Special Relativity with an Aether Velocity

[Rock Brentwood]

The issue of causal signature has importance for four major
applications:
* the design of more refined empirical tests to determine what the
spacetime signature actually is
* the retroactive development of classical theory (just to address the
question of how it would be done today)
* to formulate more generalized spacetime geometries where signature
may not be part of the background, but dynamically determined.
* to describe the transition from Lorentzian -> Euclidean signatures;
particularly, what happens on the boundary between the two.

For Lorentzian spacetimes, the metric g_{mn} and its dual g^{mn} have
signatures (+,-,-,-) or (-,+,+,+) depending on which author's
convention is adopted. For Euclidean spacetimes, the signature is (+,+,
+,+) or (-,-,-,-); and for Galilean spacetimes: g_{mn} has signature
(+,0,0,0), while g^{mn} is no longer the inverse of g_{mn} but has
signature (0,+,+,+).

(There is even an "Aristotlean" spacetime where g_{mn} has signature
(0,+,+,+) and g^{mn} the signature (+,0,0,0)).

The constitutive law for electromagnetism (and gauge theory) in their
present form are ill-suited for this purpose. However, it is possible
to repair the situation by generalizing electromagnetic theory into a
TWO parameter family of theories. One parameter V represents light
speed, the other c represents the invariant velocity. Necessarily,
this involves the inclusion of a reference velocity G.

Various limits recover the following:
* c -> infinity, V remains finite ... Maxwell's theory
* V -> c, c remains finite ... relativistic electrodynamics; i.e.,
Maxwell's theory as modified by Lorentz and later by Einstein and
Poincare'
* |G| -> V ... the answer to Einstein's question of what happens as
you "travel alongside a light beam".
* |G| -> infinity ... ??? (something weird)

In the limit V -> c, G becomes "superfluous" and drops out from
expressions ... EXCEPT if one first takes the |G| -> c limit!

It is "widely known" that gauge theory "has no Galilean limit". This
ramifies also to electromagnetism, though the comment is not properly
appreciated there. In the latter case what it means is that
permittivity (epsilon) is strictly relevativistic. This can be seen by
writing the Lorentz invariants in a suitable form:
I = 1/2 (E^2 - B^2 c^2), J = E.B, K = -I/c^2 = 1/2 (B^2 - (E/c)^2).

Assuming the dynamics for the field are given by a Lagrangian density
L, the electric displacement D and magnetic field strength H will be
the respective gradients:
D = dL/dE, H = -dL/dB.
If the Lagrangian is a function solely of the invariants, then one can
define
epsilon = dL/dI, theta = dL/dJ, mu = -dL/dK = (1/c)^2 (1/epsilon).

From these, one derives the following relations
D = epsilon E + theta B = (1/c)^2 (1/mu) E + theta B
H = epsilon c^2 B - theta E = (1/mu) B - theta E.

In the non-relativistic limit, as (1/c)^2 -> 0, the permittivity is
gone. One only has D = theta B, with no involvement of E.

A similar set of observations holds for gauge fields, though one can
recover SOME dependence on the E field for D by exploiting the cubic
Lorentz invariants. But the coupling coefficient, which plays the
analogous role to (epsilon) here by virtue of the correspondence 1/g^2
= epsilon c, would no longer be well-defined.

Yet, Maxwell wrote down a theory that was (at the time of the
treatise) Galilean. How? The answer is that Maxwell's field vectors A,
B, C (= J + dD/dt), D, E, F (= force density), H, I (= magnetization),
J ... which now has a gap where G used to be ... originally had a G.
That's the velocity with reference to the vacuum.

Making use of G, one can recover a semblance of the I invariant by
writing I' = (E + G x B)^2. Maxwell's epsilon is the derivative dL/
dI'.

Thus, the coefficients mu and epsilon are split off from one another.
In the "stationary" frame (G = 0), one has
D = epsilon E + theta B, H = (1/mu) B - theta E.
By stipulation, G transforms as a velocity, so when transforming to
the "moving" frame (non-zero G), one gets:
D = epsilno (E + G x B) + theta B,
H = (1/mu) B - theta (E + G x B) + epsilon G x (E + G x B).
The latter relation can then be written
B = mu (H - G x D) + (theta mu/epsilon) D.

