are called the contravariant components of the vector a and the a i are called the covariant components of the vector a.

Transcription

1 Non-Cartesan Coordnates The poston of an arbtrary pont P n space may be expressed n terms of the three curvlnear coordnates u 1,u,u 3. If r(u 1,u,u 3 ) s the poston vector of the pont P, at every such pont there exst two sets of bass vectors ê = r and ê = u u where the ê (subscrpts) are tangent to the coordnate curves (the axes) and the ê (superscrpts) are normal to the coordnate curves. Thus, we can wrte a vector n two ways (we change our summaton conventon so that we now sum over repeated ndces only f one s up and the other s down) a = a ê = a ê The a are called the contravarant components of the vector a and the a are called the covarant components of the vector a. For cartesan coordnate systems there s no dfference between these two sets of bass vectors, whch s why we were able to only use lower ndces. The ê are the covarant bass vectors and the ê contravarant bass vectors. are the In general, the vectors n each set are nether of unt length nor form an orthogonal bass. The sets ê and ê are, however, dual systems of vectors, so that We thus have ê ê j = δ j a ê = a j ê j ê = a j δ j = a a ê = a j ê j j ê = a j δ = a If we consder the components of hgher ran tensors n non- Cartesan coordnates, there are even more possbltes. For example, consder a second ran tensor T. Usng the outer product notaton we can wrte T n three dfferent ways T = T j ê ê j = T jê ê j = T j ê ê j Page 1

2 where T j, T j and T j are called the contravarant, mxed and covarant components of T respectvely. These three sets of quanttes form the components of the same tensor T, but refer to dfferent (tensor) bases made up from the bass vectors of the coordnate system. In Cartesan coordnates, all three sets are dentcal. (13) The Metrc Tensor Any partcular curvlnear coordnate system s completely characterzed (at each pont n space) by the nne quanttes g j = ê ê j Snce an nfntesmal vector dsplacement can be wrtten as dr = du ê we have these results (ds) = d r d r = du ê du j ê j = du du j ê ê j = g j du du j Snce (ds) s a scalar and the du are components of a contravarant vector, the quotent law says that the g j are the covarant components of a tensor g called the metrc tensor. The scalar product can be wrtten n four dfferent ways n terms of the metrc tensor a b = a ê b j ê j = ê ê j a b j = g j a b j = a ê b j ê j = ê ê j a b j = g j a b j = a ê b j ê j = ê ê j a b j = δ j a b j = a b These mply that = a ê b j ê j = ê ê j a b j = δ j a b j = a b g j b j = b and g j b j = b or that the covarant components of g can be used to lower n ndex and the contravarant components of g can be used to rase an ndex. In a smlar manner we can show that ê = g j ê j and ê = g j ê j Now snce ê and ê are dual vectors,.e., ê ê j = δ j Page

3 We have δ a = a = g j a j = g j g j a g j g j = δ In terms of matrx representatons ths says that G = g j, G = gj, I = δ j 1 GG = I G = G or the matrx formed from the covarant components s the nverse of the matrx formed from the contravarant components. The above relatons also gve the result g j = ê ê j = δ j components are dentcal Fnally, we have g = g = det[g j ] = g 1 g j g 3 ε j = ê 1 (ê ê 3 ) and dv = du 1 ê 1 (du ê du 3 ê 3 ) = ê 1 (ê ê 3 )du 1 du du 3 or dv = g du 1 du du 3 General Coordnate Transformatons and Tensors We now dscuss the concept of general transformatons from one coordnate system u 1,u,u 3 to another u' 1,u',u' 3. We can descrbe the coordnate transformaton usng the three equatons u' = u' (u 1,u,u 3 ) for = 1,,3, n whch the new coordnates u' can be arbtrary functons of the old ones u, rather than just represent lnear orthogonal transformatons (rotatons) of the coordnate axes. We shall also assume that the transformaton can be nverted, so that we can wrte the old coordnates n terms of the new ones as u = u (u' 1,u',u' 3 ) An example s the transformaton from sphercal polar to Cartesan coordnates gven by x = r snθ cosφ y = r snθ snφ z = r cosθ whch s clearly not a lnear transformaton. Page 3

