Black Holes - Twmposter 12apr09

Black holes – who needs them? Trevor Marshall, Manchester University [email protected]

http://crisisinphysics.co.uk

Laplace I have the honour to present your Majesty with my new treatise on the Mechanics of the Heavens.

Napoleon Laplace

Where is God in your mechanical system? I have no need of that hypothesis.

the conventional wisdom Schwarzschild metric dr2

The coordinate singularity at r = 2m is no problem. Just define

and we get the Eddington-Finkelstein metric

The singularity at r = 0 is unavoidable; thus a cold star with mass > 3 solar masses will collapse to r = 0 under gravity. • I have shown that this is incorrect; r = 0 lies outside the physical space.

Oppenheimer-Snyder metric [8,9] A dust ball is a system of free-falling gravitating particles under zero pressure. The metric used is comoving [4, pp.338-341]

and the stress tensor is for which the OS solution is

with F = 1 (R>1) and F(R) arbitrary for R<1, but satisfying F(1-) =1. I add F′ (1-) = 0 as a correction to Oppenheimer-Snyder. The geodesics are R=const, ds=dτ ; in particular the surface of the ball is at R=1 for all τ .

Harmonic form of Oppenheimer-Snyder Change coordinates to (t,r), where [3,4,7,10]

□ (t) = 0, □(r cosθ) = 0 and □ is the d’Alembertian operator

The solutions, for R>1, are

which give W→2m, r →m and t →+∞ as

interpretation a particle in the exterior region falls to the surface r = m in finite proper time τ but infinite external time t; it suffers an infinite red shift as it approaches r = m. The exterior (t, r) metric is

which is the S-metric with r replaced by r + m. The S-point r = 0 corresponds to the value r =-m in harmonic (H) coords and is unphysical. We shall see that the H-point r = 0 is the centre of the ball.

solution in physical space The boundary characteristic (BC) of both PDEs is defined by the ODE d τ /dR = V(τ ,R), τ (1)=2/[3√(2m)] - 4m/3. In R >1, the equation of the BC is τ (R)=2 √R³/[3√(2m)] - 4m/3. Along the BC, t → + ∞. There is no singularity inside the region τ< τ(boundary), which is why I call it physical! All geodesics, both interior and exterior, end up on the BC. In the figure I put 2m = 1, so that τ (final) = 0 at R = 1, and I put F = √[R³exp(3-3R)].

τ

t = +∞

Interior

R

exterior

the Newtonian limit As τ → - ∞, t ~ τ and r ~ [-(3/2) τ F(R)√(2m)] ⅔ this gives an initial constant density for any choice of F. As τ varies for constant R, we obtain a Newtonian collapse with dr/dt proportional to r, thus maintaining the constant density until the strong field regime is reached.

From t(R,τ ) and r(R,τ ) we get r(R,t). Position of a given dust particle, and thereby the evolution of density with t, is described by

gives a crude picture of the overall evolution. With F = √[R³exp(3-3R)] plot ufinal v. uinitial shows ● any interior particle of the ball is displaced towards the surface ● as the ball contracts, and in the final state the surface density becomes infinite.

ufinal

uinitial

Implications of the new solution 1 The topology of space, outside and within a collapsing star of any mass, is the same as that of Minkowski space: ie. no black holes [11,12]. The final state of the star, in the comoving coords (τ ,R), is described by an ODE linking τ with R. This ODE was obtained by going to harmonic coordinates, => a believer in a strong Principle of Equivalence must find a coordinate-free explanation for it.

Implications of the new solution 2 Insisting on a physical space, with orbits for each individual particle of the collapsing star, has led to a final state with a strong concentration of particles at the surface. The harmonic frame is associated, in the Logunov [2] theory, with a restoration of the distinction between inertial and gravity forces; in this frame the effect of the latter is to replace the attraction of the weak-gravity regime with repulsion in the strong-gravity regime.

Implications of the new solution 3 The harmonic frame was first used by Einstein [5] to derive his formula for the radiation from a quadrupole source. It has been demonstrated by the Logunov [7] school that Einstein’s result holds only in the harmonic frame, something Eddington [6] pointed out long ago.

Bibliography [1] N. Rosen, Phys. Rev. 57, 147-153 (1940) and Ann. Phys. (New York) 22,1(1963) [2] A.A. Logunov, The Theory of Gravity, Nauka, Moscow (2001) [3] V.A. Fock, The Theory of Space, Time and Gravitation, Pergamon, New York (1959) [4] S. Weinberg, Gravitation and Cosmology (John Wiley, New York, 1972) pp. 161-163 [5] A. Einstein, Sitzungsber. preuss. Akad. Wiss., 1, 154 (1918) [6] A.S. Eddington, The Mathematical Theory of Relativity, University Press, Cambridge, pp128-131 (1924) [7] A. Logunov and M Mestvirishvili, The Relativistic Theory of Gravitation, Mir, Moscow (1989) [8] R. C. Tolman, Proc. Nat. Acad. Sci USA, 20, 169-176 (1934) [9] J.R. Oppenheimer & H. Snyder, Phys. Rev., 56, 455-459 (1939) [10] T.W. Marshall, http://arxiv.org/abs/0707.0201 [11] A.S. Eddington, The Observatory, 58, 37-39 (1935) [12] A. Einstein, Ann. Math. 40, 922 (1939)