The Big Bang as a Cosmological Galilean Limit
Rock Brentwood
sci.physics thread ?Re: where did the big bang start??, and includes
some minor corrections and with a major addition.
So, I've divided this into the following parts: (0) a preliminary
discussion of the main issue, (1-3) the article itself:
(1) The Isotropic Signatures
(2) The Classification of Signatures and Signature-Changing Boundaries
(2.1) Generalized Wigner Classification
(2.2) Signature Domain Interfaces
(2.3) Classification of Signature Types and Interfaces
Parts 1 and 2 were based respectively on articles posted 2009 July 2
00:59:05 -0700 (PDT) and 2009 July 3 00:14:02 -0700 (PDT); ultimately
in reply to the following:
?kquir...@yahoo.com? <kquir...@yahoo.com> wrote:
>1. Do we know in what direction in our universe the big bang ?started? from?
First, the
(0) Preliminary Discussion:
An interesting point about the so-called "singularity" that lies at
the Big Bang is that the metric actually does NOT go singular. It has
the form, given by the line element
ds^2 = dt^2 - A^2 (K^2 dr^2)
where the conformal factor K = K(r) = 1/(1 - z r^2) distinguishes
between the elliptical, parabolic and hyperbolic cases (thus, giving
us the inspiration behind the question "define the universe and give 3
examples"), based on the sign of z.
The factor A -> 0 as the time t -> 0, thus reducing the metric to
dt^2.
This is the covariant metric for non-relativistic spacetime.
On the contravariant side, because Riemannian geometry and Riemann-
Cartan geometry both (tacitly) impose the condition that the covariant
and contravariant metrics be inverses to one another, this makes it
impossible to (a) deal with non-relativistic spacetime strictly within
the framework of Riemann(-Cartan) geometry and (b) impossible to
consistently account for what the contravariant metric for the
cosmological solution is doing as you approach time 0.
In fact, what it is doing ... apart from a scale transformation ... is
that it is becoming a rank 3 metric:
g^{mn} d_m d_n = (d/dt)^2 - (1/A)^2 Del^2 = -(1/A)^2 (Del^2 - A^2
(d/dt)^2).
Apart from the -(1/A)^2 factor, the metric approaches the Hodge-de
Rham operator Del^2.
This is the non-relativistic contravariant metric for a conformally
flat space (again, with the distinction between the elliptical,
parabolic and hyperbolic cases, based on the factor K).
So ... what we're actually talking about with the Big Bang is NOT a
singularity at all, but simply a domain interface between two
different space-time signature. The interface, itself -- the time 0
equal-time surface -- is a 3-space of global absolute simultaneity
(hence it makes sense to designate a single time to it). It is a slice
of non-relativistic space-time.
In other words, the Big Bang is a cosmolgical realization of the
Galilean Limit, itself.
The reason you can't talk beyond this point is that Riemann(-Cartan)
geometry suffers the constraint:
g^{mn} g_{nr} = delta^m_r.
That is: there are actually TWO degrees of conformal scaling, not just
one! One of them is the one usually identified as conformal symmetry
and acts on the metrics in equal and opposite ways:
g^{mn} -> a g^{mn}, g_{mn} -> a^{-1} g_{mn}.
The other, however, acts on both metrics the same way
g^{mn} -> a g^{mn}, g_{mn} -> a g_{mn}
and tha'ts the one that's excluded by the central premise of Riemann(-
Cartan) geometry that the two metrics remain as inverses of one
another.
But this is the mode you need to turn back on, if you want to get to
(and beyond) time 0.
This, first and foremost, requires dropping back outside of Riemann(-
Cartan) geometry to a more general framework in which both metrics may
be treated as independent. This, of course, puts the brunt of the
issue on who to relate the two metrics (never mind, how to account for
the 2nd degree of conformal scaling).
So, with that in place, this was the direct reply to the original
query:
2009 June 21 19:48
?kquir...@yahoo.com? <kquir...@yahoo.com> wrote:
> 1. Do we know in what direction in our universe the big bang ?started? from?
If a Cosmic event happened 13 1/2 billion years ago, then it is in the
direct line of sight in every direction at a distance of 13 1/2
billion light years from where you sit.