4 The two sets of bass vectors n the new coordnate system u' 1,u',u' 3 are gven by ê' = r and ê' = u' u' Consderng the frst set, we have from the chan rule that ê j = r = u' r = u' ê' u' so that the bass vectors n the old and new coordnate systems are related by ê j = u' ê' Now, snce we can wrte any arbtrary vector n terms of ether bass as u' a = a' ê' = a j ê j = a j ê' t follows that the contravarant components of a vector must transform as a' = u' a j In fact, we use ths relaton as the defnng property that a set of quanttes a must have f they are to form the contravarant components of a vector. If we consder the second set of bass vectors, ê' = u', we have from the chan rule that x = u' u' x and smlarly for / y and / z. So the bass vectors n the old and new coordnate systems are related by ê j = u' ê' For any arbtrary vector a, a = a' ê' = a j ê j = a j u' ê' and so the covarant components of a vector must transform as a' = u' a j In a smlar way to that used n the contravarant case, we tae ths result as the defnng property that a set of quanttes a Page 4

5 must have f they are to form the covarant components of a vector. We now generalze these two laws for contravarant and covarant components of a vector to tensors of hgher ran. For example, the contravarant, mxed and covarant components, respectvely, of a second-order tensor must transform as follows: contravarant components T ' j = u' u' j T l u u l mxed components T ' j = u' u u l u' T j l covarant components T ' j = u u l u' u' T j l It s mportant to remember that these quanttes form the components of the same tensor T but refer to dfferent tensor bases made up from the bass vectors of the dfferent coordnate systems. For example, n terms of the contravarant components we may wrte T = T j ê ê j = T ' j ê' ê' j We can clearly go on to defne tensors of hgher order, wth arbtrary numbers of covarant (subscrpt) and contravarant (superscrpt) ndces, by demandng that ther components transform as follows: T ' j... lm...n = u' u' j u' u d u e f u u a u b u c u' l m u' u' T ab...c n de... f Usng the revsed summaton conventon (matched contravarant and covarant ndces summed over), the algebra of general tensors s completely analogous to that of Cartesan tensors dscussed earler. For example, as wth Cartesan coordnates, the Kronecer delta s a tensor, provded t s wrtten as a mxed tensor δ j, snce δ ' j = u' u l u u' δ j l = u' u = u' = δ u u' j u' j j where we have used the chan rule to prove the thrd equalty. Snce we showed earler that g j = ê ê j = δ j δ j can be consdered as the mxed components of the metrc tensor g. In the new (prmed) coordnate system, we have Page 5

6 Usng g' j = ê' ê' j ê j = u' ê' we have u u' j u' u ê' j = u' j ê' u' j = δ jê' j = ê' = u u' ê and smlarly for ê' j. Thus, we can wrte g' j = u u l u' u' ê j ê l = u u l g u' u' j l whch shows that the g j are ndeed the covarant components of a second-order tensor (the metrc tensor g). A smlar argument shows that the quanttes g j form the contravarant components of a second-order tensor, such that g' j = u' u' j g l u u l Earler we saw that the components g j and g j could be used to rase and lower ndces n contravarant and covarant vectors. Ths can be extended to tensors of arbtrary ran. In general, contracton of a tensor wth g j wll convert the contracted ndex from beng contravarant(superscrpt) to covarant (subscrpt),.e., t s lowered. Ths can be repeated for as many ndces as requred. For example, T j = g T j = g g jl T l Smlarly, contracton wth g j rases an ndex,.e., T j = g j T = g g jl T l That these two relatons are mutually consstent, can be shown by usng the relaton g g j = δ j Dervatves of bass vectors and Chrstoffel symbols In Cartesan coordnates, the bass vectors ê are constant and so ther dervatves wth respect to the coordnates vansh. In general coordnate systems, however, the bass vectors ê and ê are functons of the coordnates. In order that we may Page 6

7 dfferentate general tensors, we must therefore frst consder the dervatves of the bass vectors. Let us consder the dervatve ê Snce ths s tself a vector, t can be wrtten as a lnear combnaton of the bass vectors ê, = 1,,3. If we ntroduce the symbol Γ j have The coeffcent Γ j to denote the coeffcents n ths combnaton, we ê = Γ ê j Usng the recprocty relaton ê ê j = δ j gven (at each pont n space) by Furthermore, we then have ê ê s smply the th component of the vector ê ê ê Γ j = ê ê ( ) = Γ mê j ê m = Γ m j δ m ê ê = ê ê, these 7 numbers are = Γ j = 0 = ê ê + ê ê = Γ mê j ê m = Γ j ê = Γ j ê The symbol Γ j s called a Chrstoffel symbol (of the second nd), but despte appearances to the contrary, these quanttes do not form the components of a thrd-order tensor. In a new coordnate system Usng we get Γ ' j = ê' ê' u' j ê' = ul u' ê l and ê' = u' u n ên Page 7