Light speed becomes a SEPARATE parameter, V = 1/(mu epsilon).

This is the Galilean version of the Lorentz relations (with an extra
axial coefficient theta included for completeness).

This generalizes to other signatures. One starts out by adopting the
same relations as before for the "stationary" frame G = 0:

D = epsilon E + theta B, H = (1/mu) B - theta E
with
epsilon mu = (1/V)^2.

Upon Lorentz transformation to a non-zero G frame, this becomes
D = epsilon E + theta B
+ epsilon (c^2-V^2)/(c^2-G^2) Gx (B - GxE/c^2)
H = (1/mu) B - theta E +
+ epsilon (c^2-V^2)/(c^2-G^2) G x (E + GxB).

In the limit as c -> infinity, one recovers Maxwell's relations
D = epsilon E + theta B + epsilon G x B
H = (1/mu) B - theta E + epsilon G x (E + G x B).

In the limit as V -> c, G becomes superfluous, and one recovers the
Lorentz relations
D = epsilon E + theta B, H = (1/mu) B - theta E.

In the limit as |G| -> V, one gets the following
D = epsilon (E + G x (B - GxE/c^2) + theta B
H = (1/mu) B - theta E + epsilon G x (E + G x B).
Interestingly, this yields a non-trivial result even if one takes V =
c.

Finally, in the limit as |G| -> infinity, one recovers the relations:
D = epsilon E + theta B
+ epsilon (1-(V/c)^2) Gx (GxE)/G^2
H = (1/mu) B - theta E +
- epsilon c^2 (1-(V/c)^2) G x (GxB)/G^2.
However, taking V -> c, the G terms will drop out.

[Dirk Bruere at NeoPax]

Rock Brentwood wrote:
> The issue of causal signature has importance for four major
> applications:
> * the design of more refined empirical tests to determine what the
> spacetime signature actually is
> * the retroactive development of classical theory (just to address the
> question of how it would be done today)

Which reminds me.
Is it possible to get the results of GR by tweaking Newtonian theory, or
does so much have to be altered that the result is effectively GR?

--
Dirk

http://www.transcendence.me.uk/ - Transcendence UK
Remote Viewing classes in London

[Oh No]

Thus spake Dirk Bruere at NeoPax <dirk.bruere@gmail.com>
>Rock Brentwood wrote:
>> The issue of causal signature has importance for four major
>> applications:
>> * the design of more refined empirical tests to determine what the
>> spacetime signature actually is
>> * the retroactive development of classical theory (just to address the
>> question of how it would be done today)
>
>Which reminds me.
>Is it possible to get the results of GR by tweaking Newtonian theory, or
>does so much have to be altered that the result is effectively GR?
>
I would say, not really. There is a fundamental conceptual difference
between a geometrical theory and a potential theory. In gtr,
gravitational redshift can be thought of as a consequence of the
changing rate of clocks wrt each other at different positions. This can
be reinterpreted locally as a change in a potential in a flat geometry
(the Newtonian correspondence).

Of course Einstein did find gtr by using the equivalence principle to
reverse the argument, so, in one sense, yes, the result is gtr, but in
the process one loses the major concepts of Newtonian theory, absolute
space and the potential itself. IOW, I don't see that gtr can be
regarded simply as a tweaked Newtonian theory. It is much deeper than
that.

Regards

--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)

http://www.teleconnection.info/rqg/MainIndex

[Einar Andreas Rodland]

"Dirk Bruere at NeoPax" <dirk.bruere@gmail.com> wrote in message
news:6c4u9pF3e0m9eU1@mid.individual.net...

> Is it possible to get the results of GR by tweaking Newtonian theory, or
> does so much have to be altered that the result is effectively GR?

While some things can be explained, although in a non-rigorous manner,
using only classical theory, some very fundamental things cannot and
would require more than just tweaking.

