Foundations of General Relativity
On the foundations of Einstein's theory of General Relativity
[L52] Theo M. Nieuwenhuizen,
[L51] Theo M. Nieuwenhuizen,
The General Theory of Relativity (General Relativity) explains a wealth of phenomena in the Universe.
It firstly incorporates Newton's attraction of bodies to the earth, of the earth to the sun,
of the moon to the earth.
But it goes far beyond that, in explaining the bending of star light around massive bodies such
as the sun, the perihelion shift of Mercury, structure of galaxies. Moreover,
it provides a framework to explain cosmology: the universe at far distances,
that is, at early times, when the light that we now observe, was emitted.
Despite this overwhelming success, some clouds have always been hanging over this theory.
The first is the question: what is the energy-momentum tensor of the gravitational field itself?
This object is utterly required to explain how, for instance, a gravitational wave can interact
with a gravitational wave detector by transferring part of its energy to the detector.
Only recently a convincing answer has been given, where the starting point it not
Einstein's view of coordinate invariance (curved space), but just the Maxwell view that
gravitation is, just like the Faraday-Maxwell electromagnetism,
a physical field in flat space (Minkowski space-time).
In plain terms: gravitation is like a gel, and, because the sun is heavy, there is quite
a bit of gel hanging around it.
A second trouble is the occurrence of singularities, that is, unlimited gravitational fields,
and unlimited redshifts.
Somehow, these should not occur in Nature, effects can be large, but not "infinite".
An example is the case of a black hole, of which it is said that "nothing can escape".
The latter statement is anyhow believed not to hold anymore in a quantum approach,
due to Hawking radiation and other quantum approaches.
Overall, the situation is not very satisfactory, since the integration of general
relativity with quantum (field) theory has never been established, and, actually,
it may just not be the path that Nature chose. Indeed, as explained at other places of this site,
quantum theory has its own problems, and perhaps we should understand them,
before being able to integrate its underlying physics with gravitation,
in a radically new way, different from straightforward quantization.
Gravitational energy momentum tensor and energy of the Universe
Given the gravitational energy momentum tensor, one may write the Einstein equations in the
Newtonian shape: (acceleration tensor) = constant * (total energy momentum tensor).
For a Universe modeled by the flat space case of the Friedman-Lemaitre-Robertson-Walker
metric, it then follows that the total energy density of the Universe vanishes.
The reason it that the gravitational energy density is negative and it cancels the one of matter.
This cancellation happens due to the Friedman equation.
In plain terms: perhaps it costed no energy to create the Universe...
The Relativistic Theory of Gravitation with a positive cosmological constant
or "Not-so-General Relativity"
In view of all of this, one may be inclined to look at a theory that is close to General Relativity,
but has as its basis a flat space. Such was attempted in the bimetric theory of Nathan Rosen.
(Rosen was a coauthor of the famous Einstein-Podolsky-Rosen paper on the foundations of quantum mechanics).
In this theory there is the Minkowski metric of flat space time and the Riemann metric that codes
the gravitational field. Logunov continued on this by adding an explicit coupling between
these two metrices in the Lagrangian - a bimetric theory.
Such a coupling breaks general coordinate invariance, so this "Relativistic Theory of Gravitation"
differs from the General Theory of Relativity by having a smaller gauge group.
One particular gauge needs to be imposed: the flat space version of the harmonic gauge.
In General Relativity energy cannot be localized, but in
this bimetric approach there occurs a localized gravitational energy density,
at least for static spherically symmetric problems [L51].
The prefactor of Logunov's bimetric term is cosmologically small, therefore it has no
sizeable influence on gravitational effects in the solar system such as light bending by
the sun or the perihelium shift of planets. Even though the theory has an additional scalar
component of the gravitational field, this component does not enter the gravitational radiation
of e.g. the Hulse-Taylor binary. Scalar components typically lead to monopole and dipole
gravitational radiation, much stronger than the observed quadrupole radiation.
The absence of these effects is a known "narrow escape", for which we realized an elegant explanation:
In General Relativity the scalar component cannot radiate, because it is a gauge artifact.
The Relativistic Theory of Gravitation just takes this result in one specific gauge, so
it has the same gravitational radiation.
The theory can still lead to the standard Lambda-Cold-Dark-Matter model of
cosmology (cosmological constant or "dark energy", normal matter and dark matter),
having a positive curvature-like term in the Friedman equation, despite the flatness of Minkowsky space.
