On Wednesday, December 28, 2022 at 11:41:36 PM UTC-6 Bruce wrote:
> On Thu, Dec 29, 2022 at 4:34 PM Brent Meeker <[email protected]> wrote:
>
>> On 12/28/2022 9:01 PM, Bruce Kellett wrote:
>>
>> On Thu, Dec 29, 2022 at 3:29 PM Brent Meeker <[email protected]> wrote:
>>
>>> Of course one reason there are "laws of physics" is what my late friend
>>> Vic Stenger called Point Of View Invariance. This was his generalization
>>> of Emmy Noether's theorem that showed every symmetry implied a conservation
>>> law.
>>>
>>
>> That is not strictly true. It is only continuous symmetries of the
>> Lagrangian that imply conservation laws -- not all symmetries. For example,
>> the symmetries of a square under rotation and reflection do not generate
>> any conservation laws. Neither do discrete symmetries like parity and
>> charge conjugation.
>>
>> So momentum is conserved because we want any law of physics to be
>>> invariant under translation of a different location. Energy is conserved
>>> because we want the laws of physics to be the same at different times, etc.
>>>
>>
>> It is not what we want, it is what we find. We find that nature is
>> invariant under these continuous transformations, so we build those
>> symmetries into our laws.
>>
>>
>> Vic called in POVI because he wanted to extend it to transformations in
>> abstract spaces, e.g. gauge invariance. Of course the invariance depends
>> on the "point of view" in a sense. Things didn't look at all space
>> translation invariant to Aristotle. Galileo said ignore that your ship is
>> moving along the shore, just look at the dynamics in the cabin. So we
>> discovered these symmetries by learning what ignore as well as what to
>> measure.
>>
>
> The real point is that the laws are discovered, not imposed. The fact that
> continuous symmetries correspond to conservation laws was discovered only
> very much later. Most of the history of physics is about discovering what
> works -- what the laws might be. POVI was thought of only very late in the
> game, and is not a fundamental insight.
>
> Bruce
>
This begins to look a bit similar to the debate over whether mathematics is
objectively real or something invented. Emmy Noether gave consideration to
that 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
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