> 2. or, do we know where in the expanding sphere following the big bang we are?
It is, in fact, not a consensus that the initial cosmic singularity is
of a nature that all of space is scrunched up at one point; but rather
that it is a singularity of the kind that the spatial part of the
metric was 0.
These are two completely different things.
To say that the spatial part of a metric is 0 is not to say that space
is scrunched up into a point. A case in point counter-example is the
geometry of non-relativistic spacetime. The metric here, is rank 1 and
can be normalized to the quadratic form dt^2.
The initial singularity is, in fact, itself a surface of global
simultaneity on which you may objectively synchronize all time to. So,
it?s more appropriate to think of it as a surface on which spacetime
becomes non-relativistic ? literally; rather than as a surface on
which space scrunches down to 0.
The problem people have had with this concept (and with even treating
Galilean spacetime as a Riemannian space) is the insistence on tying
the covariant and contravariant metrics together and insisting on them
being inverses with one another. This approach runs afoul precisely in
those places where either the covariant metric goes from rank 4 to
rank 3 or less; or where the contravariant metric does the same. And
the inability to deal with the ?singularity? of either type is
precisely based on this oversight.
The two metrics have to be treated independently, though related. The
more general approach, instead, is to assert that the spacetime
symmetry group have 6 degrees of freedom; and to define the symmetry
group as that which preserves both metrics, rather than that which
preserves only one of them.
1. The Isotropic Signatures
To simplify the discussion, we're be talking about signatures in the
context of flat (or conformally flat) spaces.
You can then proceed to classify all the possibilities which yield
isotropic quadratic forms (those where 3 of the coordinates are on the
same footing). The quadratic form governing the metric will be of the
form
g = A dr^2 - B dt^2
where
dr = (dx, dy, dz), dr^2 = dx^2 + dy^2 + dz^2;
and the contravariant metric is given by the quadratic form
g' = C del^2 - D d_t^2
where
del = (d_x, d_y, d_z), del^2 = d_x^2 + d_y^2 + d_z^2,
with d_t denoting the partial derivative operator with respect to t
(similarly for d_x, d_y, d_z respectively for x, y and z).
A symmetry which preserves both metrics as well as the quardatic form
theta = dr.del + dt d_t
is given in infinitesimal form by
(I) 0 = A dr.delta(dr) - B dt delta(dt),
(II) 0 = C del . delta(del) - D d_t delta(d_t).
(III) 0 = delta(dr).del + delta(dt) d_t + dr.delta(del) + dt delta
(d_t).
The number of independent degrees of freedom the variation delta can
have depends on what A, B, C and D are. Requiring this to be only 6,
puts restrictions on their values.
For equation (I), if A, B are both non-zero, you can write
delta(dr) = w x dr - B u dt, delta(dt) = -A u.dr
which introduces 6 parameters, w = (w_1, w_2, w_3) and u = (u_1, u_2,
u_3). These correspond, respectively, to the parameters for rotation
and boost (velocity-change).
Substituting into (III), and separating out the factors of adjoining
dr and dt, you get
delta(del) = w x del + A u d_t, delta(d_t) = B u.del.
Upon substitution into (II), you end up with
0 = (CA - DB) u d_t . del.
Since the number of degrees of freedom is already down to 6, the only
way for this condition to not yield further restrictions is if CA -
DB = 0, or D = kA, C = kB, for some factor k.
The two cases that occur are:
(1) A, B are the same sign ? this is the Lorentzian signature of
relativistic spacetime
(2) A, B are opposite signs ? this defines the signature of a 4-
dimensional space, where all the dimensions are spatial, and none are
of time ? the Euclidean signature.
So, moving on to the other cases: if A = 0 = B, then C, D are both
non-zero, or (II) would yield a result with more than 6 degrees of
freedom. This can be lumped into the case just discussed: with the
metric g and g' being inverses up to proportion (here the proportion
is 0).
The remaining cases are where A = 0 and B non-zero and B = 0 and A non-
zero.
