Emmy Noether gave consideration to a boundary term we usually discard when
deriving the Euler-Lagrange formula to show that a symmetry was involved
with this term. This symmetry and that this boundary term is zero meant a
conservation law. A law of physics considered as such is something
associated with covariant and invariant properties of space, spacetime or
an abstract space under some set of transformations. Is this principle, a
law of laws should we say, something that is discovered or is some
objective aspect of a mathematical reality?
The type D, II, III and N solutions, black holes = D and gravitational
waves = N, are vacuum solutions with the Weyl tensor C_{abcd} that wholly
determines the curvature. The Weyl curvature is an operator on Killing
vectors, such that Killing vectors are eigenvalued with the Weyl curvature
C_{abcd}K^bK^d = λK_aK_c. The type N solutions have Killing vectors that
have zero eigenvalue C_{abcd}K^d = 0. Type III spacetimes have λ = 0 and
type II and D have nontrivial eigenvalues that are unequal for C_{abcd} and
*C_{abcd}, for * the Hodge dual with C_{abcd}K^bK^d = λK_aK_c and
*C_{abcd}K^bK^d = λ’K_aK_c for λ ≠ λ’ and λλ ≠ 0. These Killing vectors
define symmetries and thus conservation laws. A timelike Killing vector
defines conservation of energy, a spacelike Killing vector defines
conservation of momentum, and a Killing bi-vector or one derived from such
defines conservation of angular momentum. That is a total of 1 + 3 + 6 = 10
Killing vectors. These eigenvalued equations should make one think of the
Schrodinger equation. Indeed for a timelike Killing vector K_t =
√(g_{tt})∂_t so that this gives a general wave equation HΨ[g] =
iK_t∂Ψ[g]/∂t, which for g_{tt} = 1 is the Schrodinger equation. The ADM
approach to general relativity give NH = 0 and the Wheeler-deWitt equation
HΨ[g] = 0. General relativity does not automatically define conservation
laws. Conservation laws only occur with certain symmetries of spacetime.
This often occurs where there is an ADM mass defined by an asymptotic
condition of flatness or some other spacetime with constant curvature at a
distance.
Conservation laws appear as asymptotic or boundary terms. The AdS/CFT
correspondence of Maldacena shows that a nonlocal quantum gravity theory
corresponds to a local conformal field theory on the conformal boundary of
the anti-de Sitter spacetime. The anti-de Sitter (AdS) spacetime has
constant negative curvature. This is a negative vacuum energy, where this
has some correspondence with string theory, such as the type I string
theory has a negative energy vacuum and its first excited state is a
negative energy state. The AdS_4 has a correspondence with black hole
physics. The AdS spacetime is not the spacetime of the observable universe.
It is though in line with the theory of Emmy Noether, also work by
Hurzebruch, and even the old Gauss-Bonnet theory.
Physical spacetime is more similar to de Sitter spacetime, and is the
Friedmann-Lemaitre-Robertson-Walker spacetime with positive energy. This
means curvature is positive, which involves how space is embedded in
spacetime, and this does not have conservation laws. If that space is a
sphere S^3 the constant vacuum energy on this space grows with the
evolution of this space and volume growth. This is one reason that people
tend to prefer the flat space model, where vacuum energy is net "infinity"
and remains so. However, there is nothing to prevent vacuum energy density
from changing. The phantom energy model leading to a big rip of the cosmos
is possible, and the curious discrepancy between CMB and SNII data, with
the Hubble constant H = 70km/sec-Mpc and H = 74km/sec-Mpc respectively,
appears to resist analysis meant to show it is zero. If the phantom energy
model should be realized then conservation of energy, even with an infinite
flat space, is gone.
The expansion of the universe also means we will not be able to observe
much physics that could be called “pre-cosmic,” or the quantum gravitation
of the pre-inflationary universe. Because of inflation and this 60-efolds
of expansion, expansion by ~ 10^{29}, a Planck scale region was expanded
from 10^{-33}cm to 10^{-4} cm. Since inflation began at 10^{30} sec in the
early universe, any Planck scale fluctuation involved with the generation
of the universe would have been 10^{-23}cm, and was expanded to 10^6 cm ---
beyond the scale of the then observable universe ~ 10cm. After inflation
the observable universe with a scale of ~ 10cm an possible Planck scale
process was stretched by more normal expansion to 10^{10} light years, and
might appear as some order anisotropy in the CMB. Using blackbody physics,
these quanta would have been a tiny aspect of the early universe. These
would be very difficult to find in the CMB. Beyond that, we cannot observe
anything. Any pre-cosmic physics emerged from something smaller than the
Planck scale and is expanded beyond any measurable scale on the CMB.
John Wheeler said that the ultimate law of physics is there is no law. We
may then have something similar to this, where what we call the laws of
physics are just local emergent pattern in the observable universe. At
large the universe may simply have no conservation laws and ultimate there
are globally no physical laws.
LC
On Sunday, January 8, 2023 at 6:52:32 AM UTC-6 johnk...@gmail.com wrote:
> This is a very good video, it describes the 4 different types of
> Multiverses that have been proposed. The first is purely a result of
> considerations from astronomical observations, the second and third come
> from considerations from both astronomy and quantum mechanics, and the
> fourth from mathematics and a large dose of speculative philosophy. It is
> possible that more than one of them is true, and type #1 almost certainly
> exists because our best measurements say space is flat over cosmic
> distances so if one denies its existence then one would have to conclude
> that the Earth really is the center of the universe. I could be wrong but I
> have the strong feeling that type 2 and 3 also exist but I don't think type
> 4 does because I think physics is more fundamental than mathematics.
>
> How Many Multiverses Are There?
> <https://www.youtube.com/watch?v=1jmNzlTd09E>
>
> John K Clark See what's on my new list at Extropolis
> <https://groups.google.com/g/extropolis>
>
> fa9
>
>
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