On Saturday, September 5, 2020 at 3:36:26 AM UTC-5, PengKuan Em wrote:
a) Material clock
What is time? This question is tricky because in relativity time-rate
changes when frame of reference changes. Time-rate changing is puzzling because it is in conflict with our intuition that time is the flow of
ticks of clocks of which the mechanical structure does not change. Then
how to clearly explain the contradiction between the constant flow of
ticks delivered by clocks and the relativistic time dilation?
We can fix that. Make it both. A coordinate time (t) and historical
time (s), which we'll identify as proper time. Throw it in as another coordinate too. The Minkowski line element for proper time
ds^2 = dt^2 - (1/c)^2 (dx^2 + dy^2 + dz^2)
then becomes
ds^2 - dt^2 + (1/c)^2 (dx^2 + dy^2 + dz^2) = 0.
Now ... let's regularize it by transforming the coordinate to the
*difference* of proper time and coordinate time, defining
u := c^2 (s - t).
Then, the line element can be rewritten as the quadratic invariant:
dx^2 + dy^2 + dz^2 + 2 dt du + (1/c)^2 du^2 = 0
and, in addition, we have a linear invariant
ds := dt + (1/c)^2 du.
which may be interpreted as a "soldiering form" which ties the
historical "flowing" time onto the space-time geometry.
In the resulting geometry, it's *all* *four* *dimensions* which
flow in time, not just space! It's both "block time" and "flowing
time" at the same time. The entire block, itself, is flowing in
time!
To find what these things mean, consider first that these both have non-relativistic limits:
dx^2 + dy^2 + dz^2 + 2 dt du = 0,
ds = dt.
In non-relativistic theory, by this account, it is justified to
treat coordinate time as the "flowing" time. On account of this,
the two may be safely confused in non-relativistic theory.
But by stepping forward into relativity and then moving back into non-relativistic form, an extra item has cropped up that wasn't
there before and now, suddenly, you also actually have a space-time
metric for non-relativistic theory ... and one that morphs continuously
into the metric for relativity by adjusting a parameter alpha from
0 to (1/c)^2, with the generic case being
dx^2 + dy^2 + dz^2 + 2 dt du + alpha du^2 and ds = dt + alpha du.
The quadratic invariant identifies a metric that *always* has a 4+1
signature, for *all* values of alpha.
Second, consider what the most general linear transforms are that
leave these invariants intact; denoting infinitesimal transforms
by D(...):
dx D(dx) + dy D(dy) + dz D(dz) + (dt + (1/c)^2 du) D(du) + du D(dt) = 0,
0 = D(ds) = D(dt) + (1/c)^2 D(du).
Substituting the second equation D(dt) = -(1/c)^2 D(du) into the
first yields:
dx D(dx) + dy D(dy) + dz D(dz) = -dt D(du).
The general solution is:
D(dx, dy, dz) = omega x (dx, dy, dz) - upsilon dt
D(du) = upsilon . (dx, dy, dz)
D(dt) = -alpha upsilon . (dx, dy, dz)
where alpha is the above-mentioned parameter and the two new vectors are:
omega = infinitesimal rotations,
upsilon = infinitesimal boosts,
() . () denotes 3-vector dot-product,
() x () denotes 3-vector cross-product.
This is the 5-dimensional representation of the Lorentz group (when
alpha = (1/c)^2) and of the Galilei group (when alpha = 0) ... which
*not* the Galilei group, but the Bargmann group!
All cases are restrictions of the symmetry group SO(4,1), for the
4+1 metric, that correspond to the little group for the linear
invariant ds.
Now, to determine what this means, consider how the mass (m),
momentum (p) and kinetic energy (H) transform in non-relativistic
theory:
D(m) = 0, D(p) = omega x p - upsilon m, D(H) = -upsilon.p.
Under the classical version of the correspondence rule
p-hat = -i h-bar del, H-hat = i h-bar @/@t
(@ denotes the curly-d partial derivative symbol)
one has the correspondence
p <-> -del, H <-> @/@t
and this leads to the consideration of the corresponding one-form:
p.dr - H dt.
What is the transform of this?
D(p.dr - H dt)
= (omega x p - upsilon m).dr + p.(omega x dr - upsilon dt)
- (-upsilon.p) dt - H (0)
= -m (upsilon.dr)
= -m D(du)
= -D(m du),
This puts the spot-light on the one-form
p.dr - H dt + m du
showing that it is actually an invariant.