For A = 0 and B non-zero, (I) gives you delta(dt) = 0. Substituting
into (III), you get
delta(dr).del + dr.delta(del) + dt delta(d_t) = 0
which has the general solution
delta(dr) = w x dr - a dr - u dt, delta(del) = a del, delta(d_t) =
u.del.
This has the 6 degrees of freedom of w and u plus the 7th of the
spatial scaling coefficient a.
Substituting into (II) yields Ca = 0 and Du = 0.
The case (C, D) gives you 7 degrees of freedom. The case C = 0, D non-
zero gives you u = 0, which drops you down to 4 degrees of freedom.
The case C and D both non-zero also gives you a = 0, dropping you down
to 3 degrees of freedom.
The only remaining possibility is D = 0, C non-zero, which eliminates
the scaling factor, giving you a = 0. So, again, all you have left are
rotations and boosts.
The metrics are then of the form
g = B dt^2, g' = C del^2 (B, C both non-zero).
That yields dt and del^2 as invariants. Since dt is an invariant, time
is an absolute and simultaneity is absolute. The del^2 operator is the
Poisson operator. This is the Galilean signature ? the signature of
non-relativistic space.
The final case B = 0, A non-zero, is the antithesis of Galileo in a
certain sense. The arguments are similar, the result is that only C =
0, D non-zero gives you 6 degrees of freedom, with the symmetries
delta(dr) = w x dr, delta(dt) = -u.dr,
delta(del) = w x del + u d_t, delta(d_t) = 0.
.
The result are the metrics
g = A dr^2, g' = D d_t^2 (A, D non-zero).
The invariant dr^2 is Euclidean distance. This defines Absolute Space.
The operator d_t is now an absolute. Consequently, motion is absolute,
as is absolute rest. This is the signature of Archimedean spacetime ?
the spacetime the Church adopted in their dispute against Galileo.
The 4 signatures are thus (after adjustment by factors)
Euclidean 4-dimensional: g = A dr^2 - B dt^2, g' = B del^2 - A d_t^2,
(AB < 0),
Lorentzian: g = A dr^2 - B dt^2, g' = B del^2 - A d_t^2, (AB > 0),
Galilean: g = dt^2, g' = del^2,
Archimedean: g = dr^2, g' = d_t^2.
So, now finally, back to the original question and issue. There are
two types of isotropic singularities for the Lorentzian signature are
where A -> 0 or B -> 0.
The case B -> 0 leads to an Archimedean signature. On it, light speed
is 0, time appears to be at a stop (because there?s no more dt in the
metric). These are the characteristics that also define a CAUSAL
HORIZON.
The case A -> 0 leads to a Galilean signature. On it, light speed
appears to have gone to infinity (hence, near the surface, you have
something that looks like what cosmologists call ?inflation?). On the
boundary, itself, dr^2 disappears from the metric, to spatial
distances appear to have scrunched up to 0. But the alternative
interpretation is that the metric is just the Galilean metric:dt^2,
and that time has become an absolute. The boundary, itself, then marks
an instant of absolute simultaneity ? a natural point to mark time 0
at.
The other side of the boundary (for either type of singularity) can be
Euclidean. In that case, the so-called ?singularity? is nothing more
than a phase boundary between two signature domains.
So the question posed in the original article of ?where? isn?t even
applicable any longer. The relevant question is "when" ? 13 1/2
billions years ago. Beyond that point, there?d be no time, but just 4
dimensions of space. The boundary would mark the point where one of
the dimensions became timelike ? the first point where so.
A more insteresting alternative is where the Galilean signature
persists on the other side at finite depth. Then you have a case where
a non-relativistic spacetime transforms into a relativistic space-
time. Then the ?initial singularity? is just one where light speed
dropped down from infinity to a finite value and where the Galilean
metric dt^2 started having a space-like part mixed up into it, while
the Poisson operator del^2 started getting a tme-like part in it and
transformed into the wave operator.