Let's take gravitational red-shift (or blue-shift). Using Newton and
Maxwell, this should not happen, but it does happen so Newton+Maxwell is
wrong in this respect. But let's instead assume we have some tweaked
version of Newton that allows graviational red-/blue-shift.

In fact, one might even argue that, given some assumptions, this
red-/blue-shift could be deduced even from Newton. E.g., if you start
with a photon with energy E=hf and let it fall a distance s through a
gravitational field with acceleration g, it should gain energy mgs where
m=E/c^2=hf/c^2, and so experience a blue-shift to f'=f*(1+gs/c^2). I'm
sure this could be deduced in numerous other ways as well, e.g. assuming
equivalence between an inertial frame and one in free fall in a
gravitational field, not assuming mass-energy duality. My point is not
how to deduce this, just that one might even argue red-/blue-shift of
light from Newtonian theory alone: after all, bending of light passing
heavy objects (e.g. the Sun) was argued ahead of testing GR.

So, if we have a laser that sends out light with frequency f, when this
beam drops a height s, the frequency changes to f'. If the one with the
laser keeps the beam on for a time T, it has sent out N=T*f complete
periods of the wave. The receiver should receive the same number of
waves: after all, the wave nature is well-established, and you could
imagine following the "tops and bottoms" of the wave from the sender to
the receiver, so the number of periods cannot change easily. However, if
the number of periods is the same for the sender and reveiver, but the
frequency is different, then the time it takes to receive must be
different: i.e. N=T*f=T'*f' where T'=T*f/f' is the time it takes the
receiver to reveice the beam. So, it follows that the two must perceive
time differently: when the sender claims to have sent a beam lasting a
time T, the receiver receivs a beam that lasted a time T'. I.e. you
have two observers at rest in a stationary space who experience time
going at "different speeds".

This basically demonstrates in a very direct way that the time-component
of the metric used in GR is required. When you start tweaking the
metric, you get GR. Thus, I suppose "tweaked Newtonian theory" would be
just GR without the geometrical interpretation, which would be
mathematically equivalent but less intuitive/pedagogical.

Einar

[Rock Brentwood]

On Jun 23, 3:34 pm, Dirk Bruere at NeoPax <dirk.bru...@gmail.com>
wrote:
> Which reminds me.
> Is it possible to get the results of GR by tweaking Newtonian theory, or
> does so much have to be altered that the result is effectively GR?

Actually I wrote down a unified tetrad formalism for Galilean &
Poincare' symmetry -- the "quasi-Galilen tetrad". It's the 3+1 answer
to Penrose's formalism. I glommed it into part 3 of

The Differential Geometry of Gauge Theory
http://federation.g3z.com/Physics/index.htm#Bleecker

Among other things, it shows that Gravito-Magnetism is NOT a
relativistic effect! Post-Newtonian, yes (playing conservatively), but
not post Galilei.

There is a defect, however. The central extension of the Galilei group
has 11 symmetry generators. To have a consistent Galilen limit, one
needs to extend the Poincare' group likewise with an 11th generator. I
show how this is done in

The Wigner Classification of Galilei/Poincare'/Euclid
http://federation.g3z.com/Physics/index.htm#GeneralizedWigner

(c.f., Annals of Physics, 323(5) 1191-1214, which I just sighted).

This is where things get interesting. The 10th and 11th symmetry
generators take the place of time translation. When moving over to the
Galilei limit, one of them becomes separated out as a U(1) field --
the one associated with time translation ... because Newton-Cartan
spacetimes are foliated by time. The other, the 11th, however becomes
mixed in with the other 9.

When going in the opposite direction toward Poincare', the U(1)
generator becomes mixed in with the other 9 generators because
Lorentzian spaces are not foliated by time. So, with the (former) U(1)
generator, you get the usual 6 degrees of freedom for the Lorentz
connection + 4 for the frame. The 11th, which was mixed in with
everyone else in the Galilei limit, is now separated out as a U(1)
field in the Poincare' limit.

