On Penrose’s ‘before the big bang’ ideas
Abstract.
We point out that algebraically special Einstein fields with twisting rays exhibit the basic properties of conformal Universes considered recently by Roger Penrose.
MSC classification: 83C15, 83C20, 83C30, 83F05
According to Penrose [6], there might have been a prebigbang era, in which the Universe was equipped with a conformal Lorentzian structure, rather than with the full Lorentzian metric structure. This conformal structure was conformally flat, and pretty much the same as the conformal structure our Universe will have at the very late stage of its evolution.
Penrose argues that the observed positive sign of the cosmological constant, , forces our Universe to last forever. It will last long enough that all the massive particles will mannage to disintegrate, either by finding their antimatter counterparts with which to annihilate, or because of their finite half life. This will produce only massless particles, such as photons, which after all the matter has been disintegrated, will be the only content of the late Universe, except for massive black holes. These massive black holes will be the remnants of the galaxies, and perhaps, of very massive stars.
Since the Universe will last forever, and since it will be expanding, it will generally cool down, trying to reach the absolute zero temperature at its final state. Thus, there will be a time in its evolution such that the temperature of the Universe will be lower than the temperature of even the most massive black holes, which at this stage will hide the only mass of the Universe. This will create a thermodynamic instability forcing all the black holes to evaporate by radiating massless particles^{1}^{1}1Penrose assumes that the black holes lose information about the properties of the masses hidden in them, and that in the process of evaporation only massless particles are being radiated.. After these evaporations the dying Universe will be totally filled by massless particles.
Penrose argues that massless particles, whatever they are, have no way of defining clocks’ ticks. This leads to the conclusion that the Universe at its late age, being filled with only massless observers, will ultimately lose the information about its conformal factor. Thus it will become similar to the Universe in the ‘prebigbang’ era that preceeded its creation. This late state of the Universe Penrose likes to interprete as the ‘prebigbang’ era of the new Universe. Let us adopt this point of view.
It is a well known fact that during the big bang (such as in the FriedmannLemaitreRobertsonWalker
models) the only metric
singularity is in the vanishing of the conformal factor, leaving the conformally rescaled metric perfectly regular
(actually conformally flat). Another well known fact is that
the field equations for the massless particles are conformally
invariant. This shows that
the massless particles (the observers) of the dying conformal
Universe, or as we
interpret it now, of the ‘prebigbang’
era of the new conformal Universe, will not feel the big
bang singularity. They will happily pass through it from the
dying conformal Universe to
another conformally flat manifold^{2}^{2}2It should be noted that
this passage is only possible conformally. In the full Lorentzian
metric of the old Universe, the process of expansion will last
infinitely long. But after conformal rescaling, this proces
takes finite time, and enables us to speculate what will be after.. Penrose interprets this conformal
object as an
‘after big bang’ conformal era of the new Universe. It will eventually
acquire a new conformal factor, and possibly some distortion
(meaning nonzero Weyl),
promoting this conformal remnant of the old Universe to
a new Lorentzian metric Universe.
Our exact solutions of Einstein’s equations, discussed below,
exhibit the main features of Penrose’s ‘before the
big bang’ argument. Although these solutions form a very thin
set in all the
possible Einstein metrics, and although they were
obtained on purely mathematical grounds by the mere assumption that the
corresponding spacetimes admit a twisting congruence of null and shearfree
geodesics, it is a remarkable coincidence that the
pure mathematics of Einstein’s equations, imposed on such
spacetimes, forces the solutions to fit to Penrose’s ideas.
Let us discuss the mathematics of our solutions first. After we do it, we will
indicate the parallels between our solutions and Penrose’s Universes.
Theorem 0.1.
Let be a 4dimensional Lorentzian spacetime which satisfies the Einstein equations
(0.1) 
with being a null vector tangent to a twisting congruence of null and shearfree geodesics. Then its metric factorizes as
(0.2) 
where is periodic in terms of the null cooordinate along .
More explicitly, (see [2, 5] and, especially, [3] for details) we showed that if satisfies (0.1), then is a circle bundle over a 3dimensional strictly pseudoconvex CR manifold , and that
(0.3) 
with
(0.4) 
Here (real) and (complex) are 1forms on such that , , and, as a result of the Einstein equations (0.1), the functions (complex) and (real) are independent of the null coordinate : . A part of the Einstein equations in (0.1) can be explicitly integrated, obtaining:
(0.5)  
Here the independent complex function is defined via
(0.6) 
and the operators are vector fields on , which are algebraic dual to the coframe on the CR manifold . Note that the function is defined uniquely, once a CR manifold has been chosen, and thus is considered as a known function in the process of solving (0.1).
The remaining Einstein equations in (0.1) reduce to a system of two PDEs on , for the functions and , which are the only unknowns. These PDEs are:
(0.7)  
(0.8) 
These are the only equations which need to be solved in order to make satisfy (0.1). Once these equations are solved, the Einstein metric has an energy momentum tensor describing the ‘pure radiation’ of a mixture of massless particles, moving with the speed of light along the null direction , in a spacetime with cosmological constant . The spacetime is algebraically special; the Weyl spin coefficients being , .
Note that at , where the conformal factor for becomes zero, the Weyl coefficient vanishes,
Although the formulae for and are quite complicated, they also share this property, i.e.
The above quoted result enables us to interpret the hypersurfaces as the respective scris of the spacetime . The Weyl tensor is conformally flat there: for all .
Since the conformally rescaled spacetime is
periodic and regular in , it gives a
fullEinsteintheory realization of Penrose’s idea [6]
that there was a ‘before the big bang era’ of the Universe.
In this context the following remarks are in order:

