Relational quantum gravity

It has been put to me that instead of giving an overview of Relational
Quantum Gravity as I did in a recent post, I would be better to give the
actual calculation which illustrates that gravity can be regarded as a
perturbation to Minkowski metric resulting from a modification to qed
which prohibits two interactions of an electron within any small
interval of proper time. That way, if anyone wants to criticise it, at
least there is a chance they will criticise what I say, rather than what
they guess I am saying. I would welcome such criticism, particularly
from differential geometers. To define notations etc I will first define
teleparallelism.

Teleparallel Displacement.
For each point x in a manifold U, choose primed locally Minkowski
coordinates with an origin at x, and define the matrix

k^alpha_beta = x^alpha'_,beta(x) (1)

k is defined using different primed locally Minkowski coordinates for
each origin x, so that (A1.2) applies pointwise; it is not a
differential equation. k is not unique under this definition, and
depends upon the choice of primed locally Minkowski coordinates at each
point x; for a Lorentzian manifold we may choose locally Minkowski
coordinates at each point such that k is differentiable.

For the vector, A^alpha, at x, a coordinate space vector, barred to
distinguish it from an ordinary, or physical, vector, is defined by

A^bar^alpha(x) = k^alpha_beta(x) A^beta(x) (2)

This ensures that the coefficients of a coordinate space vector are
equal to the coefficients of the corresponding vector in the primed,
Minkowski, coordinates and preserves the inner product in this sense:
for vectors A and B at x,

yta_mu_nu A^bar^mu B^bar^nu

= yta_mu_nu k^mu_alpha k^nu_beta A^alpha B^beta

= yta_mu'_nu' x^mu'_,alpha x^nu'_,beta A^alpha B^beta

= g_alpha_beta A^alpha B^beta (3)

(3) is true for any vectors A, B at x. So

g_alpha_beta(x) = yta^mu_nu k^mu_alpha(x) k^nu_beta(x). (4)

(4) gives the physical metric in terms of the variable scale
coefficients, k^mu_alpha(x), of coordinate space vectors compared to
physical vectors. (note: substituting (1) into (4) is a statement that
locally Minkowski coordinates can be defined with an origin at any
point, x, not that space is flat because different primed coordinates
are associated with each origin).

A short rod placed at x is described by vector, A(x). An identical short
rod is placed at y, so that its coordinate space vector is parallel to
A(y). It is described by a vector, A(y), whose length is unchanged,
A^2(x)=A^2(y).

Define: A(y) is teleparallel to A(x) if and only if the coordinate space
components are equal

k^alpha_beta(y) A^beta(y) = A^bar(y)
= A^bar(x)
= k^alpha_beta(x) A^beta(x). (5)

If coordinate space is tangent to local Minkowski space at x, axes can
be chosen such that k^alpha_beta=delta^alpha_beta and (5) reduces to

k^alpha_beta(y) A^beta(y) = A^bar(y)
= A^bar(x)
= A^alpha(x). (6)

Multiply both sides of (6) by yta_alpha_gamma k^gamma_mu(y) and use (4):

A_mu(y) = A_gamma(x)k^gamma_mu(y).

Parallel transport is a sequence of infinitesimal parallel displacements
in the form (3.2.3)

Quantum Coordinates:
In relational quantum gravity quantum coordinates are defined in which
the speed of light is 1 and free wave functions obey plane wave motion
in the radial direction, so that momentum remains teleparallel. It is an
assumption that this is true (I have not proved a theorem, but I think
this is true in general relativity when cosmological expansion is
neglected. It is not true in gtr when expansion is taken into
consideration, and leads to different predictions for cosmological
redshift). In quantum coordinates the wave function takes the form

<x|f> = (2pi)^-3/2 integral d^3p^bar <p^bar|f> exp(-i x^bar.p^bar) (7)

Both photons and electrons obey wave functions of this general form. An
orthodox interpretation of quantum mechanics is used in which the wave
function does not directly describe reality between measured states, and
is simply a device for calculating probabilities of measurement results.

