How much weigh a pound in gravitational field 4 1 A.I.Nikishov ∗ 0 2 n I.E.Tamm Department of Theoretical Physics, a J P.N.Lebedev Physical Institute, Moscow, Russia 2 2 January 28, 2014 ] h p - n e Abstract g s. Theforceintheequationofmotionofaparticleshouldbeinaccordancewithenergy c conservationinaconstantgravitationalfield. Itturnsoutthatthisispossibleonlyifthe i s force is given by thechange of momentum perunitof coordinate (notproper)time. We y h discuss the consequences of this fact. In particular it turns out that the spring balance, p which keeps the body at rest in (strong) gravitation field, shows the finite value even [ at infinitely close approach to horizons of black holes, if they exist in Nature. 1 v There are several ways of introducing force in the equation of motion of a particle in gravi- 7 6 tational field. So M¨oller elaborates three forces ”to describe all aspects of nongravitational 9 interaction”, see 10.4 in [1]. Landau and Lifshitz define the gravitational force as 3-space 6 § . covariant derivative of momentum over (synchronized) proper time, see 88, Problem 1 in 1 § 0 [2]. Not all of these forces respect the energy conservation law. Those which do it are not 4 covariant. We consider some of these approaches. 1 : In a constant gravitational field the covariant component of momentum p0 is conserved: v (see 88 in [2]) i X § ar p0 = mqc12√−gvc0220, vα = √g00(dxcdαx−αgαdxα), gα = −gg00α0, v2 = γαβvαvβ. (1) For particle at rest mc2 p0 = mc2√g00, dp0 = 2√g00g00,αdxα. (2) Hence the coordinate nongravitational force, which adiabatically changes p0 is n.g. dp0 mc2 F α= dxα = 2√g00g00,α. (3) When only space coordinates are transformed, Fα is 3-vector and dp0 is an invariant and as such it can be integrated, see equation (10.116) in [1]. ∗ E-mail: [email protected] Next we assume that in a small region of space the gravitational force is directed against the axis 1. Then only dx1 is nonzero: mc2 mc2 g dp0 = 2√g00g00,1dx1 = 2 g0000,1g11 dl1, dl1 = √g11dx1. (4) p| | From here it is natural to assume that the 3-invariant nongravitational force, holding particle at rest in constant gravitational field, is the force measured by a spring balance: mc2g F = 00,1 . (5) 2 g00g11 p| | According to heuristic approach to gravity [3] one may expect that g00g11 = 1. In general | | relativity this is true only in linear approximation in isotropic coordinates in Schwarzschild solution, which has the form 2 0 2 1 2 2 2 3 2 ds = g00(dx ) +g11[(dx ) +(dx ) +(dx ) ]. (6) In any case g00g11 is always finite. Then it follows from (6) that the invariant force acting on | | particle at rest remains finite even when the particle is approaching the black hole horizon. This remains true also when g0α are nonzero because the Coriolis-type force is not acting on particle at rest. On the other hand according to eq. (2.2.6) in [4] when vα = 0 the gravitational force (and nongravitational one holding the particle at rest) is d2xα K = m γ aαaβ, aα = , α = 1,2.3. (7) q αβ ds2 For the metric (6) we find K = mc2 g00,α 1 (8) 2 g00g11 √g00 p| | Contrary to (5) this quantity grows unlimitedly at the approach to black hole horizon. The contradiction with (5) disappears if the derivative of momentum over the proper time is replaced by the derivative over coordinate time. The derivative over coordinate time naturally appears in Lagrange formalism, see 10.4 § in [1]. Indeed, using dxidxk L = mc(g )1/2, (9) ik − dt dt we have ∂L mcg u˜l dxk ∂L mcg u˜lu˜m p = = µl , u˜k = , = lm,µ . (10) µ ∂u˜µ − g u˜iu˜k dt ∂xµ − 2 g u˜iu˜k p ik p ik Then the Euler- Lagrange equations are dp ∂L mcg u˜lu˜m g. µ = = lm,µ =F . (11) dt ∂xµ − 2 g u˜iu˜k µ p ik g. g. n.g. For particle at rest u˜i = cδi and F agrees with (3).( for slowly moving particle F F ). 0 µ µ ≈ − Introducing a nongravitational force we have dp g. n.g. µ =F + F . (12) µ µ dt . Now we dwell briefly on Problem 1 in 88 in [2]. In the second edition of this book § (Moscow, 1948) the problem is treated for small values of velocity. The force is obtained in the form fα = ddpτα = mc2[−21gg0000,α +√g00(gβ,α −gα,β)vcβ], gα = −gg00α0. (13) Replacing the differentiation over τ by differentiation over t i.e. multiplying both sides of (13) by √g00 we get √g00fα =Fg.α= mc2[−12√g0g0,α +g00(gβ,α −gα,β)vcβ]. (14) 00 For particle at rest this agrees with (3). For arbitrary velocity the equation (3) in Problem 1 in 88 in [2] should agree with equations (10.75)-(10.77) in [1] up to a common factor, but § this is not seen. It is worthwhile to note that by itself the equivalence principal and the requirement of general covariance do not give the equation (3) or (5), which gives the weight of a body in constant gravitational field. So it is not a sin to try to find an algorithm for the metric, which is different from Einstein equations. In conclusion I thank V.I.Ritus and M.I.Zelnikov for stimulating discussions. 1 References 1.C. M¨oller, The Theory of Relativity, Clarendon Press, Oxford (1972). 2.L.D.Landau and E.M.Lifshitz, The classical theory of fields, Moscow, (1973) (in Russian). 3. H.Dehnen, H.H¨onl, and K.Westpfahl, Ann. der Phys. 6, 7 Folge, Band 6, Heft 7-8, S.670 (1960). 4.I.D.Novikov, V.P.Frolov, Physics of black holes, Moscow (1986) (in Russian).