8 Γ ' j = u' u n = u' u n = u' u n = u' u n ên ên u l u' j u' ê l u l u' j u' ê l + ul u' u l u' j u' ên ê l + u' u n u l u' j u' δ n l + u' u n u l + u' u' j u' u n = u' u l Ths result shows that the Γ j ê l u' j u l u' ên ê l u' j u l u m u' u l u m u' u' Γ n j lm ê u' j ên l u' m do not form the components of a thrd-order tensor because of the presence of the frst term on the rght-hand sde. We note that n Cartesan coordnates t s clear from the relaton Γ j = ê ê that Γ j = 0 for all values of the ndces, j and. In a gven coordnate system we can, n prncple, calculate the usng the relaton Γ j Γ j = ê ê In practce, however, t s often qucer to use an alternatve expresson, whch we now derve, for the Chrstoffel symbol n terms of the metrc tensor g j and ts dervatves wth respect to the coordnates. Frst, we note that the Chrstoffel symbol Γ j s symmetrc wth respect to the nterchange of ts two subscrpts and j. Ths s easly shown, snce ê = r = r = ê j u u u Ths gves Page 8

9 ê = Γ ê j = Γ j ê = ê j u Γ ê j ê l = Γ j ê ê l l l Γ j = Γ j To obtan an expresson for Γ j the dervatve g j u = ê u ê j + ê ê j u we then use g j = ê ê j and consder = Γ l l ê l ê j + ê Γ j ê l l l = Γ g lj + Γ j g l By cyclcally permutng the free ndces,j, n ths relaton we obtan two further equvalent relatons g j u l = Γ j l g l + Γ g jl g l = Γ j g l + Γ l j g l where we have used the symmetry propertes of both Γ j and g j. Contractng both sdes wth g m leads to the requred expresson for the Chrstoffel symbol n terms of the metrc tensor and ts dervatves, namely Γ m j = 1 g j gm + g g j u u Example: cylndrcal polar coordnates (u 1,u,u 3 ) = (ρ,φ,z) ds = dρ + ρ dφ + dz = g j du du j g 11 = 1,g = ρ,g 33 = 1,all others = 0 Ths mples that the only non-zero Chrstoffel symbols are Γ 1 = Γ 1 and Γ 1. These are gven by Γ 1 = Γ 1 = g g u 1 = 1 g g ρ = 1 ρ ρ ρ = 1 ρ 1 Γ = g11 g u = 1 g 1 g 11 ρ = 1 ρ ρ = ρ Alternatvely, we can use Page 9

10 ê 1 = ê ρ = cosφê x + snφê y ê = ê φ = snφê x + cosφê y ê 3 = ê z ê ρ φ = 1 ρ êφ ê 1 u = 1 u 1 ê Γ 1 = 1 u = 1 1 ρ = Γ 1 as expected. ê φ φ = ρê ρ ê u = u1 1 ê 1 Γ = u 1 = ρ Covarant dfferentaton For Cartesan tensors, we noted that the dervatve of a scalar s a (covarant) vector. Ths s also true for general tensors, as may be shown by consderng the dfferental of a scalar dφ = φ u du Snce the du are the components of a contravarant vector, and dφ s a scalar, we have by the quotent rule that the quanttes φ u must form the components of a covarant vector. It s straghtforward to show, however, that (unle n Cartesan coordnates) the dfferentaton of the components of a general tensor, other than a scalar, wth respect to the coordnates does not, n general, result n the components of another tensor. For example, n Cartesan coordnates, f the v are the contravarant components of a vector, then the quanttes v x j form the components of a second-order tensor. In general coordnates, however, ths s not the case. We may show ths drectly by consderng ' v = v' = u v' u' j u' j u = u u' v l u' j u u l = u u' u' j u l v l u + u u' j u' u u l vl Page 10