If we adopt these same quantities for Relativity and *continue* to
assume that this is the case for the relativistic form, by turning
on the parameter alpha = 0 to alpha = (1/c)^2, then the assumption
that this be invariant leads to:
0 = D(p.dr - H dt + m du)
= Dp.dr + p.(omega x dr - upsilon dt)
- DH dt - H (-alpha upsilon.dr)
+ Dm du + m (upsilon.dr)
= (Dp - omega x p + upsilon (m + alpha H)).dr
- (DH + upsilon.p) dt + (Dm) du
then we obtain the following transforms
Dp = omega x p - upsilon M
DH = -upsilon.p
Dm = 0
which singles out the "moving" mass M = m + alpha H as the mass
that goes with the momentum in the formula (momentum = mass times
velocity). Its transform, derived from those above, is
DM = -alpha upsilon.p
and, as a consequence, we find the following as the two invariants
under these transforms:
Linear invariant: mu := M - alpha H = m,
Quadratic invariant: lambda := p^2 + 2MH - alpha H^2 = 0.
The coordinate u is conjugate to m and in a quantized theory, m
would be represented as i h-bar @/@u. The coordinates (s,u), when
used in place of (t,u) produce the one-form
m du - H dt = m du - H (ds - alpha du) = (m + alpha H) du - H ds = M du - H ds.
So the corresponding operator forms would be, respectively,
M <-> i h-bar (@/@u)_s, H <-> -i h-bar (@/@s)_u.
The energy term H - which is the relativistic form of the *kinetic*
energy (not the total energy) is conjugate to the proper time s,
provided that it and u be taken together as the coordinates.
That generalizes the non-relativistic prescription of taking H to
be the generator of "flowing time".
This clean, continuous deformation from non-relativistic to
relativistic form is obscured because in Relativity, one normally
takes the *total* energy E, instead of the kinetic energy H as the
relevant component of momentum.
Here, that arises from the fact that the 4D sub-representation (M,p)
of the 5D representation (M,p,H) closes under the transforms, so
(M,p) forms a Minkowski 4-vector. M is, of course, converted to E
by way of the equation
E = M c^2.
So the corresponding transforms would read:
D(p) = -(1/c)^2 upsilon E, D(E) = -upsilon.p
and we may find the rest-mass as the square of the "mass shell"
invariant, which can be constructed from the quadratic invariant
lambda and linear invariant mu as:
mu^2 - alpha lambda
= (M - alpha H)^2 - alpha (p^2 - 2MH + alpha H)
= M^2 - alpha p^2
= (E/c^2)^2 - (p/c)^2 = m^2.
Strictly speaking, this construction goes BEYOND relativity, since
it has 5 components. The difference can be brought out clearly by
considering what the rest-frame form of the 5-vector (H,p,M) is in non-relativistic theory
(H,p,M) -> (U,0,m), in the rest frame; U = internal energy.
If we adopt the same assumption here, then the respective invariants
would, generalize to:
lambda = p^2 - 2MH + alpha H^2 -> 0^2 - 2mU + alpha U^2,
mu = M - alpha H -> m - alpha U.
The constructs of Relativity are obtained by constraining U = 0.
The inclusion of a non-zero U corresponds to the inclusion of a 5th
coordinate (be it s or u) and of the splitting of the energy E into
two components: moving mass M and kinetic energy H.
This generalization allows one to consider more general systems
that may not have a rest-frame, and it continues to make sense in
those context, while the notion of "rest mass" (m) no longer carries
any meaning, except for systems whose mass shell invariant is
non-negative M^2 >= alpha p^2 (i.e. tardions, luxons and the
"vacuons", a.k.a. homogeneous states where M = 0, p = 0).
For curved space-times ...
If you repeat all of the same processes above with the Schwarzschild
solution:
proper time metric:
ds^2 = dt^2 (1 + 2 alpha U)
- alpha (dr^2/(1 + 2 alpha U) + r^2 ((d theta)^2 + (sin theta d phi)^2))
and
soldiering form: ds = dt + alpha du
where U = -2GM/r is the gravitational potential of a body of mass M...
and substitute and regularize, you obtain:
dr^2/(1 + 2 alpha U) + r^2 ((d theta)^2 + (sin theta d phi)^2)
+ 2 dt du + alpha du^2 - 2U dt^2 = 0
In the non-relativistic form of this - for alpha = 0 - the spatial
coordinates reduce to Euclidean form and can be replaced by Cartesian coordinates to yield the metric:
dx^2 + dy^2 + dz^2 + 2 dt du - 2U dt^2 = 0.