The black hole problem gets fundamentally modified with respect to the Schwarzschild solution.
At the horizon no singularities occur, and the inner of the black hole is not principly
different from usual.
True, the gravitational fields are pretty large there (a lot of gel).
But one can still escape with a very strong rocket,
that must overcome the huge but not infinite gravitational fields there.
Interior of a black hole
In [L51] I consider a black hole which principally has the same structure as a star, but a
very large but finite redshift at the horizon (i.e., its surface). The approach is made
for the stiff equation of state: energy density = pressure + constant, which contains
as special case the vacuum euqation of state: energy density = - pressure = constant.
The metric is solved analytically and allows one parameter, the pressure at the surface.
This has to be fixed from a fundamental theory for the matter that fills the
black hole.
Supermassive black holes as giant Bose-Einstein condensates
A supermassive black hole is located in the center of probably each galaxy. Our Galaxy has
one with 3.2 million solar masses, most other ones involve several billion solar masses,
the heaviest weighs 18 billion solar masses. The density in that case is comparable
to air - the heavier they become, the less dense. Indeed, since the Schwarzschild radius
is proportional to the mass, the density M/R^3 is proportional to 1/M^2.
One would expect that such black holes mainly consist of hydrogen, the most abundant element
in the Universe. If that is indeed the case, the interior of a supermassive black holes is not
completely different from air here - alas, don't forget to carry oxigen if you go there.
It us natural to insist that such large but non-dense objects should be describable by a known theory.
This appears possible in a specific setup: assume that the hydrogen atoms are
in their Bose-Einstein condensed ground state.
This problem can be modeled by a quartic quantum field theory in the classical gravitational metric.
A coupling of the quantum field to the curvature scalar keeps the matter confined by its own gravitation.
The strength of this coupling should follow from a microscopic treatment,
but for now it is taken as a phenomenological parameter, which turns out to be large.
It then appears that the metric can be solved within the Relativistic Theory of Gravitation,
in case the quantum field is in its Bose-Einstein condensed ground state,
while General Relativity would fail to solve the so posed problem [L52].
Can black holes have hair?
Surprisingly, this black hole solution [L52] has one "hair", a parameter that describes the interior.
The mass observed at infinity can be any fraction of the mass of the hydrogen that constitutes
the black hole, in other words: the binding energy can be any fraction of the zero point
energy of the constituting matter. In the extreme limit there would be a huge black hole that,
seen from the outside, has no mass, all energy being radiated away, thus causing 100 % binding
energy. An interesting picture for our Universe, embedded in a huge Minkowski background ???
Black holes with hair are forbidden by the "no hair" theorem of General Relativity, but they
may exist in the Universe. Observations of a quasar were already connected to a "magnetic hair",
a large magnetic field that pushes the matter in the surrounding accretion disk
out to some 70 Schwarzschild radii, much farther than non-hairy black holes could do
(Schild et al, Astronomical Journal 132 (2006) 420).
If this hairy picture is confirmed, it provides another reason to abandon General Relativity
and replace it by another theory, the Relativistic Theory of Gravitation ("Not-so-General Relativity")
being the first candidate.
Supermassive Black Holes as Giant Bose-Einstein Condensates
Europhys. Lett. 83 (2008) 10008
Exact solution for the interior of a black hole
Fluct. Noise Lett. 8 (2008) L141-153;
arXiv:0805.4169
[C43] Th. M. Nieuwenhuizen,
The Relativistic Theory of Gravitation and its Application to
Cosmology and Macroscopic Quantum Black Holes ,
AIP Conf. Proc. 962: Quantum Theory: Reconsideration of Foundations-4,
Guillaume Adenier, Andrei Yu. Khrennikov, Pekka Lahti, Vladimir I. Man'ko and Theo M. Nieuwenhuizen
, eds, (Am. Inst. Phys., Melville, NY, 2007), pp 149-161.
[L49] Theo M. Nieuwenhuizen,
Einstein versus Maxwell: Is gravitation a curvature of space,
a field in flat space, or both?
Europhys. Lett. 78 (2007) 10010, 1-5.
[C39] Theo M. Nieuwenhuizen,
On the Field Theoretic Description of Gravitation,
Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity,
edited by H. Kleinert, R.T. Jantzen and R. Ruffini,
World Scientific, Singapore, 2008, to appear.