So let's take this a step further. If time slows down as you approach a singularity and eventually stops at the event horizon, what is time like inside a singularity? We aren't actually dividing by zero here, we're dividing by infinity. It would be convenient if it just hit zero because beyond that point we could pick up the number curve with negative numbers. But what happens when you go past infinity? This is what they mean when they talk about the math breaking down. We don't actually have numbers to describe what's going on in there.
https://www.quora.com/What-do-physicists-mean-by-math-breaking-down-in-a-singularity
Professor Shahn Majid points out that in fact due to noncommutativity the reverse time antigravity pressure balances out the gravitational collapse of spacetime! He says - he replied to me an email - of course this isn't proven empirically....
A Rainbow from water droplets is due to the logarithmic singularity of the light broken into a prism from refraction.
I'm going to defer to Nobel Physicist Roger Penrose on this topic. He wrote the below in 1979:
One of the consequences of the hypothesis that I shall set forth in the
next section is that it rules out the white hole’s singularity as an unac-ceptable boundary condition. The hypothesis is time-asymmetric, but this
is necessary in order to explain the other arrows of time. When we add
this hypothesis to the discussion of equilibrium within the perfectly
reflecting container, we see explicitly what time-asymmetric physics has
been ‘smuggled’ in. For the hypothesis is designed not to constrain the
behaviour of black holes in any way, but it forbids white holes and
therefore renders irrelevant the extraordinary scenario that we seem to
need in order to produce one!
But what is it in the nature of the big bang that is of ‘low entropy’? At
first sight, it would seem that the knowledge we have of the big bang
points in the opposite direction. The matter (including radiation) in the
early stages appears to have been completely thermalized (at least so far
as this is possible, compatibly with the expansion). If this had not been so,
one would not get correct answers for the helium abundance, etc.”
And it is often remarked upon that the ‘entropy per baryon’ (i.e. the ratio
of photons to baryons) in the universe has the ‘high’ value of ~10”.
Ignoring the contribution to the entropy due to black holes, this value has
remained roughly constant since the very early stages, and then
represents easily the major contribution to the entropy of the universe —
despite all the ‘interesting’ processes going on in the world, so important
to our life here on Earth, that depend upon ‘small’ further taking up of
entropy by stars like our Sun. The answer to this apparent paradox — that
the big bang thus seems to represent a state of high entropy — lies in the
unusual nature of gravitational entropy. This I next discuss, and then
show how this relates to the structure of singularities.
first sight, it would seem that the knowledge we have of the big bang
points in the opposite direction. The matter (including radiation) in the
early stages appears to have been completely thermalized (at least so far
as this is possible, compatibly with the expansion). If this had not been so,
one would not get correct answers for the helium abundance, etc.”
And it is often remarked upon that the ‘entropy per baryon’ (i.e. the ratio
of photons to baryons) in the universe has the ‘high’ value of ~10”.
Ignoring the contribution to the entropy due to black holes, this value has
remained roughly constant since the very early stages, and then
represents easily the major contribution to the entropy of the universe —
despite all the ‘interesting’ processes going on in the world, so important
to our life here on Earth, that depend upon ‘small’ further taking up of
entropy by stars like our Sun. The answer to this apparent paradox — that
the big bang thus seems to represent a state of high entropy — lies in the
unusual nature of gravitational entropy. This I next discuss, and then
show how this relates to the structure of singularities.
We may suppose that, as is apparently the case with the actual universe, the
entropy in the initial matter is high. The kinetic energy of the big bang,
also, is easily sufficient (at least on average) to overcome the attraction
due to gravity, and the universe expands. But then, relentlessly, gravity
begins to win out. The precise moment at which it does so, locally,
depends upon the degree of irregularity already present, and probably on
various Other unknown factors. Then clumping occurs, resulting in
clusters of galaxies, galaxies themselves, globular clusters, ordinary stars,
planets, white dwarfs, neutron stars, black holes, etc.
entropy in the initial matter is high. The kinetic energy of the big bang,
also, is easily sufficient (at least on average) to overcome the attraction
due to gravity, and the universe expands. But then, relentlessly, gravity
begins to win out. The precise moment at which it does so, locally,
depends upon the degree of irregularity already present, and probably on
various Other unknown factors. Then clumping occurs, resulting in
clusters of galaxies, galaxies themselves, globular clusters, ordinary stars,
planets, white dwarfs, neutron stars, black holes, etc.
this clumping, whereby the gravitational potential energy begins to be
taken up and the entropy can consequently begin to rise above the
apparently very high value that the system had initially. This clumping
must be expected to increase; more black holes are formed; smallish
black holes swallow material and congeal with each other to form bigger
ones. This process accelerates in the final stages of recollapse when the
average density becomes very large again, and one must expect a very
irregular and clumpy final state.
There is a slight technical difficulty in that the concept of a black hole is
normally only defined for asymptotically flat (or otherwise open)
spacetimes. This difficulty could affect the discussion of the final stages of
collapse when black holes begin to congeal with one another, and with the
final all-embracing singularity of recollapse. But I am not really con-
cerned with the location of the black holes’ event horizons, and it is only
in precisely defining these that the aforementioned difficulty arises. A
black hole that is formed early in the universe’s history has a singularity
that is reached at early proper times for observers who encounter it;”’ for
holes that are formed later, they can be reached at later proper times. On
the basis of strong cosmic censorship (cf. section 12.3.2), one expects all
these singularities eventually to link up with the final singularity of
recollapse.”’ I do not require that the singularities of black holes be, in
any clear-cut way, distinguishable from each other or from the final
singularity of recollapse. The point is merely that the gravitational
clumping which is characteristic of a state of high gravitational entropy
should manifest itself in a very complicated structure for the final
singularity (or singularities).