11 The presence of the second term on the rght-hand sde shows that the v do not form the components of a second-order tensor. Ths term arses because the "transformaton matrx" u' / changes wth poston n space. Ths s not true n Cartesan coordnates, for whch the second term vanshes, and v s a second-order tensor. We can, however, use the Chrstoffel symbols to defne a new covarant dervatve of the components of a tensor, whch does result n the components of another tensor. Let us frst consder the dervatve of a vector v wth respect to the coordnates. Wrtng the vector n terms of ts contravarant components v = v ê, we fnd v = v u ê ê j + v where the second term arses because, n general, the bass vectors ê are not constant (ths term vanshes n Cartesan coordnates). Usng the defnton of the Chrstoffel symbol we can wrte v = v u ê j + v Γ ê j Snce and are dummy ndces n the last term on the rght-hand sde, we may nterchange them to obtan v = v u ê j + v Γ j ê = v + v Γ j ê The reason for nterchangng the dummy ndces s that we may then factor out ê. The quantty n the bracet s called the covarant dervatve, for whch the standard notaton s v ; j = v v + Γ j where the semcolon denotes covarant dfferentaton; a smlar short-hand notaton also exsts for the smple partal dervatve, n whch a comma s used nstead of a semcolon. For example Page 11

12 so that v, j v ; j = v = v, j + Γ j v v = v ; j ê = v Usng the quotent rule, t s then clear that the v ; j (mxed) components of a second-order tensor. are the In Cartesan coordnates, all the Γ j are zero, and so the covarant dervatve reduces to the smple partal dervatve v Example: cylndrcal polar coordnates Contractng the defnton of the covarant dervatve we have v ; = v, + Γ v = v + Γ u v Usng the Chrstoffel symbols we wored out earler we fnd Γ 1 = Γ Γ 1 + Γ 3 13 = 1 ρ and v ; Γ Γ 3 1 = Γ 1 1 = Γ 31 + Γ + Γ Γ Γ 33 = 0 = 0 = vρ ρ + vφ φ + vz z + 1 ρ vρ = 1 (ρv ρ ) + vφ ρ ρ φ + vz z whch s the standard expresson for the dvergence of a vector feld n cylndrcal polar coordnates. So far we have consdered only the covarant dervatve of the contravarant components of a vector. The correspondng result for the covarant components v may be found n a smlar way, by consderng the dervatve of v = v ê. We obtan v ; j = v Γ j v Followng a smlar procedure we can obtan expressons for the covarant dervatves of hgher-order tensors. Expressng T n terms of ts contravarant components, we have Page 1

13 T jê j T ( ê j ) = T u = u u ê ê ê j + T j u ê j + T jê ê j u Usng the defnton of the Chrstoffel symbols we can wrte j T T = u u ê ê j + T j Γ l ê l ê j + T j l ê Γ j ê l Interchangng dummy ndces and l n the second term and j and l n the thrd term on the rght-hand sde ths becomes j T T = u u + Γ lt lj + Γ j l T l ê ê j where the expresson n bracets s the requred covarant dervatve T j j T ; = u + Γ lt lj + Γ j l T l = T j, + Γ l T lj + Γ j l T l In a smlar way we can wrte the covarant dervatve of the mxed and covarant components. Summarzng we have j T ; = T j, + Γ l T lj + Γ j l T l T j; = T j, + Γ l T l l j Γ j T j; = T j, Γ l l T lj Γ j T l We note that the quanttes T j ;, T j; and T j; are the components of the same thrd-order tensor T wth respect to dfferent tensor bases,.e., T = T jê ; ê j ê = T j; ê ê j ê = T j; ê ê j ê We conclude by consderng the covarant dervatve of a scalar. The covarant dervatve dffers from the smple partal dervatve wth respect to the coordnates only because the bass vectors of the coordnate system change wth poston n space (hence for Cartesan coordnates there s no dfference). However, a scalar functon φ does not depend on the bass vectors at all, so ts covarant dervatve must be the same as ts partal dervatve,.e., φ ; j = φ = φ, j (17) Vector Operators n tensor form We now use tensor methods to obtan expressons for the grad, dv, curl and Laplacan that or vald n all coordnate systems. Gradent. The gradent of a scalar φ s smply gven by T l Page 13

