun in our own Solarsystem. As one of two particularly simple, yet physicallyrelevant examples of a non-trivial metric (the other be-ing the FLRW spacetime of an expanding universe), it isparticularly well-suited for teaching about general tech-niques of “reading” and interpreting a spacetime metric.Consider undergraduate courses where students are in-troduced to selected concepts and results from generalrelativity without exposing them to the full mathemat-ical formalism. Such courses have the advantage of in-troducing students to one of the two great fundamentaltheories of 20th century physics early on (the other beingquantum mechanics); they also profit from subject mat-ter that meets with considerable interest from students. Using the terminology of Christensen and Moore, inthe “calculus only” approach pioneered by Taylor andWheeler, spacetime metrics are not derived, but takenas given, and the focus is on learning how to interpreta given spacetime metric. Similar presentations can befound in the first part of the “physics first” approach ex-emplified by Hartle’s text book, where the concepts ofthe metric and of geodesics are introduced early on, andtheir physical consequences explored, while the mathe-matics necessary for the Einstein equations is only intro-duced at a later stage.Whenever the approach involves an exploration of sim-ple metrics such as the Schwarzschild solution, but stopsshort of the formalism required for the full tensorial formof Einstein’s equations, access to a simple derivation ofthe Schwarzschild solution that does not make use of theadvanced formalism can be a considerable advantage.Simplified derivations of the Schwarzschild solu-tion have a long tradition within general relativityeducation, although specific simplifications have metwith criticism. This article presents a derivation whichrequires no deeper knowledge of the formalism of dif-ferential geometry beyond an understanding of how to interpret a given spacetime metric d s . The deriva-tion avoids the criticism levelled at attempts to derivethe Schwarzschild solution from the Einstein equivalenceprinciple in combination with a Newtonian limit, relyingas it does on a simplified version of the vacuum Einsteinequation.More specifically, I combine the restrictions imposedby the symmetry with the simple form of Einstein’sequations formulated by Baez and Bunn. That samestrategy was followed by Kassner in 2017, but in thistext, I use the “infalling coordinates” that are com-monly associated with the Gullstrand-Painlev´e form ofthe Schwarzschild metric, not the more commonSchwarzschild coordinates. That choice simplifies the ar-gument even further. In the end, what is required is nomore than the solution of an ordinary differential equa-tion for a single function, which yields to standard meth-ods, to obtain the desired result. II. COORDINATES ADAPTED TO SPHERICALSYMMETRY AND STATICITY
Assume that the spacetime we are interested in isspherically symmetric and static. In general relativity,a symmetry amounts to the possibility of being able tochoose coordinates that are adapted to the symmetry,at least within a restricted sub-region of the spacetimein question. That the spacetime is static is taken tomean that we can introduce a (non-unique) time coor-dinate t so that our description of spacetime geometrydoes not depend explicitly on t , and that space and timeare completely separate — in the coordinates adapted tothe symmetry, there are no “mixed terms” involving d t times the differential of a space coordinate in the metric.If we use t to slice our spacetime into three-dimensionalhyperplanes, each corresponding to “space at time t ,”then each of those 3-spaces has the same spatial geom-etry. A mixed term would indicate that those slices ofspace would need to be shifted relative to another in or-der to identify corresponding points. The mixed term’sabsence indicates that in adapted coordinates, there isno need for such an extra shift. In those coordinates, wecan talk about the 3-spaces as just “space,” without the a r X i v : . [ phy s i c s . e d - ph ] A ug need for specifying which of the slices we are referring to.In the case of spherical symmetry, we can introducespherical coordinates that are adapted to the symmetry:a radial coordinate r and the usual angular coordinates ϑ, ϕ , so that the spherical shell at constant r has thetotal area 4 πr . In consequence, the part of our metricinvolving d ϑ and d ϕ will have the standard form r (d ϑ + sin θ d ϕ ) ≡ r dΩ , (1)where the right-hand side defines dΩ , the infinitesimalsolid angle corresponding to each particular combinationof d ϑ and d ϕ .The radial coordinate slices space into spherical shells,each corresponding to a particular value r = const. Therotations around the origin, which are the symmetrytransformations of spherical symmetry, map each of thosespherical shells onto itself, and they leave all physicalquantities that do not explicitly depend on ϑ or ϕ invari-ant.In what follows, we will use the basic structures intro-duced in this