• #### SR versus Electrodynamics

From Nicolaas Vroom@21:1/5 to Nicolaas Vroom on Fri May 15 14:06:42 2020
Copied from the tread "in making a new theory cannot be postulated what is patentely false" in sci.physics.relativity

On Thursday, 14 May 2020 00:55:50 UTC+2, tjrob137 wrote:

**** Start of posting.
On 5/13/20 11:00 AM, Nicolaas Vroom wrote:
On Friday, 17 April 2020 02:49:37 UTC+2, tjrob137 wrote:
Indeed, the second postulate is not needed in a modern derivation of SR.

However, I also have a problem with this sentence.
Do you mean that the second postulate is not needed at all in relation to SR? or only in relation to a modern derivation of SR?

First remember that Einstein's 1905 paper was about ELECTRODYNAMICS.
Since then we have found it convenient to split his subject into two
separate theories:
1. SR, which models the (local) geometry of spacetime.
2. Electrodynamics, which models electromagnetic interactions.

Einstein's second postulate involves light, putting it FIRMLY in theory
#2. So it cannot be used in theory #1.

Today we understand group theory (which Einstein did not in 1905). We
can use group theory to show that there can be only three transformation
groups among inertial frames, and that just one of them, the Lorentz
group [#], agrees with all the experiments.

So Einstein's second postulate is not needed at all in theory #1 above,
SR as understood today by physicists.

Not all popular articles and elementary textbooks adhere
to the separation above. But that separation is necessary
to go from SR to GR -- that relies on the invariant speed
of SR, and has nothing to do with light.

[#] Well, the Poincare' group. Which is the same as the
inhomogeneous Lorentz group.

The second postulate discusses the speed of light and that seems very important in relation to SR, specific in relation to the invariant:
ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2

Do not be deceived into thinking that "c" there is the speed of light.
IT ISN'T. It represents the invariant speed of the Lorentz group.

Historically, "c" was used to represent the vacuum speed of light, and
"c" was also used to represent the invariant speed of the Lorentz group.
This has caused countless confusion, but it is only a confusion about
history and nomenclature -- the two theories above are distinct and
quite clear about the meanings of their symbols.

The remarkable thing is that the two meanings of "c" in those two
theories have the same numerical value, established experimentally to
high precision (yes, separate sets of experiments).

So in a very real sense, Einstein "lucked out" in 1905.
Had it happened that those two meanings of "c" were
significantly different, the history of physics would
have been VERY different.

not be updated?

Perhaps. Even probably. I have not looked at it.

Tom Roberts
****** End of posting

Nicolaas Vroom

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• From Ned Latham@21:1/5 to Nicolaas Vroom on Sat May 16 09:52:41 2020
Nicolaas Vroom wrote:

Copied from the tread "in making a new theory cannot be postulated
what is patentely false" in sci.physics.relativity

tjrob137 wrote:
Nicolaas Vroom wrote:
tjrob137 wrote:

Indeed, the second postulate is not needed in a modern
derivation of SR.

However, I also have a problem with this sentence.
Do you mean that the second postulate is not needed at all
in relation to SR? or only in relation to a modern derivation
of SR?

First remember that Einstein's 1905 paper was about
ELECTRODYNAMICS. Since then we have found it convenient to split
his subject into two separate theories:
1. SR, which models the (local) geometry of spacetime.
2. Electrodynamics, which models electromagnetic interactions.

Einstein's second postulate involves light, putting it FIRMLY in
theory #2. So it cannot be used in theory #1.

The second postulate is not even *relevant* to SR?

Today we understand group theory (which Einstein did not in 1905).
We can use group theory to show that there can be only three
transformation groups among inertial frames, and that just one of
them, the Lorentz group [#], agrees with all the experiments.

So Einstein's second postulate is not needed at all in theory #1 above,
SR as understood today by physicists.

Not all popular articles and elementary textbooks adhere
to the separation above. But that separation is necessary
to go from SR to GR -- that relies on the invariant speed
of SR, and has nothing to do with light.

[#] Well, the Poincare' group. Which is the same as the
inhomogeneous Lorentz group.

The second postulate discusses the speed of light and that seems very important in relation to SR, specific in relation to the invariant:
ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2

Do not be deceived into thinking that "c" there is the speed of light.
IT ISN'T. It represents the invariant speed of the Lorentz group.

Historically, "c" was used to represent the vacuum speed of light, and
"c" was also used to represent the invariant speed of the Lorentz group. This has caused countless confusion, but it is only a confusion about history and nomenclature -- the two theories above are distinct and
quite clear about the meanings of their symbols.