Schwarzschild
Bondi's k-calculus finds the metric for special relativity by
postulating instantaneous reflection of radar at an event whose time and
position is to be determined. There is no empirical basis for such an
assumption and a natural generalisation is to hypothesise a small time
delay between absorption and emission in proper time of a fundamental
charged particle (electron or quark) reflecting electromagnetic
radiation. Such a delay affects any empirical definition of space-time
measurement (e.g. SI units), and I seek to analyse the geometric
implications. The metric is determined as in the k-calculus for special
relativity, from the minimum time for the return of information
reflected at an event. But now this minimum net time depends not only on
the maximum theoretical speed of information, c, but also on the least
proper time between absorption and emission in the reflection of a
photon. We postulate a proper time delay 4GM, where M is the mass of the
reflecting particle and 4G is a constant of proportionality. I will show
that this results in a Schwarzschild geometry, and that G may be
identified with the gravitational constant. Special relativity can be
recovered in the limit in which G goes to zero (Allowing G to go to zero
introduces the Landau pole, so this limit may not be valid. I have shown
elsewhere that if 0<4GM is small the model yields Maxwell's equations in
the classical correspondence and is identical in its predictions with
Scharf's finite QED, up to the perturbation of the metric described
here).

For the purpose of analysis, I consider a static system and calculate
the metric at particular time, so expansion is ignored. I consider only
the effect on geometry caused by the time delay in reflection at a
single gravitating particle at O, as measured by a distant observer at
A, where O and A are stationary in quantum coordinates. An isolated
elementary particle in an eigenstate of position has spherical symmetry
and spacetime diagrams may be used to show a radial coordinate in n
dimensions without loss of generality. A spacetime diagram (figure 1) is
drawn using quantum coordinates, (t,r,theta,phi), with an origin at O,
so that light is shown at 45deg and lines of equal time are horizontal.
The observer uses primed coordinates (t',r',theta',phi') with respect to
a clock at A and also using an origin at O, and such that the axes
coincide, theta'=theta and phi'=phi. The primed coordinates are defined
locally by the observer at A using the radar method, and are locally
Minkowski. Then the primed and unprimed coordinates are related by

x^mu' = x^mu'_,nu x^nu

where x^mu'_,nu = k^mu_nu is diagonal.

figure 1

t_2 |photon returns |
|\ |
| \ |
| \ |
| \ |
| \ |
| \ |
| \ |
|coordinate \ |
|distance r-2GM \photon reflected with delay 4GM
|________________| \ Apparent distance r
| | /
| /photon absorbed
| / |
| / |
| / |
| / |
| / |
| / |
| / |
|/ |
t_1 |photon emitted |
| |
|Observer A |particle O

The radar method is used to define coordinates, according to standard
formulae

r = (t_2-t_1)/2 and t = (t_2 + t_1)/2

Consider a photon emitted at t'=0 from displacement r' from the observer
(figure 2).

|t'=k^0_0 r'^bar
|\
| \
| \
| \
|________\____
| r'^bar = k^1_1 r'

According to (7) the photon is emitted as though from the corresponding
coordinate space displacement vector, r'bar. Since light is drawn at
45deg, it is observed at time

t'=k^0_0 r' = k^0_0 k^1_1 r.

But the observer is using Minkowski coordinates in which t'=t. So

k^0_0 k^1_1 = 1.

Let k=k^0_0. Then

k^1_1 = 1/k.

Due to spherical symmetry there is no change in the relationship between
metric distance and coordinate distance associated with rotation, so

k^2_2 = k^3_3 = 1

Using radar referred to a clock at A, the observer measures a radial
coordinate,

r' = (t'_2 - t'_1)/2 = k^0_0 (t_2 - t_1)/2 = k^0_0 r.

But the position coordinate of A in quantum coordinate space is

r - 2GM = r'/k^1_1.

Hence

k^0_0 r = k^1_1(r-2GM),

so that

k^ = 1 - 2GM/r. (8)

From (4) and (8) we find the Schwarzschild metric:

ds^2=(1-2GM/r)dt^2 - (1 - 2GM/r)^-1 dr^2 + r^2(dtheta^+sintheta dphi^2)

Of course this argument only applies for an idealised experiment,
reflection off an electron in an eigenstate of position. General
relativity is not a linear theory, so there is still a bit of work to do
to justify going from Schwarzschild, being a particular solution of the
EFE to establishing the EFE in the general case. But I'll leave that for
another day.

Regards

--
Charles Francis
substitute charles for NotI to email

.

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