way — the slices of simultaneous t , the ra-dial directions within each slice, the angular coordinatesspanning the symmetry–adapted spherical shells of area4 πr — as auxiliary structures for introducing spacetimecoordinates. For now, let us write down the shape thatour metric has by simple virtue of the spherical symme-try, the requirement that the spacetime be static, andthe adapted coordinates, namelyd s = − c F ( r )d t + G ( r )d r + r dΩ . (2)Students familiar with “reading” a spacetime metric willimmediately recognize the sign difference between theparts describing space and describing time that is charac-teristic for spacetime, and the speed of light c that givesus the correct physical dimensions. That there is no ex-plicit dependence on ϕ and ϑ in the remaining functions F and G is a direct consequence of spherical symmetry.That the factor in front of dΩ is r is a consequence ofour coordinate choice, with spherical angular coordinatesso that the area of a spherical surface of constant radius r is 4 πr . That there is no explicit dependence on t is oneconsequence of the spacetime being static; the absence ofthe mixed term d t · d r is another. We are left with twounknown functions F ( r ) and G ( r ). In the following, letus call t and r the static coordinates .Note that, since G ( r ) is as yet undefined, we have notyet chosen a specific physical meaning for the length mea-surements associated with our r coordinate. But becauseof the dΩ part, it is clear that whatever choice we make,the locally orthogonal lengths r · d ϑ and r · sin ϑ · d ϕ willhave the same physical interpretation as for the lengthmeasurement corresponding to d r . III. INFALLING OBSERVER COORDINATES
Now that we know what the radial directions are, ateach moment of time t , we follow Visser as well as Hamilton and Lisle in defining a family of radially in-falling observers. Observers in that family are in free fallalong the radial direction, starting out at rest at infinity:In mapping each observer’s radial progression in terms ofthe static coordinate time t , we adjust initial conditions,specifically: the choice of initial speed at some fixed time t , in just the right way that the radial coordinate speedgoes to zero for each observer in the same way as r → ∞ . It is true that talking about “infalling” observers al-ready reflects our expectation that our solution shoulddescribe the spacetime of a spherically symmetric mass.As we know from the Newtonian limit, such a mass at-tracts test particles in its vicinity. It should be noted,though, that all our calculations would also be compat-ible with the limit of no mass being present. In thatcase, “infalling” would be a misnomer, as our family ofobservers would merely hover in empty space at unchang-ing positions in r .We can imagine infinitesimal local coordinate systemsassociated with our observers — think of the observermapping out space and time by defining three orthogo-nal axes, and by measuring time with a co-moving clock.We assume all such little coordinate systems to be non-rotating — otherwise, we would break spherical symme-try, since rotation would locally pick out a plane of ro-tation that is distinguishable from the other planes. Theradial direction is a natural choice for the first space axisof those little free-falling systems. The other directions,we take to point to observers falling side by side with ourcoordinate-defining observer — and to remain pointed ata specific such other observer, once the choice of directionis made.We assume our infalling observers’ clocks to be syn-chronised at some fixed radius value r . By sphericalsymmetry, those clocks should then be synchronised at all values of r . Anything else would indicate direction-dependent differences for the infalling observers and theirclocks, after all. Hence, at any given static time t , all theinfalling observers who are at radius value r show thesame proper time T on the ideal clocks travelling alongwith them.Once our definition is complete, our static, sphericallysymmetric spacetime is filled with infalling observersfrom that family: Whenever we consider an event E ,there will be an observer from that family passing byat that time, at that location.Now, consider the coordinate speed of those infallingobservers. If we position ourselves at some constant ra-dius value r and watch the falling observers fly by, thenwe can express both their proper time rate and their co-ordinate speed in the r direction in terms of r and t . Wecan combine the two pieces of information to obtain therate of change in radial position r with proper time T forthose infalling observers. But since the initial conditionsfor those observers are the same, and since our spacetimeis, by assumption, static, the resulting function can onlydepend on r , and not explicitly on t . Let us rescale thatfunction with the speed of light to make it dimension-less, give it an overall minus sign to make it positive forinfalling particles, and call it β ( r ), β ( r ) ≡ − c d r d T ( r ) . (3)Recall from section II