According to Aristotle [Poetics VII
1450b27-29], the Beginning is that which does not have anything necessarily
before it, but does have something necessarily following from it. However, the
only way we could think of the Beginning is to place it on a time line, and fix
a finite time interval, 13.7 billion years, that is "expanding" along the
cosmological time arrow. To make a 'peaceful co-existence' of these two
contradictory requirements, we postulate a dual age of the universe. Hence the Beginning stays always in the global mode of spacetime, and
cannot be reached for any finite time interval measured in the local mode
-- neither 'back to the Beginning' nor 'forward to the end'. Got a headache?
Subject: The first second of the Universe
Dear Dr. Schwarz,
In your third version of "The first second of the Universe",
you wrote: "Let me finally stress that this review represents my personal
point of view and, there are certainly more issues than discussed here
that are relevant for a complete understanding of the early Universe."
(t_2 - t_0) - (t_1 - t_0) = t_2 - t_1 = 1 min
I will appreciate your critical comments, as well as those from your colleagues. Will keep them private and confidential.
Also, I will be happy to send you and your colleagues my CD ROM "Physics of Human Intention"; info at my web site.
Note: I believe t_0 is utterly important for cases in which we are not concerned with an increment of some physical quantity but the "whole" of it, say, when we apply the requirement for unitarity.
Concrete example: vacuum energy in QED and the discrepancy between GR and the standard model, after R. Feynman [Ref. 1]. It's huge, and calls for new physics beyond the standard model. Another case for t_0 is the so-called coincidence problem. Norbert Straumann has written a lot about it.
The next case is another unresolved issue, background-independent theories and subsequently the inner product problem, i.e., the problem of fixing the inner product in the Hilbert space of physical states by requiring that it is both invariant under Diff(M) and conserved in "time" [Ref. 2]. In what "time"? Surely we're talking about the global mode of spacetime, t_0 .
But I think there is a clear cut case for t_0 here. Note the remark by Mike Turner, "the inherent possibility for W and Z bosons to become massive was realized."
Let me stress again that we can not, even in principle, think of any finite interval without t_0 . It shows up inevitably as a sliding cut-off, and is well hidden in the infinitesimal, as we known from Thompson's paradox. Can we solve it with Non-Standard Analysis? I'm not aware of any successful efforts.
To cut the long story short, I believe the global mode of spacetime, t_0 , is what we need to develop a complete theory of quantum gravity without any closed time curves and time machines [Ref. 3], and to reveal the ultimate reality "that is neither mental nor material (or, equivalently, is both), for it is conceptually prior to the mind-matter splitting" [Ref. 4, p. 267]. See also Leibnitz and Pauli & Jung [Ref. 5]. This is what the doctrine of trialism is all about.
We're certainly missing something
in our formulation of the theory of gravity [Ref. 1].
"In the quantum field theories, there
is an energy associated with what we call the vacuum in which everything
has settled down to the lowest energy; that energy is not zero-according
to the theory. Now gravity is supposed to interact with every form of energy
and should interact then with this vacuum energy. And therefore, so to
speak, a vacuum would have a weight -- an equivalent mass energy -- and
would produce a gravitational field. Well, it doesn't! The gravitational
field produced by the energy in the electromagnetic field in a vacuum --
where there's no light, just quiet, nothing -- should be enormous, so enormous,
it would be obvious. The fact is, it's zero! Or so small that it's completely
in disagreement with what we'd expect from the field theory. This problem
is sometimes called the cosmological constant problem. It suggests that
we're missing something in our formulation of the theory of gravity."
"A related issue is the Hilbert-space
problem: What is the appropriate inner product that encodes the probability
interpretation and that is conserved in time?"
"The answer is straightforward. To
make a loop a curve must somewhere leave the null cone as shown in Fig.
1. A particle with such a world line would exceed the speed of light and,
insofar as no tachyons have been found yet, the time machine is impossible
in special relativity. In general relativity the situation is much less
[Ref. 5] Harald Atmanspacher, Hartmann Römer, and Harald Walach, Weak Quantum Theory: Complementarity and Entanglement in Physics and Beyond, Foundations of Physics, 32(3), March 2002.
"Another, even more speculative,
approach to the relationship between mind and matter was put forward by
Jung and Pauli (1952). (...) Concrete and detailed indications concerning
the substance of such a scenario are, at least
to our knowledge, not available so far."
origin of the arrow of time: The Cauchy PROBLEM
Dear Drs. Chen and Nikolic,
I believe first we have to cure this generic pathological behavior of the pseudo-Riemannian manifold, and then proceed with our speculations, say, whether exists some turning Cauchy hypersurface (hep-th/0410270, Fig. 9) or not (hep-th/0411115).
