Research Note 2: Sign Error in General Relativity, a Problem in Einstein Field Equations

I. INTRODUCTION

General Relativity (GR) is currently being questioned for a number of reasons. Among the large range of objections, some authors challenge the role of the energy-momentum tensor

Tμν=ρuμuν,

where ρ is the mass-energy density and uμ=dxμ/ds is the μ-component of the four-velocity, as the unique source of the gravitational field, and suggest additional sources such as the trace T of the energy-momentum tensor, or functions f(T) of the trace.

Indeed, the energy-momentum tensor as it stands appears to introduce contradictions into Einstein’s theory. In addition to often-cited violations of energy conservation, as well as GR’s failure to explain cosmic acceleration, there also arise conflicts with observational tests at the solar system scale. At least one of these conflicts, to the knowledge of this author, has so far not been investigated. Such an investation is the purpose of this research note.

II. ORIGIN OF SIGN ERROR IN GENERAL RELATIVITY

The most critical contradiction in GR seems to arise from an apparent sign error in the pressure components of Tμν, which will be explained. For brief background, references in the literature show that Einstein’s Field Equations (EFE)

Rμν − (1/2)gμνR = κTμν

where Rμν is the Ricci tensor and R the scalar curvature, reduce, for the spherical static non-vacuum, to the following pair of simultaneous differential equations:

κρ(r) = ─ 1/r2 ─ 1/r2g11  + g11′/g112r        (1a)

─ κp(r) =  ─ 1/r2 ─ 1/r2g11 ─ g00′/g00g11r.         (1b)

Here, ρ is mass-energy density, p is pressure, κ=-8πG/c2 is a constant, primes denote derivatives with respect to r, and the metric is given by the spherical line element:

ds2 = g00dt2 + g11dr2 ─ r22,

with signature (+ ─ ─ ─), ie. such that  g00>1 and g11<1. Taking the difference of Eqs (1a) and (1b), it is easy to show that g11=─1/g00 if and only if an Equation of State p(r) = ─ρ(r) is assumed. This Equation of State (EoS) indicates that for positive mass-energy density ρ(r), the pressure p is negative, an apparently unphysical result.

Yet despite that this EoS seems unphysical, many static spherical metrics, including the well-tested Schwarzschild metric and the commonly used Scwharzschild-de Sitter metric, are of the form  g11=-1/g00. This implies that an unrealistic EoS is required to obtain realistic and observationally tested metrics in GR, pointing to a contradiction in Einstein’s theory as it is currently interpreted.

Supposing such a contradiction can be shown to exist, it is then natural to ask: which assumptions in GR give rise to this contradiction? Toward this end, note that the left-hand sides of Eqs (1) correspond to the first two diagonal elements of the mixed orthogonal energy-momentum tensor Tμν=diag(ρ,-p,-p,-p). This assumed form of Tμν  is therefore the origin of the unphysical sign of pressure p in the EoS. It thus becomes necessary to examine the theoretical justification for the definition of the diagonal components of Tμν .

But first, it is important to note that if one uses the more physical EoS ρ(r)=p(r), which is said to apply to “stiff matter”, Eqs (1) become far more difficult to solve. Indeed, the authors of the compendium Exact Solutions of Einstein’s Field Equations say few solutions are known.

In contrast, were the sign of p reversed on the left side of Eq (1b) so that p becomes ─p , the modified equations are easy to solve for the realistic EoS ρ=p. Indeed, I propose below a method for solving this modified form of EFE for any spherical matter distribution in the range r=0 to infinity. Most crucially, it can be shown that some of these solutions─indeed the most well-tested solutions─give correct physical results.

On the other hand, were we to assume GR is valid as it stands, we have the unacceptable situation in which a “sign error” gives the right physical answer, while the “correct” sign gives the wrong physical answer. Hence, it may be appropriate to doubt the validity of this particular aspect of GR.

III. WHEN IS A METRIC THEORY VALID?

The definitive test of general relativity, or any metric theory of gravity, is whether it can produce the Schwarzschild solution for static spherically symmetric configurations. This solution is the only one that has been thoroughly tested by repeated observations in a theory-independent way.  However GR, as it stands, produces the Schwarzschild solution only for the  universal vacuum, in other words, only for a universe containing no matter whatsoever, in which the energy tensor Tμν is identically zero. Such a universe is unrealistic.

A correct metric theory of gravity should produce the Schwarzschild solution in vacuum regions that lie within or surrounding spherical non-zero mass distributions, such that the corresponding energy tensor Tμv is not identically zero, but contains, in the T00 element, a spherically symmetric matter density function ρ(r). If a metric theory of gravity fails to accomplish this, the theory cannot be considered complete.

The question then becomes: Can general relativity produce the Schwarzschild metric for vacuum regions embedded in a universe containing spherical mass distributions? The answer is apparently no, as will be shown. This means either general relativity is not a complete theory, or that it contains a fundamental flaw.

For reference, the Schwarzschild solution consists of the metric given by the squared line element:

ds2 = (1-2m/r)dt2 ─ (1-2m/r)-1dr2 ─ r22

When this metric is obtained by solving Einstein’s Field Equations (EFE) for the universal vacuum, there are at least three reasons why the solution is trivial, and therefore does not comprise a complete test of GR. First, a universal vacuum T00=ρ(r)=0 for all r is unphysical, as mentioned above. Second, Einstein’s Field Equations simplify significantly for Tμv=0, meaning that a universal vacuum solution does not test all of the terms of the equations. Third, the quantity 2m in the vacuum Schwarzschild solution is put in by hand as a constant of integration using Newtonian gravity as a boundary condition. Thus, the quantity 2m, as important as it is, does not arise automatically out of Einstein’s Field Equations. A complete metric theory of gravity should produce the quantity 2m uniquely and automatically, without resorting to Newtonian gravity.