that we also still have the free-dom to decide on the physical meaning of r . We makethe choice of making d r the physical length measured byone of our infalling observers at the relevant location inspacetime, at constant time T . Via our angular coordi-nates, that implies that length measurements orthogonalto the radial direction, r · d ϑ and r · sin ϑ d ϕ inherit thesame physical interpretation.As a next step, we transform our metric (2) from thestatic form into the form appropriate for our coordinatechoice r and T . We do so by writing the static time coor-dinate as a function t ( T, r ) in terms of infalling observertime and radius value. In consequence,d t = ∂t∂T · d T + ∂t∂r · d r, (4)and our new metric now has the formd s = − c F ( r ) (cid:18) ∂t∂T (cid:19) d T − c F ( r ) (cid:18) ∂t∂T (cid:19) (cid:18) ∂t∂r (cid:19) d T d r + (cid:34) G ( r ) − c F ( r ) (cid:18) ∂t∂r (cid:19) (cid:35) d r + r dΩ . (5)At face value, this looks like we are moving the wrongway, away from simplification, since we now have morefunctions, and they depend on two variables instead ofone.But in fact, this new formulation paves the way foran even simpler form of the metric. Consider a specificevent, which happens at given radius value r . In a smallregion around that event, we will introduce a new coordi-nate ¯ r to parametrize the radial direction. We want thiscoordinate to be co-moving with our infalling observersat r ; each such observer then has a position ¯ r = const. that does not change over time.Key to our next step is that we know the metric forthe local length and time measurements made by anyone of our free-falling observers. By Einstein’s equiva-lence principle, the metric is that of special relativity.Locally, namely whenever tidal effects can be neglected,spacetime geometry for any non-rotating observer in freefall is indistinguishable from Minkowski spacetime as de-scribed by a local inertial system.Since we have chosen both the time coordinate T andthe physical meaning of the radial coordinate r so as toconform with the measurements of the local infalling ob-server, the transformation between ¯ r and r is particularlysimple: It has the form of a Galilei transformationd¯ r = d r + β ( r ) c d T. (6) In that way, as it should be by definition, radial coordi-nate differences at constant T are the same in both sys-tems, while for an observer at constant ¯ r, with d¯ r = 0,the relation between d r and d T is consistent with thedefinition of the function β ( r ) in (3).Are you surprised that this is not a Lorentz trans-formation, as one might expect from special relativity?Don’t be. We are not transforming from one local iner-tial coordinate system to another. The T is already thetime coordinate of the infalling observers, so both coor-dinate systems have the same definition of simultaneity,and time dilation plays no role in this particular transfor-mation. Also, we have chosen r intervals to correspond tolength measurements of the infalling observers, so thereis no Lorentz contraction, either. It is the consequence ofthese special choices that gives the relation (6) its simpleform.Last but not least, when we analyse specifically an in-finitesimal neighbourhood of the point r, ϑ, ϕ , let us makethe choice that directly at our point of interest, we make ¯ r coincide with r . Since before, we had only fixed the differ-ential d¯ r , we do have the remaining freedom of choosinga constant offset for ¯ r that yields the desired result.By Einstein’s equivalence principle, the metric in termsof the locally co-moving coordinates T, ¯ r, ϑ, ϕ is thespherical-coordinate version of the Minkowski metric,d s = − c d T + d¯ r + ¯ r dΩ . (7)This version can, of course, be obtained by taking themore familiar Cartesian-coordinate versiond s = − c d T + d X + d Y + d Z , (8)applying the definition of Cartesian coordinates X, Y, Z in terms of spherical coordinates ¯ r, ϑ, ϕx = ¯ r sin ϑ cos ϕ, y = ¯ r sin ϑ sin ϕ, z = ¯ r cos ϑ, (9)to express d X, d Y, d Z in terms of d¯ r, d ϑ, d ϕ , and substi-tute the result into (8).By noting that we have chosen ¯ r so that, at the specificspacetime event where we are evaluating the metric, ¯ r = r , while, for small radial coordinate shifts around thatlocation, we have the relation (6), we can now write downthe same metric in the coordinates T, r, ϑ, ϕ , namely asd s = − c (cid:2) − β ( r ) (cid:3) d T + 2 cβ ( r )d r d T + d r + r dΩ . (10)Since we can repeat that local procedure at any eventin our spacetime, this result is our general form of themetric, for all values of r . This, then is the promisedsimplification: By exploiting the symmetries of our solu-tions as well as the properties of infalling observers, wehave reduced our metric to a simple form with no morethan one unknown function of one variable, namely β ( r ).So far, what I have presented is no more than a long-form version of the initial steps of the derivation givenby Visser in his heuristic derivation of the Schwarzschildmetric. In the next section, we will deviate fromVisser’s derivation.