A simple way to cure the pathologies from the Cauchy PROBLEM is outlined at
Otherwise we may never understand how the "initial" state is being chosen, and will be discussing the number of angels you can place on the head of a pin.
Note: I believe
there exist generic pathologies in the 3-D
space in geometrodynamics, as modeled with a Cauchy surface, which
boil down to causal geodesic incompleteness in the case of
Cauchy problem, exhibited in closed time curves (CTCs)
and spacetime singularities.
Since the initial value conditions or 'Cauchy
data' cannot be uniquely specified in Einstein's GR due to the so-called
diffeomorphism freedom (R. Geroch), I will gently
remind the conceptual solution from the "relational"
Suppose you take on Route 66 from Chicago to Los Angeles, a 2448-mile long trip, but in Flagstaff you suddenly realize that you can't go further, because there is no road ahead. Would this be a correct example of the Cauchy problem?
Nope. You would know that there is no road ahead after Flagstaff, you would know that are in Arizona, and you could calculate how many miles lie ahead to LA, had Route 66 been extended to your final destination. Most importantly, you can go back to Chicago and figure it out what went wrong (possibly with some help from Bob Geroch).
But in Einstein's GR, you may wind up in a truly pathological situation in which you know nothing. You can't even go back to Chicago and find out exactly how was Route 66 configured from Chicago to LA, that is, what were the initial conditions for your trip, and to what extent these conditions were flexible due to the gauge freedom. Why? Because any movement ahead, any infinitesimal timelike displacement along your geodesic is prohibited. You don't have this luxury anymore, because you don't know which is 'past' and which is 'future', and the 'road ahead' may not belong to your trip anymore. Suddenly, the 'road ahead' after Flagstaff has been "cut off" from Route 66, and all you could do is to hope and pray that some magic "cosmic censorship" would help you get back to Chicago.
Of course, you may say that this is a very general and incomplete sketch of the Cauchy problem for Einstein's equation, firstly, and secondly -- in many cases we can bypass it with a careful selection of the concrete case examined, under the stipulations that some unknown symmetry law may be operating in Nature, such that all our problems with negative energy densities could be solved (only we don't yet know how to do it). It's a bit like that: you want to purchase an Audi quatro, but the dealer tells you that the car has been designed to run only on smooth dry roads, very slowly, and you must never take it on Route 66 because it will certainly crash. Would you buy it?
If something wrong can happen in the universe, it will happen. Let's keep in mind this Murphy's Law for quantum cosmology, and seek some modification of the Hamiltonian dynamics, which would produce the world as we see it, without any ad hoc "cosmic censorship" or positivity mass "theorems" inserted by hand from the outset.
It's all here. All we need is to build a geodesic relationally, by following the rule 'think globally, act locally'. See also my email to Fred Cooperstock and a brief online paper on the nature of "points". More on quantum gravity here.
Panta rei conditio sine qua non
References and notes
Global dynamical properties of solutions of the Einstein equations,
Alan Rendall, The Cauchy problem
for the Einstein equations,
Alan Rendall, Theorems on existence and global dynamics for the Einstein equations, Living Rev. Rel. 5 (2002) 6; gr-qc/0203012
Helmut Friedrich, Allan D. Rendall, The Cauchy Problem for the Einstein Equations, Lect. Notes Phys. 540 (2000) 127-224; gr-qc/0002074
Three lectures delivered at 50 years of the Cauchy problem in General Relativity, Electronic proceedings of the summer school on mathematical general relativity and global properties of solutions of Einstein's equations, Cargèse, Corsica, France, July 29 - August 10, 2002. (A Birkhäuser book with DVD can be ordered from here.) To get into the mood of the ideas of R. Geroch, listen to the audio file (RealMedia format, .rm) from the first, introductory lecture, available form here (son.rm, 16,314,278 bytes). Pay special attention to the fibre bundle presentation of Einstein's GR, as outlined in the 16th minute of the first talk on July 29, 2002 (16:01 - 16:30): in what time does Mother Nature make the continuous surjective projection map? Read also the lecture notes by R. Geroch below.
Robert Geroch, Gauge, Diffeomorphisms,
Initial-Value Formulation, Etc
R. Geroch: "Einstein’s equation as
it stands does not admit an initial-value