So, the configuration variables of the field are easy to lay out
(frame, torsion, connection, curvature). But the dynamics are somewhat
up in the air. This has partly to do with the fact that the single
metric of the Poincare' based symmetry group splits into 2 separate
metrics in the Galilei based theory. So, there should really be two
separate sets of stress tensors, frame fields, torsions; and a
multiplicity of different connections ... one for each arrangement of
indices up or down.

I don't know who's sorted out that mess. There are Newton-Cartan
theories, but I haven't seen much of any (only one) that even got as
far as noticing the 11th-parameter oversight.

I'm not totally sure what the 11-parameter Poincare' gauge theory is
-- Poincare + Dilation, maybe? If so, then that gives you 6 connection
1-forms for the Lorentz group, 1 for dilation, and 4 frame 1-forms
associated with translation. That's the most general connection
compatible with spinors -- in the sense that the covariant derivative
of the "soldiering" form sigma^mu_{AA'} remains 0.

Interestingly, both the "Generalized Wigner" article and the Annals of
Physics article (independently) developed the same generalized spinor
formalism for Galilei. So, it should actually be possible to expand
the above into a spinor form for Galileii gravity.

[Rock Brentwood]

> Actually I wrote down a unified tetrad formalism for Galilean &
> Poincare' symmetry -- the "quasi-Galilen tetrad". It's the 3+1 answer
> to Penrose's formalism. I glommed it into part 3 of
>
> The Differential Geometry of Gauge Theory
> http://federation.g3z.com/Physics/index.htm#Bleecker

Oops. Wrong cite. It's
http://federation.g3z.com/Physics/index.htm#Solutions

> On Jun 23, 3:34�pm, Dirk Bruere at NeoPax <dirk.bru...@gmail.com>
> wrote:
> > Which reminds me.
> > Is it possible to get the results of GR by tweaking Newtonian theory, or
> > does so much have to be altered that the result is effectively GR?

Expanding on my previous reply: yes.

In fact, the fixes required resolve a mystery that's been in my mind
for a long time. You see, back when I first "learned" SR -- actually
BEFORE I learned it -- I came up with my own transformation which is
quite unlike the Lorentz transformation in that it effectively leaves
time intact, while doing something strange with the spatial
coordinates. In effect, I came up with 2 time-like axes; one which
remains absolute and the other which shifts "locally".

I never really understood what I did (in geometric terms), until
resolving the issue you're asking about. In fact, the extra coordinate
is required to match up with the 11th generator of the Galilei group.
In turn, this is needed to pass over to a consistent Galilean limit --
not just to the 10-parameter Galilean group, but its 11-parameter
central extension.

Without it, you can't resolve the correspondence limit! In that sense,
GR is not the correct inverse correspondence limit of the Newtonian
theory of gravity -- it only works with the 10 parameter group, not
the 11 parameter centrally extended group.

The 11th generator of the Galilei group is accommodated, which fixes
the "missing hole" left behind by the (1/c)^2 -> 0 limit. Without this
fix, then because you lose the ability to change the direction of the
equal-time planes in Galilean physics, (since simultaneity is absolute
in non-relativistic physics), you lose an important degree of freedom
in the corresponding gravitational field equations.

In fact, this has held up efforts by people in the past to come up
with a field law for Newtonian gravity in the same mould as Einstein's
equations that has a seamless limit with respect to it.

The fix means that the stress tensor becomes a 4x5 quantity T^{mu}_a;
mu = 0, 1, 2, 3; a = 1, 2, 3, 4, 5; representing the current density
for the P_a, the ath component of momentum; defining
P_0 = E = total energy
P = (P_1, P_2, P_3) = momentum
P_4 = H = kinetic energy
P_5 = M = relativistic mass.
where M = E/c^2, E - H = constant.

The mass shell P^2 - E^2/c^2 has to be changed to P^2 - 2MH + z H^2,
where z is the parameter that distinugishes Galilean spacetime (z = 0)
from Lorentzian space-time (z = (1/c)^2).

This leads to a 5-metric
g^{ab} P_a P_b = P^2 - 2MH + z H^2.
The most general linear transformations that preserve both this and
the 1-form
mu = M - z H
is the 11-parameter Galilei group, itself.

