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On Isotropic Coordinates and Einstein's Gravitational Field July, 2006 PROGRESS IN PHYSICS Volume 3 On Isotropic Coordinates and Einstein’s Gravitational Field Stephen J. Crothers Queensland, Australia E-mail: [email protected] It is proved herein that the metric in the so-called “isotropic coordinates” for Einstein’s gravitational field is a particular case of an infinite class of equivalent metrics. Furthermore, the usual interpretation of the coordinates is erroneous, because in the usual form given in the literature, the alleged coordinate length dx2 + dy2 + dz2 is not a coordinate length. This arises from the fact that the geometrical relations between the components of the metric tensor are invariant and therefore bearp the same relations in the isotropic system as those of the metric in standard Schwarzschild coordinates. 1 Introduction interacting comparable masses. In fact, it is not even known if Einstein’s field equations admit of solutions for multi- Petrov [1] developed an algebraic classification of Einstein’s body configurations, as no existence theorem has even been field equations. Einstein’s field equations can be written as, adduced. It follows that there is no theoretical sense to concepts such as black hole binaries, or colliding or merging 1 Rαβ Rgαβ = κ Tαβ λgαβ , black holes, notwithstanding the all too common practice − 2 − of assuming them well-posed theoretical problems allegedly where κ is a constant, and λ the so-called cosmological substantiated by observations. constant. If Tαβ gαβ, the associated space is called an The metric for Einstein’s gravitational field in the usual Einstein space. Thus,∝ Einstein spaces include those described isotropic coordinates is, in relativistic units (c = G = 1), by partially degenerate metrics of this form. Consequently, 2 1 m such metrics become non-Einstein only when ds2 = − 2r dt2 m 2 − 1 + 2r (1a) g = det gαβ = 0 . m 4 k k 1 + dr2 + r2 dθ2 + sin2θ dϕ2 = − 2r A simple source is a spherically symmetric mass (a mass h i 2 island), without charge or angular momentum. A simple m 4 1 2r 2 m 2 2 2 source giving rise to a static gravitational field in vacuum, = − 2 dt 1 + dx + dy + dz , (1b) 1 + m − 2r where space is isotropic and homogeneous, constitutes a 2r Schwarzschild space. The associated field equations external having set r = x2 + y2 + z2. This metric describes to the simple source are a Schwarzschild space. p 1 By virtue of the factor (dx2 + dy2 + dz2) it is usual that R Rg = 0 , αβ − 2 αβ 0 6 r < is taken. However, this standard range on r is due entirely∞ to assumption, based upon the misconception or, more simply, that because 0 r < is defined on the usual Minkowski R = 0 . 6 αβ metric, this must also∞ hold for (1a) and (1b). Nothing could Thus, a Schwarzschild space is an Einstein space. There be further from the truth, as I shall now prove. are four types of Einstein spaces. The Schwarzschild space is a type 1 Einstein space. It gives rise to a spherically 2 Proof symmetric gravitational field. The simple source interacts with a “test” particle, which Consider the standard Minkowski metric, has no charge, no angular momentum, and effectively no ds2 = dt2 dx2 dy2 dz2 mass, or so little mass that its own gravitational field can be − − − ≡ neglected entirely. A similar concept is utilised in electro- dt2 dr2 r2(dθ2 + sin2 θdϕ2) , (2) dynamics in the notion of a “test” charge. ≡ − − 0 r < . The only solutions known for Einstein’s field equations 6 ∞ involve a single gravitating source interacting with a test The spatial components of this metric describe a sphere of particle. There are no known solutions for two or more radius r > 0, centred at r = 0. The quantity r is an Efcleeth- S. J. Crothers. On Isotropic Coordinates and Einstein’s Gravitational Field 7 Volume 3 PROGRESS IN PHYSICS July, 2006 ean∗ distance since Minkowski space is pseudo-Efcleethean. where the prime indicates d/dh. This gives, in the usual way, Now (2) is easily generalised [2] to 2 1 m 2 2 2 2 2 2 2 2 2h 2 ds = dt dr (r r ) (dθ + sin θ dϕ ) = (3a) ds = − 2 dt 0 1 + m − − − − 2h (7) (r r )2 = dt2 − 0 dr2 r r 2(dθ2 + sin2θ dϕ2) , (3b) m 4 − r r 2 − | − 0| 1 + dh2 + h2 dθ2 + sin2θ dϕ2 , | − 0| − 2h 2 2 2 2 2 2 = dt d r r0 r r0 (dθ + sin θ dϕ ) , (3c) h i − | − | − | − | from which the admissible form for h( r r ) and the value | − 0| 0 6 r r0 < . of the constant r0 must be rigorously ascertained from the | − | ∞ intrinsic geometrical properties of the metric itself. The spatial components of equations (3) describe a sphere Now the intrinsic geometry of the metric (2) is the same of radius R (r) = r r , centred at a point located any- c 0 on all the metrics given herein in terms of the spherical where on the 2-sphere| −r .