14 φ = φ ; ê = φ u ê snce the covarant dervatve of a scalar s the same as ts partal dervatve. Dvergence. The dvergence of a vector feld v n a general coordnate system s gven by v = v ; Usng = v u + Γ v Γ = 1 g l gl u + g l u The last two terms cancel because g u l = 1 g l gl u g g l l u g = g l u l g = g l u l where n the frst equalty we have nterchanged the dummy ndces and l, and n the second equalty we have used the symmetry of the metrc tensor. We also have g u = g ggj j u Fnally, we get Γ = 1 g l gl u = 1 g g u = 1 g g u whch gves the result v = v ; = v + 1 g u g u v = 1 ( gv ) g u Laplacan. φ = 1 φ gg j g u Curl. The specal vector form of the curl of a vector feld exsts only n three dmensons. We therefore consder ts more general form, whch s also vald n hgher-dmensonal spaces. In a general space the operaton curl v s defned by (curl v) j = v ; j v j; whch s an antsymmetrc covarant tensor. The dfference of dervatves can be smplfed snce Page 14

15 v ; j v j; = v Γ l j v l v j + Γ l u j v l = v v j u usng the symmetry propertes of the Chrstoffel symbols. Thus, (curl v) j = v v j = v u, j v j, Absolute dervatves along curves We now consder the problem of calculatng the dervatve of a tensor along a curve r(t) parameterzed by some varable t. Let us begn by consderng the dervatve of a vector v along the curve. If we ntroduce an arbtrary coordnate system u wth bass vectors ê, = 1,,3, then we can wrte v = v ê, and we have dv dt = dv dê ê + v dt dt = dv ê ê + v du dt u dt where we have used the chan rule to rewrte the last term on the rght-hand sde. Now, usng the defnton of the Chrstoffel symbols we obtan dv dt = dv du ê + Γ j v ê j dt dt Interchangng the dummy ndces and j n the last term we get dv dt = dv dt + Γ du jv j dt ê The expresson n the bracets s called the absolute (or ntrnsc) dervatve of the components v along the curve r(t) and s usually denoted by δv dv δt dt + Γ du jv j dt so that dv dt = δv du ê = v ; δt dt u + Γ j = v v j du dt Smlarly, we can show that the absolute dervatve of the covarant components v of a vector s gven by δv δt v ; du dt ê = v ; du dt Page 15

16 and the absolute dervatves of the contravarant, mxed and covarant components of a second-order tensor T are δt j j du T ; δt dt δt j δt δt j T j; du dt du T j; δt dt The dervatve of T along the curve r(t) may then be wrtten n terms of, for example, ts contravarant components as j dt δt j du = ê ê j = T ; ê ê j dt δt dt (19) Geodescs As an example of the use of the absolute dervatve, we conclude our dscusson of tensors wth a short dscusson of geodescs. A geodesc n real three-dmensonal space s a straght lne, whch has two equvalent defnng propertes. Frst, t s the curve of shortest length between two ponts and, second, ts tangent vector always ponts along the same drecton (along the lne). Although we have explctly consdered only the famlar three dmensonal space n our dscussons, much of the mathematcal formalsm developed can easly be generalzed to more abstract spaces of hgher dmensonalty n whch the famlar deas of Eucldean geometry are no longer vald. It s often of nterest to fnd geodesc curves n such spaces by usng the propertes of straght lnes n Eucldean space that defne a geodesc. Consderaton of these more complcated space s left for a future semnar n general relatvty. Instead, we wll derve the equaton that a geodesc n Eucldean three dmensonal space(.e., a straght lne) must satsfy, n a suffcently general way that t may be appled wth lttle modfcaton, to fnd the equatons satsfed by geodescs n more abstract spaces. Let us consder a curve r(s), parameterzed by the arc length s from some pont on the curve, and choose as our defnng property for a geodesc that ts tangent vector t = d r ds Page 16

17 always ponts n the same drecton everywhere on the curve,.e., dt ds = 0 Ths s called parallel transport of the tangent vector,.e., the vector s always moved parallel to tself along the curve, whch s the same as ts drecton not changng for a straght lne. If we now ntroduce an arbtrary coordnate system u wth bass vectors ê, = 1,,3, then we can wrte t = t ê, and we have dt ds = t du ; ds ê = 0 Wrtng out the covarant dervatve, we obtan dt ds + Γ du jt j ds ê = 0 But snce t j = du j ds we fnd that the equaton satsfed by a geodesc s d u du j du + Γ ds j ds ds = 0 Example: cartesan coordnates All Chrstoffel symbols are zero. Therefore, the equatons of a geodesc are d x ds = 0, d y ds = 0, d z ds = 0 whch correspond to a straght lne. Example: cylndrcal polar coordnates The only non-zero Chrstoffel symbols are 1 Γ = ρ and Γ 1 The geodesc equatons are then = Γ 1 = 1 ρ Page 17