(2) Classification of Signatures and Signature-Changing Boundaries
(2.1) Generalized Wigner Classification
All of the 4 cases which yield bi-orthogonal symmetry groups with 6
generators fit in the general form:
g = beta dt^2 - alpha dr^2, g' = beta del^2 - alpha d_t^2
up to overall factors. The cases are
alpha beta > 0: Lorentzian (i.e. Relativistic)
alpha beta < 0: Euclidean 4-dimensional
alpha = 0, beta non-zero: Galilean (i.e. Non-Relativistic), and
alpha non-zero, beta = 0: Archimedean (i.e. pre modern)
These all yield symmetry groups with 6 generators. The case (alpha,
beta) = (0, 0) yields a symmetry group of maximal size (16
generators), and so is excluded by the above criterion.
For the other 4 cases, the symmetries are all of the same form:
delta(dr) = w x dr - b u dt, delta(dt) = -a u.dr,
delta(del) = w x del + a u d_t, delta(d_t) = b u.del
where the vectors w and u represent the parameters, respectively, for
infinitesimal rotations and boosts.
The Archimedean signature has ? as its invariants ? (Euclidean
distance) dr^2 and d_t (which ultimately gives us definition of
absolute rest and absolute motion).
If you were to go ahead and do a Wigner classification of the
?particles?/?elementary systems? in this signature, the three major
classes you?d get are
(0) translation-invariant media (vacuons; Wigner type 3);
(1) media at absolute rest (statons; combines Wigner types 1 and 2,
combined); and
(2) media in absolute motion (tachyons; Wigner type 4).
The only systems with rest states are those at absolute rest.
So, in a signature-changing boundary between Lorentz and Euclid of the
Archimedean type, everything would appear to be stationary or frozen;
including light.
This is characteristic of a causal horizon. The staton is the
Archimedean limit of both the tardion & luxon from the Lorentzian
signature.
The Galilean signature has the opposite set of invariants: dt
(absolute time) and del^2 (Poisson operator); but no absolute motion
(hence, the Galilean principle of relativity).
A Wigner classification yields the following types
(0) translation-invariant media, as before (vacuons, Wigner type 3);
(1) media which support instantaneous action at a distance (synchrons,
Wigner types 2 and 4, combined); and
(2) media in motion at finite speed or at rest (tardions, Wigner type
1; possessing a rest or staton state).
Synchrons correspond to instantaneous transfer of impulse across space
and have an infinite speed (i.e they don?t move at all, but are simply
there at an instant, everywhere all at once on the line across which
the transfer of impulse takes place).
Here, since the invariant speed is infinite, as you approach a
signature-changing boundary of the Galilean type, spatial distances
appear to go to 0, relative to the corresponding temporal distance.
Time is globally synchronized on the boundary, itself (a natural ?time
0? marker), and things can move arbitrarily fast at times near the
boundary (?inflation?).
For the Lorentzian signature, the Wigner types are
(0) vacuon, type 3;
(1) tardion, type 1 (slower than light, possessing a rest state or
staton state);
(2) luxon, type 2;
(3) tachyon, type 4 (media which support relativistic action at a
distance: systems which possess an infinite speed or synchron frame).
The difference between the Lorentzian and Galilean types 4, is in the
former case, the ?instantaneousness? of the action is only with
respect to a distinguished set of frames (the tachyon?s ?synchron?
frames).
The best-known guru (Sudarshan) ? like most people ? think of all the
Wigner classes except 3 as ?particles? (the misguided ?irreducible
representation [irrep] = particle? creed). This creed doesn?t fit
(e.g. if irreps are particles, then where is the ?vacuum particle??)
Generally, class 3 is arbitrarily excluded from the equation irrep =
particle. But, neither should the Galilean type 2/4 or Lorentzian type
4 be considered as anything more than specialized media; not
particles. So, the argument ?tachyons don?t exist because nobody?s
ever found a tachyon particle? makes no more sense than ?the vacuum
doesn?t exist because nobody?s ever found a vacuum particle?. It?s
based on the false premise that irrep = particle.
The Euclidean signature has only
(0) vacuon, type 3; and
(1) tardion, types 1, 2 and 4 combined.
(2.2) Signature Domain Interfaces
A more insteresting alternative is where the Galilean signature
persists on the other side at finite depth.