This is how the recent journal reference I cited also handles the
issue (it proceeds to write down a path integral form for the Galilean
Dirac equation).

The generalized Lorentz transformations consists of 5 translations and
a 6-parameter homogeneous group. The infinitesimal form of the latter
is:
delta(P) = omega x P - u M
delta(M) = z delta(H) = -z u.P
delta(H) = -u.P
delta(mu) = 0

A coordinate form can be erected by writing out the corresponding
canonical 1-form:
theta = P.dr - H ds + M du = P.dr - H dt + mu du.
t = s - z u
dr = (dx, dy, dz)
The action of the Lorentz group on the coordinates is uniquely
determined by the requirement that it leave theta invariant. This
yields the following infinitesimal form:
delta(dr) = omega x dr - u dt
delta(du) = u.dr
delta(dt) = -z u.dr
delta(ds) = 0

You can see, here, that t is just the "real time"; while s is a
vestige of the "Galilean time" s that lingers on even when z > 0. The
coordinate u = (s - t)/z is conjugate to the mass. For the Galilean
limit, t -> s, and u loses its moorings from s and t and becomes
independent.

That's the non-trivial element that has to be added to GR to make it
have a Galilean limit. The coordinate s plays no direct role in the
dynamics ... in effect, space-time is foliated in 4-dimensional
sections along surfaces of constant s. However, as z -> 0, all the
variability you see on the right-hand side of
delta(dt) = -z u.dr
is lost and passes over to u:
delta(du) = u.dr.

This extra element has to remain in place when writing down the
Galilean form of GR.

So .. now you can write down the configuration variables for a
gravitational field dynamics. One has FIVE frame 1-forms, which I'll
call *x* = (theta^1, theta^2, theta^3); s = theta^4, u = theta^5;
along with t = theta^0 = s - z u.

One has FIVE torsion 1-forms, derived from these:
*X* = (Theta^1, Theta^2, Theta^3); S = Theta^4, U = Theta^5
along with T = Theta^0 = S - z U.

The most general connection that resolves the metric and 1-form has 6
independent components, which can be labelled *sigma* = (sigma_1,
sigma_2, sigma_3); and *alpha* = (alpha_1, alpha_2, alpha_3). One can
then write down the equations for the torsion and 6 curvature 2-forms
(*Sigma* and *Alpha*), by rote-following the Cartan structure
equations
d*x* - *sigma* x *x* + *alpha* t = *X*
dt - z *alpha*.*x* = T = S - z U; (t = s - z u)
du + *alpha*.*x* = U
ds = S
d*sigma* - 1/2 (*sigma* x *sigma*) + z/2 (*alpha* x *alpha*) =
*Sigma*
d*alpha* - *sigma* x *alpha* = *Alpha*
where I use 3-vector notation in conjunction with the differential
forms' wedge product (e.g. *alpha* t = (alpha_1 ^ t, alpha_2 ^ t,
alpha_3 ^ t); and *sigma* x *alpha* = *alpha* x *sigma*).

One can formulate the question: what is the most general algebraic
combination of these configuration coordinates that is Lorentz
invariant (hence, independent of *sigma* and *alpha*) that yields 4-
forms suitable for a Lagrangian?

Without the extra element (u and s), there are 6 combinations. The
extra elements make for 2 more. This leads to a potentially richer
field law --- though I haven't yet looked at it in detail.

Most importantly, the laws corresponding to the u and U fields are
CRITICALLY involved in formulating a consistent set of laws for the
stress tensor (especially the energy density). Lose this, and you lose
the ability to consistently ramp down from z = (1/c)^2 > 0 -> z = 0.
Newtonian gravity would have to be written in "by hand" instead of
derived, in this way, as a z -> 0 limit.

An interesting feature: one of the invariants is linear in the torsion
and is not present in the ordinary Lorentz-based gravity theory (X.x +
Ts)u, I think it is. This can lead to a non-zero torsion, even in the
absence of external sources -- another issue I haven't looked at in
detail yet.