| Only if r = 0 does (3) describe a 0 0 coordinates of Minkowski space, namely, the radius of cur- sphere centred at the origin of the coordinate system. With vature R in the space described by the metric is always respect to the underlying coordinate system of (3), R (r) is c c the square root of the coefficient of the angular terms ofthe the radial distance between the 2-spheres r = r and r = r . 0 0 metric and the proper radius R is always the integral of The usual practice is to supposedly generalise (2) as6 p the square root of the component containing the differential ds2 = A(r)dt2 B(r) dr2 + r2dθ2 + r2 sin2θ dϕ2 (4) element of the radius of curvature. Thus, on (2), − r to finally obtain (1a) in the standard way, with the assumption Rc(r) r, Rp(r) dr = r Rc(r) , that 0 r < on (2) must hold also on (4), and hence ≡ ≡ ≡ 6 Z0 on equations∞ (1). However, this assumption has never been proved by the theoreticians. The assumption is demonstrably and on (3), R (r) r r , false. Furthermore, this procedure does not produce a gene- c ≡ | − 0| ralised solution in terms of the parameter r, but instead a r r particular solution. | − 0 | Rp(r) dr = r r Rc(r) , Since (3) is a generalisation of (2), I use it to generalise ≡ | − 0| ≡ Z0 (4) to whereby it is clear that Rc(r) and Rp(r) are identical, owing ds2 = eν dt2 eμ dh2 + h2dθ2 + h2 sin2θ dϕ2 (5) − to the fact that the spatial coordinates of (2) and (3) are Efcleethean. h = h(r) = h( r r0 ), ν = ν (h(r)), μ = μ(h(r)) . | − | Now consider the general metric of the form Note that (5) can be written in the mixed form ds2 = A(r)dt2 B(r)dr2 C(r)(dθ2 + sin2θ dϕ2) (8) dh 2 − − ds2 = eν dt2 eμ dr2 +h2dθ2 +h2 sin2θdϕ2 , (6) − dr A, B, C > 0. " # from which the particular form (4) usually used is recovered In this case, if h( r r ) = r. However, no particular form for h( r r ) | − 0| | − 0| should be pre-empted. Doing so, in the routine fashion of Rc(r) = C(r),Rp(r) = B(r) dr . the majority of the relativists, produces only a particular p Z p solution in terms of the Minkowski r, with all the erroneous I remark that although (8) is mathematically valid, it assumptions associated therewith. is misleading. In the cases of (2) and (3), the respective Now (5) must satisfy the energy-momentum tensor equa- metrics are given in terms of the radius of curvature and its tions for the simple, static, vacuum field: differential element. This is not the case in (8) where the first 2 and second components are in terms of the parameter r of μ μ0 μ0ν0 μ0 + ν0 0 = e− + + the radius of curvature, not the radius of curvature itself. I 4 2 h therefore write (8) in terms of only the radius of curvature 2 μ μ00 ν00 ν0 μ0 + ν0 on (8), thus 0 = e− + + + 2 2 r 2h 2 2 2 2 ds = A∗ C(r) dt B∗ C(r) d C(r) μ μ0 2μ0 − − 0 = e− μ00 + + , (9a) 4 h p C(rp)(dθ2 + sinp2θ dϕ2) , − ∗Due to Efcleethees, incorrectly Euclid, so the geometry is rightly Efcleethean. A∗,B∗, C > 0 . 8 S. J. Crothers. On Isotropic Coordinates and Einstein’s Gravitational Field July, 2006 PROGRESS IN PHYSICS Volume 3 m 4 m 4 Note that (9a) can be written as, 1 + dh2 h2 1 + dθ2 + sin2θ dϕ2 , (11) 2 − 2h − 2h 2 2 d C(r) 2 ds =A∗ C(r) dt B∗ C(r) dr − dr − h = h(r) = h( r r0 ) . p (9b) | − | p p Since the geometrical relations between the components of C(r) dθ2 + sin2θ dϕ2 , − the metric tensor are invariant it follows that on (11), A∗,B∗, C > 0 , m 2 R (r) = h(r) 1 + , (12a) and by setting c 2h(r) 2 d C(r) 2 B∗ C(r) = B (r) , m dr R (r) = 1 + dh(r) = p p 2h(r) equation (8) is recovered,p proving that (8) and equations (9) Z m2 1 are mathematically equivalent, and amplifying the fact that = h(r) + m ln h(r) + K, (8) is a mixed-term metric. Note also that if C(r) is set − 2 h(r) equal to r2, the alleged general form used by most relativists where K = constant, is obtained. However, the form of C(r) should not be pre- h(r) m2 1 empted, for by doing so only a particular parametric solution Rp(r) = h(r) + m ln + K2 (12b) is obtained, and with the form chosen by most relativists, the K1 − 2 h(r) properties of r in Minkowski space are assumed (incorrectly) where K1 and K2 are constants.
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