The remarkable thing is that the two meanings of "c" in those two
theories have the same numerical value, established experimentally to
high precision (yes, separate sets of experiments).

So in a very real sense, Einstein "lucked out" in 1905.
Had it happened that those two meanings of "c" were
significantly different, the history of physics would
have been VERY different.

not be updated?

Perhaps. Even probably. I have not looked at it.

Tom Roberts

Nicolaas Vroom

A question, actually.

SR "models the (local) geometry of spacetime": to what purpose?

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• From Nicolaas Vroom@21:1/5 to Ned Latham on Tue May 19 21:05:15 2020
On Saturday, 16 May 2020 11:52:44 UTC+2, Ned Latham wrote:
Nicolaas Vroom wrote:

tjrob137 wrote:

Einstein's second postulate involves light, putting it FIRMLY in
theory #2. So it cannot be used in theory #1.

The second postulate is not even *relevant* to SR?

See below.

A question, actually.

SR "models the (local) geometry of spacetime": to what purpose?

The concept: SR "models the (local) geometry of spacetime" is not clear.
IMO it makes more sense to claim that:
GR "models the (local) geometry of spacetime"
But even that is not clear.
How can any theory have any influence on the behaviour of free
moving objects through space?
i.e. on the behaviour of the movement of the planets around the Sun.

The only thing that we humans can do is to describe the movements
of these objects and if these movements are stable to describe them
in a more mathematical way.

Anyway (My opinion) spacetime is not a physical concept.
does not mention the word spacetime.
It should be mentioned that there is nothing wrong to draw
in one figure an x-axis and a time-axis in order to draw the path
of a particle through time but that does not mean that there exist
something like spacetime.

When you want to understand anything the first step is to identify
the process that you want to explain.
For example a clock
The next is what are the details. For example, a clock using light signals.
If that is the case it must be clear that the laws that describe
that behaviour, in principle, can not be used for any other clock
for example an atomic clock (wich 'uses' its own laws)

The next step is to describe at least one specific experiment
that you have used with a clock (using light signals).

The next step is to mention if your experiment requires certain assumptions. The most important assumption is that the speed of light,
locally, is considered the same, in all directions.

One final remark is that the description of your experiment
should be as detailed as possible, from start to end.
It is not enough to write that one clock is at rest and that a second
clock moves with a constant speed.

Nicolaas Vroom

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• From rockbrentwood@gmail.com@21:1/5 to Nicolaas Vroom on Thu Jun 18 05:55:54 2020
On Friday, May 15, 2020 at 4:06:46 PM UTC-5, Nicolaas Vroom wrote:
Today we understand group theory (which Einstein did not in 1905). We
can use group theory to show that there can be only three transformation groups among inertial frames, and that just one of them, the Lorentz
group [#], agrees with all the experiments.

14, not 3. There are 14 kinematic groups consistent with a general set
of assumptions - a 3-parameter family of groups distinguished by the
sign of each parameter, which I'll call here (alpha,beta,kappa), such
that the (-alpha,-beta,-kappa) group is isomorphic to the
(alpha,beta,kappa) group.

They are:
Static: (alpha,beta,kappa) = (0,0,0)
Galilei: (alpha,kappa) = (0,0), beta != 0
Carroll: (beta,kappa) = (0,0), alpha != 0
Para-Galilei: (alpha,beta) = (0,0), kappa != 0
Newton-Hooke(+/-): alpha = 0, zeta > 0 (+) or zeta < 0 (-)
Para-X: beta = 0, lambda > 0 (X = Poincare') or lambda < 0 (X = Euclid)
X: kappa = 0, gamma > 0 (X = Poincare') or gamma < 0 (X = Euclid-4D) deSitter(+/-): gamma > 0; lambda > 0 (+) or lambda < 0 (-)
Hyper-X: gamma < 0; zeta > 0 (X = bolic), zeta < 0 (X = spherical)

where gamma = alpha beta and lambda = alpha kappa and zeta = beta kappa.

The Bacry Levi-Leblond (BLL) 1968 classification.

Roughly speaking: alpha = 0 corresponds to c = infinity, beta = 0
corresponds to c = 0, with beta/alpha = c^2, when alpha beta > 0. The 3
groups where alpha beta < 0 correspond to geometries with signature 4+0
and were excluded in the 1968 paper ... though they are relevant for
geometries with signature-changing metrics.