The geodesics for this metric are *precisely* the motions of a body
moving under the influence of an energy potential U per unit mass;
i.e. moving under the influence of a potential energy in a way that
respects the equivalence principle. For U = -GM/r, that's the field
given by Newton's law of gravity.
I'm not the only one doing things this way. As I discovered a short
while ago there are these...
5D Generalized Inflationary Cosmology
L. Burakovsky∗ and L.P. Horwitz
https://arxiv.org/abs/hep-th/9508120
Their tau is my s. They take it out to a more general context -
curved space-times. I've toyed with this before, and they got the
same expression (equation 2.6) as I encountered, for the 5D form
of the radiation-dominant case of the FRW metric.
This *might* all tie into the MacDowell-Mansouri formulation - they
succeed in wrapping up the connection and frame field into a single
gauge field for gravity which works whenever the cosmological
coefficient Lambda is non-zero.
MacDowell–Mansouri Gravity and Cartan Geometry
Derek K. Wise
https://arxiv.org/abs/gr-qc/0611154
One of the reasons I say it's probably related, is because Mansouri
is already known as one of the people involved with dealing with
"signature changing" geometries. The FRW Big Bang metrics - especially
the radiation dominant one - has a null surface at time t = 0 and
its metric passes continually from one corresponding to alpha > 0
for t > 0, to alpha = 0 at t = 0, to Euclidean 4D form alpha < 0
at t < 0. Mansouri showed that the sectors of a signature-changing
metric with a null initial surface can only be consistently stitched
together under a "junction condition" that *forces* the cosmology
to be inflationary.
Signature Change, Inflation, and the Cosmological Constant
Reza Mansouri and Kourosh Nozari
https://arxiv.org/pdf/gr-qc/9806109.pdf
Also in the same category and potentially related:
Particles as Wilson lines of gravitational field
https://www.researchgate.net/publication/1971397_Particles_as_Wilson_lines_of_gravitational_field
which works within the MacDowell-Mansouri gravity gauge-theory
formulation.
I've used this as a means to continuously deform electromagnetic
and gauge theory from relativistic to non-relativistic form. It
just so happens that mostly the same result of that process is
described here:
Galilean Geometry in Condensed Matter Systems
Michael Geracie
https://arxiv.org/abs/1611.01198
(Note, particularly, the discussion of Bargmann Geometry - the
alpha = 0 case of the geometry I described. He also associates the
extra coordinate with mass, using M as the coordinate index.)
Curved non-relativistic spacetimes, Newtonian gravitation and massive
matter
Michael Geracie, Kartik Prabhu, Matthew M. Roberts
https://arxiv.org/abs/1503.02682
also works with Bargmann geometry and the extra coordinate.
Newton-Cartan Gravity Revisited
Roel Andringa
https://www.rug.nl/research/portal/files/34926446/Complete_thesis.pdf
Section 4.4 deals with the Bargmann algebra. There are 11 generators.
It may be derived as the algebra associated with the transforms of
the coordinates that I described above:
D(dx,dy,dz) = omega x (dx,dy,dz) - upsilon dt,
D(dt) = -alpha upsilon.(dx,dy,dz),
D(du) = upsilon.(dx,dy,dz).
by integrating, which produces constants of integration:
D(x,y,z) = omega x (x,y,z) - upsilon t + epsilon,
Dt = -alpha upsilon.(x,y,z) + tau
Du = upsilon.(x,y,z) + psi
that correspond to:
epsilon: infinitesimal spatial translations,
tau: infinitesimal time translations,
psi: infinitesimal translations along the u direction.
That adds in 5 extra generators: 3 for the components of the vector,
epsilon, and 1 each for tau and psi.
That's in section 4.5. Their Z is my "mu".
The attempt to use this as a device to continuously morph between
General Relativity (suitably extended to 5D) and Newton-Cartan
Gravity (of a Bargmann geometry) -- along with the issues that crop
up when doing so -- is described here:
Bargmann Structures and Newton-Cartan Theorem
Duval, Burdet, Kuenzel, Perrin
Physical Review D 31(8), 1985 April 15
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.31.1841
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