The picture is not altogether dissimilar for a universe that continues to
expand indefinitely away from its big bang. We still expect local clumping,
and (provided that the initial density is not altogether too low or too
uniform for galaxies to form at all) a certain number of black holes should
arise. For the regions inside these black holes, the situation is not
essentially different from that inside a collapsing universe (as was remar-
ked upon in section 12.2.6), so we expect to find, inside each hole, a very
complicated singularity corresponding to a very high gravitational
entropy. ...
taken up and the entropy can consequently begin to rise above the
apparently very high value that the system had initially. This clumping
must be expected to increase; more black holes are formed; smallish
black holes swallow material and congeal with each other to form bigger
ones. This process accelerates in the final stages of recollapse when the
average density becomes very large again, and one must expect a very
irregular and clumpy final state.
There is a slight technical difficulty in that the concept of a black hole is
normally only defined for asymptotically flat (or otherwise open)
spacetimes. This difficulty could affect the discussion of the final stages of
collapse when black holes begin to congeal with one another, and with the
final all-embracing singularity of recollapse. But I am not really con-
cerned with the location of the black holes’ event horizons, and it is only
in precisely defining these that the aforementioned difficulty arises. A
black hole that is formed early in the universe’s history has a singularity
that is reached at early proper times for observers who encounter it;”’ for
holes that are formed later, they can be reached at later proper times. On
the basis of strong cosmic censorship (cf. section 12.3.2), one expects all
these singularities eventually to link up with the final singularity of
recollapse.”’ I do not require that the singularities of black holes be, in
any clear-cut way, distinguishable from each other or from the final
singularity of recollapse. The point is merely that the gravitational
clumping which is characteristic of a state of high gravitational entropy
should manifest itself in a very complicated structure for the final
singularity (or singularities).
The picture is not altogether dissimilar for a universe that continues to
expand indefinitely away from its big bang. We still expect local clumping,
and (provided that the initial density is not altogether too low or too
uniform for galaxies to form at all) a certain number of black holes should
arise. For the regions inside these black holes, the situation is not
essentially different from that inside a collapsing universe (as was remar-
ked upon in section 12.2.6), so we expect to find, inside each hole, a very
complicated singularity corresponding to a very high gravitational
entropy. ...
So the entropy of the system as a whole increases with time even though the matter itself is in
thermal equilibrium during an initial stage of the expansion. There is, in
fact, a contribution to the entropy from R (and R), which must be
regarded as a dynamical variable in the model. (This arises because of the
phenomenon of bulk viscosity.*°)
One can view what is involved here as basically a transfer of potential
energy from the global structure of the universe (gravitational potential
energy) into the local energy of the matter,...
thermal equilibrium during an initial stage of the expansion. There is, in
fact, a contribution to the entropy from R (and R), which must be
regarded as a dynamical variable in the model. (This arises because of the
phenomenon of bulk viscosity.*°)
One can view what is involved here as basically a transfer of potential
energy from the global structure of the universe (gravitational potential
energy) into the local energy of the matter,...
I propose,’ then, that there should be a complete lack of chaos in
the initial geometry. We need, in any case, some kind of low-entropy
constraint on the initial state. But thermal equilibrium apparently held (at
least very closely so) for the matter (including radiation) in the early
stages. So the ‘lowness’ of the initial entropy was not a result of some
special matter distribution, but, instead, of some very special initial
spacetime geometry.
the initial geometry. We need, in any case, some kind of low-entropy
constraint on the initial state. But thermal equilibrium apparently held (at
least very closely so) for the matter (including radiation) in the early
stages. So the ‘lowness’ of the initial entropy was not a result of some
special matter distribution, but, instead, of some very special initial
spacetime geometry.
I have merely asserted that certain of the
laws are not in fact time-symmetric — and worse than this, that these
asymmetric laws are yet unknown! [This is before Roger Penrose's embrace of noncommutativity math].
laws are not in fact time-symmetric — and worse than this, that these
asymmetric laws are yet unknown! [This is before Roger Penrose's embrace of noncommutativity math].
I would contend, in any case, that the arguments I
have been presenting (notably those of sections 12.2.7 and 12.3.3 which
most directly relate to the Bekenstein—Hawking formula) point towards
some new theory which is time-asymmetric. Accordingly, whatever
nonlinear physicst eventually replaces suddenly collapsing wave
functions may well turn out to involve an essential time-asymmetry. ...
have been presenting (notably those of sections 12.2.7 and 12.3.3 which
most directly relate to the Bekenstein—Hawking formula) point towards
some new theory which is time-asymmetric. Accordingly, whatever
nonlinear physicst eventually replaces suddenly collapsing wave
functions may well turn out to involve an essential time-asymmetry. ...
The puzzle then becomes: why does Nature choose to hide this
time-asymmetry so effectively? As we do not yet know the principles that
govern Nature’s choices of physical law, we cannot yet answer this
question. But perhaps we should not be so surprised at a situation in
which a fundamental asymmetry lies hidden deep beneath a facade of
apparent symmetry. "
time-asymmetry so effectively? As we do not yet know the principles that
govern Nature’s choices of physical law, we cannot yet answer this
question. But perhaps we should not be so surprised at a situation in
which a fundamental asymmetry lies hidden deep beneath a facade of
apparent symmetry. "
Pretty stunning. It's a chapter of a book on Einstein that includes this awe-inspiring quote: Bryce DeWitt:
“According to the general theory of relativity, spacetime itself is the medium” and “in principle, gravitational radiation could be used as a propellant.”(page 681)
book Title: General Relativity : An Einstein Centenary ...Publisher: Cambridge University Press Publication Date: 1980.
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