Both the c = 0 and c = infinity cases can occur together: alpha = 0 and
beta = 0 (the Static and Carroll groups). Static and Carroll have the
same central extension, even though one is trivial; Galilei has the
Bargmann group as its central extension.

If the symmetries are denoted:
* Spatial translations P = (P1,P2,P3)
* Spatial rotations J = (J1,J2,J3)
* Time translations H
* Boosts K = (K1,K2,K3)
then the groups are those with the Lie brackets:
* [Ja,Vb] = Vc, where V is (J,K,P), (a,b,c)=(1,2,3), (2,3,1) or (3,1,2)
* [J,H] = 0
* [Ka,Kb] = -gamma Jc; [Pa,Pb] = lambda Jc, with (a,b,c) as above
* [Ka,H] = beta Pa, [Pa,H] = kappa Ka
* [Ka,Pb] = alpha delta_{a,b} H where delta_{a,b} = 1 if a = b, 0 else
where gamma = alpha beta, lambda = alpha kappa. The combination zeta =
beta kappa is also useful, but does not appear as a structure
coefficient.

This set is derived from the assumptions:
* isotropy to fix all the J brackets as above
* consistency with the discrete transforms (J,K,P,H) -> (J,aK,bP,abH)
for time reversal (a,b)=(-,+) and parity reversal (a,b)=(+,-), which
fixes the forms of the brackets,
* Jacobi identities (from which follow gamma = alpha beta, lambda =
alpha kappa).
A slightly more general classification can be derived by seeking out all possible deformations of the (alpha,beta,kappa) = (0,0,0) case.

The dimensions of the various quantities (using M,L,T,V respectively for
mass, length, time, speed) are:

alpha: T/VL, beta: L/TV, kappa: V/LT
gamma: 1/VV, lambda: 1/LL, zeta: 1/TT
J: MLV, K: ML, P: MV, H: MLV/T

The V dimension has to be treated as independent of L/T for consistency
with beta -> 0, though it can be normalized to L/T for non-zero beta by
setting beta = +1 or beta = -1.

All of the kinematic groups have a central extension of the form:

[Ka,Pb] = M delta_{ab}, where M def= mu + alpha H and mu is the central
charge.

Both M and mu have the dimension M. The central extension is only
non-trivial when alpha = 0; it is "trivial" otherwise; i.e. the group,
for non-zero alpha, splits into (J,K,P,H,mu) -> (J,K,P,M) x (mu). In
this case, E = M/alpha can be used in place of M. This corresponds to
"total energy" in Relativity, while M corresponds to "relativistic
mass".

For uniformity, however, it's necessary to use (J,K,P,H,M) or (J,K,P,H,mu); since E is ill-defined for alpha = 0.

Two of the groups coincide under central extensions, so the number then
drops to 13.

The transformations in infinitesimal form are
* rotations: omega
* boosts: upsilon
* spatial translations: epsilon
* time translations: tau
* extra translation: psi
which have dimensions:

omega: 1, upsilon: V, epsilon: L, tau: T, psi: LV

Their actions on (J,K,P,H,M,mu) are all given by

delta X = [X, J.omega + K.upsilon + P.epsilon - H tau + mu psi]

leading to the following infinitesimal transformations:

delta J = omega x J + upsilon x K + epsilon x P
delta K = omega x K - gamma upsilon x J + epsilon M - beta tau P
delta P = omega x P - upsilon M + lambda epsilon x J - kappa tau K
delta H = -beta upsilon.P - kappa epsilon.K
delta M = -gamma upsilon.P - lambda epsilon.K
delta mu = 0
where ()x() denotes vector cross-product and ().() vector dot product.

They can be integrated to finite form to see how each transforms under the respective symmetry.

Rotations: (J,K,P,H,M,mu) -> (RJ,RK,RP,RH,RM,R mu)
where R is a rotation operator given by

RV = V + sin(theta) n x V + (1 - cos(theta)) n x (n x V),
where the unit vector n is the axis of rotation, theta the rotation angle

Time Translations by time-shift s:
(J,H,M,mu) -> (J,H,M,mu)
(K,P) -> (K - beta s P)/r, (P - kappa s K)/r)
where r = root(1 + zeta s^2) and zeta s^2 > -1

For zeta < 0, the transform has an horizon at |s| = root(-1/zeta); time
is hyperbolic. For zeta > 0, time is circular and the transform is
actually a rotation and you can take either the + or - sign of the
square root, root (1 + zeta s^2).