People in the Hawking-Hartle crowd also do the same construction for
the other type of boundary (the causal horizon); this leads to the
Euclidean wormhole solution. This consists of a Lorentzian domain, an
Archimedean boundary (the event horizon) and a Euclidean domain on the
other side. The Euclidean part is compact, and has a second
Archimedean boundary which has, on its other side, a second Lorentzian
domain.
You could also combine all 4 in one master stroke:
(0) a compact Euclidean domain;
(1) a Galilean boundary on one side (the ?time 0? 3-surface);
(2) a Lorentzian domain, spatially compact and sandwiched between the
Galilean and Archimedean boundaries;
(3) an Archimedean boundary between the Lorentzian domain and a
Euclidean domain;
(4) the two Euclidean domains are one and the same.
This is a compact 4-space with a Lorentzian spacetime embedded in it.
To make it interesting, make the Galilean and Archimedean boundaries
of finite depth, so that the 4-space actually is non-relativistic in
the Galilean zone and Archimedean in Archimedean zone, rather than
just on 3-surface boundaries.
(2.3) Classification of Signature Types and Interfaces
Finally, to fully exhaust the range of possibilities for signatures,
remove the isotropy assumption and allow the signature of each of the
4 dimensions to vary independently.
The most general case of a metric g and dual metric g' which together
yield 6-dimensional symmetry group is where
g = aWW + bXX + cYY + dZZ, g' = Aww + Bxx + Cyy + Dzz,
such that
aA = bB = cC = dD,
where (W, X, Y, Z) is a co-tangent frame dual to the tangent frame (w,
x, y, z); each frame being a linearly independent set.
Other solutions exist that yield 6 degree-of-freedom symmetry groups,
but the symmetry groups in those cases leave 1 or more dimensions
fixed, and so would be excluded too.
Excluding the cases g = 0 and g' = 0, this gives you 11 distinct
signature families. Denote by (r, r') the ranks respectively of (g,
g'). The families are:
(r, r') = (4, 4): 4+0 (Euclidean), 3+1 (Lorentzian), 2+2
(?Penrosian?).
The 2+2 signature has 2 space-like and 2 time-like dimensions. Second,
we have
(r, r') = (3, 1): Quasi-Archimedean; g can have signature 3+0
(Archimedean) or 2+1.
The case where g has signature 2+1, there are three types of
dimensions and the metrics have forms that can be normalized to:
g = WW + XX - YY, g' = zz.
The w/x directions can be designated ?spacelike?; the z direction
?timelike?, while the y direction is a third kind of dimension.
Third, we have
(r, r') = (2, 2): Quasi-Penrosian: g and g' can each have signatures
2+0 or 1+1.
The 2+0/0+2 case has 2 types of dimensions; while the 2+0/1+1 and
1+1/0+2 cases have 3; and the 1+1/1+1 has 4.
Finally, there are the following:
(r, r') = (1, 3): Quasi-Galilean.g' can have signature 0+3 (Galilean)
or 1+2.
In a similar way as in the previous discussion, we can discuss the
types of interfaces between signature domains; where we continue to
enforce the symmetry group condition across interface boundaries.
Then, by virtue of this condition, we can write every interface in the
form
g = beta g_1 - alpha g_2, g' = beta g_2^{-1} - alpha g_1^{-1}
where (alpha, beta) is not (0, 0), g_1 and g_2 are both non-singular
metrics that apply on mutually orthogonal subspaces. The types of
interfaces depend on the ranks and signatures of the respective
metrics and may be classified as follows:
Euclidean: g_1 = diag(+, +, +, +), g_2 = 0;
Lorentzian: g_1 = diag(+, +, +, -), g_2 = 0;
Penrosian: g_1 = diag(+, +, -, -), g_2 = 0;
Isotropic: g_1 = diag(+, +, +), g_2 = diag(+);
Quasi-Isotropic: g_1 = diag(+, +, -), g_2 = diag(+);
Spacelike Penrosian: g_1 = diag(+, +), g_2 = diag(+, +);
Timelike Penrosian: g_1 = diag(+, -), g_2 = diag(+, -);
Mixed Penrosian: g_1 = diag(+, +), g_2 = diag(+, -).