18 d u 1 ds + Γ d u 1 du ds + Γ 1 ds du 1 ds du ds = 0 d ρ ds du ds = 0 d φ ds + ρ ρ dφ ds d u 3 = 0 d z ds ds = 0 On the surface of a cylnder gven by ρ = constant we have dρ ds = 0 dφ ds = 0 d ρ ds = 0, d φ ds = 0, d z ds = 0 whch also corresponds to a straght lne. Thn f unrollng the cylnder. It s then just a plane! Example: sphercal polar coordnates The metrc tensor s g = 0 r r sn θ The non-zero Chrstoffel symbols are 1 Γ 1 = r, Γ 33 = r sn θ, Γ 33 = snθ cosθ Γ 1 = Γ 1 = 1, Γ = Γ 31 = 1 3 3, Γ 3 = Γ 3 = cotθ r r The correspondng geodesc equatons on the surface of the sphere r = constant are d r ds = 0 d θ snθ cosθ dφ ds d φ dφ dθ + cotθ ds ds ds = 0 whch correspond to the equatons of a great crcle! ds = 0 Page 18

19 Parallel Transport and the Remann Tensor If a vector s parallel transported along a curve, the geodesc equatons tells us how the vector components change durng the transport. d u du j du + Γ ds j ds ds = 0 It also can be shown that for a covarant vector feld A β we have ths result A β;µ;ν A β;ν;µ = R α βµν A α that s, n a general curved spacetme the covarant dervatves do not commute (order s mportant). In a Cartesan or flat space the dfference would be zero. Thus the fourth-ran tensor α, whch s called the Remann curvature tensor s a measure R βµν of the curvature of spacetme. It s gven by α R βµν α = Γ βµ,ν α + Γ βν,µ Γ σ βν Γ α σµ Γ σ α βµ Γ σν If a vector feld s parallel transported around a closed path n a curved spacetme, the vector components do not return to the same values at the end (as they would do n flat space). In fact, parallel transport around a parallelogram gves the result ΔA α = R βµν A β dξ ν dξ µ where the dξ µ represent the sdes of the parallelogram. Thus, once agan the Remann tensor serves as a measure of the curvature of spacetme. The second-ran Rcc tensor s defned by a contracton over the frst and last ndces of the Remann tensor α R βµ = R βµα In addton we defne the curvature scalar R by R = R β αβ β = R βα Ensten Feld Equaton for Metrc Coeffcents The gravtatonal feld equatons developed by Ensten are R µν 1 g R = 8πG µν T c 4 µν Page 19

20 where the ncluson of the Remann scalar term s necessary for energy-momentum conservaton. where T µν s a second-ran tensor that gves the energy-momentum content of spacetme. It represent 16 coupled dfferental equatons for the metrc coeffcents g µν. An alternatve form of these feld equatons orgnally proposed by Ensten but later dscarded by hm as hs worst mstae, s now comng bac nto favor. It contans the so-called cosmologcal constant Λ. R µν 1 g R + Λg = 8πG µν µν T c 4 µν It predcts the exstence of a repulsve gravtatonal force on a cosmologcal scale and s of nterest now that data seems to ndcated that the unversal expanson s acceleratng. Schwarzschld Soluton For a sphercally symmetrc pont mass at the orgn, the fled equatons are gven by (for r > 0) R µν 1 g R = 0 µν Schwarzschld solved these equatons n Hs soluton wrtten as the square of the spacetme nterval loos le ds = 1 GM r c dt where M s the central mass. dr 1 GM r r (dθ + sn θdφ ) Ths soluton accounts for bendng of lght around the sun, the advance of the perhelon of mercury, gravtatonal redshft, radar tme delays from sgnals bounced off of planets, precesson of spnnng satelltes n earth orbt and blac holes, where the radus r=gm s the radus of the event horzon or the boundary where nothng can escape the mass, even lght. Further wor n the General relatvty Semnar - Physcs 130 Page 0