IV. β ( r ) FROM TIDAL DEFORMATIONS
In the previous section, we had exploited symmetriesand Einstein’s equivalence principle. In order to deter-mine β ( r ), we need to bring in additional information,namely the Einstein equations, which link the mattercontent with the geometry of spacetime. For our solu-tion, we only aim to describe the spacetime metric out-side whatever spherically-symmetric matter distributionresides in (or around) the center of our spherical sym-metry. That amounts to applying the vacuum Einsteinequations .More specifically, we use a particularly simple and in-tuitive form of the vacuum Einstein equations, which canbe found in a seminal article by Baez and Bunn: Con-sider a locally flat free-fall system around a specific event E , with a time coordinate τ , local proper time, where theevent we are studying corresponds to τ = 0. In that sys-tem, describe a small sphere of freely floating test parti-cles, which we shall call a test ball . The particles need tobe at rest relative to each other at τ = 0. Let the volumeof the test ball be V ( τ ). Then the vacuum version ofEinstein’s equations states thatd V d τ (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = 0 . (11)In words: If there is no matter or energy inside, the vol-ume of such a test ball remains constant in the first order(those were our initial conditions) and the second order(by eq. [11]).If you are familiar with Wheeler’s brief summary ofEinstein’s equations, “spacetime grips mass, telling ithow to move” and “mass grips spacetime, telling it howto curve”, you will immediately recognise that this is aspecific way for the structure of spacetime telling the testball particles how to move. The calculation later in thissection provides the second part: It will amount to using(11) to determine the structure of spacetime, namely thestill missing function β ( r ), and that is the way for mass,in this case: for the absence of mass, to tell spacetimehow to curve.Note that equation (11) also holds true in Newtoniangravity. So in a way, this version of Einstein’s equationcan be seen as a second-order extension of the usual Ein-stein equivalence principle: Ordinarily, the equivalenceprinciple is a statement about physics in the absence oftidal forces. Equation (11) adds to this that the lowest-order correction for tidal forces in a freely falling refer-ence frame is that specified by Newtonian gravity. Thismakes sense, since by going into a free-fall frame, andrestricting our attention to a small spacetime region, wehave automatically created a weak-gravity situation. Insuch a situation, tidal corrections are approximately thesame as those described by Newton. This argument canserve as a heuristic justification of (11).In 2017, Kassner made use of the Baez-Bunn form ofEinstein’s vacuum equation to derive the Schwarzschildsolution, starting from what we have encountered as the static form of the metric (2). We follow the same gen-eral recipe, but using the infalling coordinates introducedin section III, which makes our derivation even simpler.Consider five test particles in a small region of space.Let the motion of each be the same as for the local repre-sentative from our coordinate-defining family of infallingobservers. We take the central particle C to be at ra-dial coordinate value r = R at the time of the snapshotshown in Fig. 1. The other four are offset relative to thecentral particle: As described in the local inertial systemthat is co-moving with the central particle, one of theparticles is shifted by ∆ l upwards in the radial direction,another downward, while two of the particles are offsetorthogonally by the same distance. The ∆ l is meant to l l l lCD RL U FIG. 1. Five test particles in our spherically-symmetric space-time be infinitesimally small, so while Fig. 1 is of course show-ing a rather large ∆ l so as to display the geometry of thesituation more clearly, we will in the following only keepterms linear in ∆ l .Consider a generic particle, which moves as if it werepart of our coordinate-defining family of infalling ob-servers, and which at the time T is at r = r . By aTaylor expansion, that particle’s subsequent movementis given by r ( T ) = r + d r d T ( T ) · ∆ T + 12 d r d T ( T ) · ∆ T (12)where ∆ T ≡ T − T . We know from (3) that the deriva-tive in the linear term can be expressed in terms of β ( r );by the same token,d r d T = − c d β d T = − cβ (cid:48) d r d T = c β · β (cid:48) , (13)where the prime denotes differentiation of β with respectto its argument. Since, in the following, the product of β and its first derivative will occur quite often, let usintroduce the abbreviation B ( r ) ≡ β ( r ) · β (cid:48) ( r ) . (14)With these results, can