Moreover the scris are conformally flat, and therefore can be identified with the respective surfaces of the big bang () and the surface of the conformal infinity in the future ().

The Universes corresponding to our solutions are either empty (), or are filled with a dust of massless particles () moving with the speed of light along .

Since going from the ‘big bang’ to the ‘infinity in the future’ corresponds to a passage from to , and since the conformal metric is periodic in , we see that our conformal solutions are repetitive in the variable.

Thus our solutions give conformal Universes which periodically reproduce themselves, and smoothly pass through the ‘big bang’ and the ‘future infinity’.
To be more explicit we discuss the following example^{3}^{3}3Note
that among our solutions there are many interesting, well known, solutions of
Einstein’s field equations. Actually our metrics include all
algebraically special vacuums and the aligned pure radiation
gravitational fields. Thus, our solutions include for example the celebrated
rotating black hole solution of Kerr. This solution, however,
is beyond the class of solutions relevant for Penrose’s
ideas since it has (and ). which is
a solution to our equations (0.7)(0.8).
We choose the CR manifold to be the Heisenberg group CR manifold. This may be represented by the 1forms and . Here are the standard coordinates on the Heisenberg group ( is real, is complex).
Obviously and , so that the function in (0.6) is . This immediately leads to a solution for (0.7)(0.8). Indeed: take and , then equation (0.7) is automatically satisfied, and equation (0.8) gives . Thus we take , . This leads to the conformal metric
with ,
and the physical metric satisfying all the equations
(0.1). Actually, the physical metric satisfies more: It is a
solution to the Einstein equations ,
thus . Its Weyl tensor has
everywhere, with the only nonvanishing Weyl coefficient
This means that the
metric corresponding to this solution is of Petrov type
everywhere, except along the scris, , where it is conformally
flat^{4}^{4}4This solution is the classical TaubNUT solution with
cosmological constant[1, 9]. It includes the TaubNUT
() solution [4] as
a special case. When the solution is a vacuum metric with a
cosmological constant, which has two Robinson congruences [7] as two
distinct principal null dierctions.. Restricting
to we get a 2parameter family of solutions with spacelike scris,
which has a periodic conformal metric . This, in addition to
being periodic in , is
regular everywhere on any hypersurface transversal to .
As a more complicated example we take a CR structure parameterized by coordinates and represented by forms
It has . Then assuming that and , we immediately get the following solution for equation (0.7):
Having this, the only remaining Einstein equation to be solved is (0.8). It is equivalent to an ODE:
for the functions and . Since this is a single ODE for two
real functions of one real variable , one can use one of these functions to arrange
the energy of the corresponding pure radiation to be nonegative for
positive .
We believe that many more solutions with appealing
physical properties may be found in our class, the main reason being
that the class consists of all (known and unknown) algebraically special
solutions with twisting rays.
We close the paper with the following mathematical comment.
It is interesting to give an interpretation to the only nontrivial Einstein equation (0.8). If one considers the metric , with as in (0.3), and functions as in (0.4)(0.5), then the equation (0.8) is the Yamabe equation (see e.g. [8], p. 332) saying that the rescaled metric has constant Ricci scalar .
References
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