rewrite the Taylor expansion (12)as r ( T ) = r − cβ ( r ) · ∆ T + 12 c B ( r ) · ∆ T . (15)In order to find r C ( T ) for our central particle, we simplyinsert r = R into that expression. If, on the other hand,we want to write down the time evolution for particles U and D , let us denote it by r U,D ( T ), we need to evaluatethe expression (15) at the initial location r = R ± ∆ l .Since ∆ l is small, we can make a Taylor expansion of β ( r )and its derivative around r = R , and neglect everythingbeyond the terms linear in ∆ l . The result is r U,D ( T ) = R ± ∆ l − c [ β ( R ) ± β (cid:48) ( R )∆ l ] ∆ T + c (cid:2) B ( R ) ± B (cid:48) ( R )∆ l (cid:3) ∆ T (16)In consequence, the distance between the upper and lowerparticle, d (cid:107) ( T ) ≡ r U ( T ) − r D ( T ) , changes over time as d (cid:107) ( T ) = 2∆ l (cid:20) − cβ (cid:48) ( R )∆ T + 12 c B (cid:48) ( R )∆ T (cid:21) . (17)Next, let us look at how the distance between the parti-cles L and R changes over time. The initial radial coor-dinate value for each of the particles is r ( T ) = (cid:112) R + ∆ l = R (cid:34) (cid:18) ∆ lR (cid:19) (cid:35) ≈ R, (18)that is, equal to R, as long as we neglect any terms thatare higher than linear in ∆ l . In consequence, r L,R ( t )is the same function as for our central particle, givenby eq. (15) with r = R . The transversal (in Fig. 1:horizontal) distance d ⊥ ( T ) between the particles L and R changes in proportion to the radius value, d ⊥ ( T ) = 2∆ l · r L ( T ) R = 2∆ (cid:20) − cβ ( R ) R ∆ T + c B ( R ) R ∆ T (cid:21) . (19)With these preparations, consider the vacuum Einsteinequation (11) for the volume of a test ball. Initially, ourparticles C, U, D, L, R define a circle, which is deformedto an ellipse. By demanding rotational symmetry aroundthe radial direction, we can construct the associated el-lipsoid, which is initially a spherical surface. That el-lipsoid has one axis in radial direction, whose length is d (cid:107) ( T ), and two axes that are transversal and each have the length d ⊥ ( t ). But that ellipsoid is not quite yet thetest ball we need. After all, the particles of the test ballneed to be at rest initially, at time T , in the co-movingsystem defined by the central particle C . Our definingparticles are not, as the terms linear in ∆ T in both (17)and (19) show, where the coefficients of ∆ T correspondto the particles’ initial velocities.In order to define our test ball, we need to considerparticles at the same location, undergoing the same ac-celeration, but which are initially at rest relative to thecentral particle C .We could go back to the drawing board, back to Fig. 1,make a more general Ansatz that includes initial veloc-ities which measure the divergence of the motion of ourtest ball particles from that of the infalling-observer par-ticles, and repeat our calculation while including thoseadditional velocity terms. But there is a short-cut. Theonly consequence of those additional velocity terms willbe to change the terms linear in ∆ T in equations (17)and (19). And we already know the end result: We willchoose the additional terms so as to cancel the terms lin-ear in ∆ T in the current versions of (17) and (19). Butby that reasoning, we can skip the explicit steps in be-tween, and write down the final result right away. Thetime evolution of the radial-direction diameter of our testball, let us call it L (cid:107) ( T ), must be the same as d (cid:107) ( T ), butwithout the term linear in ∆ T . Likewise, the time evo-lution L ⊥ ( T ) of the two transversal diameters must beequal to d ⊥ ( T ), but again without the term linear in ∆ T .The result is L (cid:107) ( T ) = 2∆ l (cid:20) c B (cid:48) ( R )∆ T (cid:21) (20) L ⊥ ( T ) = 2∆ l (cid:20) c B ( R ) R ∆ T (cid:21) . (21)Thus, our test ball volume is V ( T ) = π L (cid:107) ( T ) L ⊥ ( T ) (22)= 4 π l (cid:20) c (cid:18) B ( r ) r + B (cid:48) ( r )2 (cid:19) ∆ T (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = R (23)For the second time derivative of V ( T ) to vanish at thetime T = T , we must have B ( r ) r + B (cid:48) ( r )2 = 0 (24)for all values of r . This is readily solved by the standardmethod of separation of variables: We can rewrite (24)as d BB = − rr , (25)which is readily integrated to giveln( B ) = − ln( r ) + const. ⇒ ln( Br ) = C (cid:48) , (26)with a constant C (cid:48) , which upon taking the exponentialgives us Br = C, (27)with a constant C . Note that the constant C can be neg-ative — there is no reason the constant C (cid:48) needs to bereal; only our eventual function B ( r ) needs to be that,and it is clear that (27) satisfies the differential equa-tion (24) for any constant C , positive, zero, or negative.By (14), the solution (27) corresponds to the differentialequation β ( r ) β (cid:48) ( r ) = Cr (28)for our function β ; with another separation of variables,we can re-write this as β · d β = C d rr . (29)Both sides are readily integrated up; we can solve theresult for β ( r ) and obtain β ( r ) = (cid:114) − Cr + 2 D, (30)where D is the second integration constant, and where wehave chosen the proper sign, since we know that β ( r ) > r , the description provided by our solutionshould correspond to the results from Newtonian gravity.First of all, we note that our initial condition for theinfalling observers, which had those observers start outat zero speed at infinity, means that we must choose D =0. Then, as we would expect, β ( r ) for large values of r becomes very small, corresponding to small speeds. Butat slow speeds, time and length intervals as measured bythe infalling observer will become arbitrarily close to timeand length intervals as measured by an observer at restin our static coordinate system at constant r , using thestatic time coordinate t . As is usual, we identify thesecoordinates with those of an approximately Newtoniandescription. In that description, the radial velocity is v ( r ) = (cid:114) GMr , (31)which follows directly from energy conservation forthe sum of each observer’s kinetic and Newtonian-gravitational potential energy. This fixes the remainingintegration constant as C = − GMc , (32)and the final form of our function β ( r ) becomes β ( r ) = (cid:114) GMrc . (33) Inserting this result in (10), we obtain the metricd s = − c (cid:20) − GMrc (cid:21) d T +2 (cid:114) GMr d r d T +d r + r dΩ . (34)This is known as the Gullstrand-Painlev´e version of theSchwarzschild metric. A last transformation stepbrings us back to the traditional Schwarzschild form. Re-call our discussion in sec. II, leading up to the explicitlystatic form (2) of the metric? The main difference be-tween our current form and the static version is the mixedterm containing d r d T in (34). Everything else alreadyhas the required shape. Inserting the Ansatzd T = d t + ξ ( r )d r (35)into the metric (34), it is straightforward to see that themixed term vanishes iff our transformation isd T = d t + (cid:112) GM/rc (cid:0) − GMrc (cid:1) d r. (36)Substitute this into (34), and the result is the familiarform of the Schwarzschild metric in Schwarzschild’s orig-inal coordinates t, r, ϑ, ϕ ,d s = − c (cid:18) − GMc r (cid:19) d t + d r (cid:0) − GMc r (cid:1) + r dΩ . (37) V. CONCLUSION
Using coordinates adapted to the symmetries, we wereable to write down the spherically symmetric, staticspacetime metric. On this basis, and using the fam-ily of infalling observers that is characteristic for theGullstrand-Painlev´e solution, we wrote down the metricin the form (10), with a single unknown function β ( r ).From the simplified form (11) of the vacuum Einsteinequations, as applied to a test ball in free fall alongsideone of our family of observers, we were able to deter-mine β ( r ), up to two integration constants. By using theEinstein equation, we escape the restrictions imposed onsimplified derivations by Gruber et al. From the initial condition for our infalling observers,as well as from the Newtonian limit at large distancesfrom our center of symmetry, we were able to fix thevalues of the two intergration constants. Our derivationdoes not require knowledge of advanced mathematicalconcepts beyond the ability to properly interpret a givenmetric line element d s . Even our analysis of tidal ef-fects proceeds via a simple second-order Taylor expan-sion, leading to differential equations for β ( r ) that arereadily solved using two applications of the method ofseparation of variables.What is new about the derivation presented here isthe combination of the Baez-Bunn equations with theinfalling coordinates typical for the Gullstrand-Painlev´eform of the metric — this combination is what, in theend, makes our derivation particularly simple. In turn,this simplicity is what should make the derivation partic-ularly useful in the context of teaching general relativityin an undergraduate setting.The derivation proceeds close to the physics, and givesample opportunity to discuss interesting properties ofEinstein’s theory of gravity. Students who are presentedwith this derivation, either as a demonstration or as a(guided) exercise, will come to understand the way thatsymmetries determine the form of a metric, the deduc- tions that can be made from Einstein’s equivalence prin-ciple, and last but not least that we need to go beyondthe equivalence principle, and consider tidal forces, tocompletely define our solution. ACKNOWLEDGEMENTS
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