Spatial Translations by shift-vector a:
(mu,J0,K1,P0) -> (mu,J0,K1,P0)
(J1,P1) -> ((J1 + a x P1)/r, (P1 + lambda a x J1)/r)
(K0,M -> ((K0 + a M)/r, (M - lambda a.K0)/r)
H -> H - kappa a.K0/r + M/r kappa a^2/(1 + r)
where r = root(1 - lambda a^2) and lambda a^2 < 1.

The components (J0,K0,P0) are those parallel to epsilon, while
(J1,K1,P1) are perpendicular to epsilon.
J = J0 + J1, K = K0 + K1, P = P0 + P1
epsilon.V1 = 0 = epsilon x V0, where V is (J,K,P).

This has an horizon at a = 1/root(lambda) if lambda > 0; the spatial
dimensions are hyperbolic if lambda > 0, circular if lambda < 0 and flat
if lambda = 0. For lambda < 0, both signs of the root can be used and
the transformations are ordinary sinusoidal rotations.

Boosts by a velocity vector v:
(J0,K0,P1,mu) -> (J0,K0,P1,mu)
(J1,K1) -> ((J1 + v x K1)/r, (K1 - gamma v x J1)/r)
(P0,M) -> ((P0 - v M)/r, (M - gamma v.P0)/r)
H -> H - beta v.P/r + M/r beta v^2/(1 + r)
where r = root(1 - gamma v^2) and gamma v^2 < 1.

This has an horizon at v = 1/root(gamma) if gamma > 0 and reduces to a
circular rotation if gamma < 0. For gamma = 0, it reduces to the form:

(J,K) -> (J + v x K, K)
(P,H,M,mu)-> (P - v M, H - beta v.P + beta v^2 M/2, M, mu)

If there is a "rest frame" (i.e. where P -> 0), then in terms of the transformed values in that frame (P,H,M) = (0,U,m), one has

0 = (P0 - v M)/r + P1
m = (M - gamma v.P0)/r
U = H - beta v.P/r + M/r beta v^2/(1 + r)

which, when inverted, yields:

P = vm/r, M = m/r, H = U + m/r beta v^2/(1 + r),
mu = M - alpha H = m - alpha U

which simultaneously generalizes both the non-relativistic and
relativistic formulas for (relativistic) mass M, momentum P and kinetic
energy H ... except that U is absent in relativity.

The appearance of an "internal energy" U term in the relativistic case
is directly connected with the inclusion of the 11th generator mu. If
setting U = 0, then both mu and the rest mass m coincide. Otherwise, mu
would have to be given a separate name, e.g. "Intrinsic Mass".

To emphasize: the case U != 0 and mu != m is a GENERALIZATION of Special Relativity that is NOT equivalent to it, but only includes it as a
special case!

There are, in the general cases of (alpha,beta,kappa), 3 invariants
under the transforms:
The Intrinsic Mass: mu = M - alpha H
The Extended Mass Shell: Phi_2 = beta P^2 - 2MH + alpha H^2 - kappa K^2 +
alpha beta kappa J^2
The "Spin" shell: Phi_4 = W^2 - gamma W0^2 + lambda W4^2

There is also the derived invariant:
The Mass Shell: mu^2 - alpha Phi_2 = M^2 - gamma P^2 + lambda K^2 - lambda gamma J^2

The 5-vector (W0, W = (W1,W2,W3), W4) is the generalization of the Pauli-Lubanski vector, and is given by

W0 = P.J, W = MJ + P x K, W4 = K.J.

It transforms as

delta W0 = -upsilon W - beta tau W4
delta W = omega x W - gamma upsilon W4 - lambda epsilon W4
delta W4 = epsilon W - kappa tau W0

If there is a rest frame P = 0, then in it, it reduces to (W0,W,W4) -> (0,mS,m^2 R.S).

This, along with the fact that there are 5 translation generators (P1,P2,P3,H,mu) in the central extension, instead of 4, strongly
suggests that the geometry naturally suited for this is 5 dimensional,
not 4.

It's something one of us found a couple years ago.

The centrally extended members of the BLL family all have a uniform
geometric coordinate representation with the following features:
* It is non-linear if kappa != 0
* It is 5 dimensional, not 4.

Coordinates: (t, u, x, y, z)
Differential Operators: denoted (T, U, X, Y, Z)

J = (z Y - y Z, x Z - z X, y X - x Y)
K = beta t (X,Y,Z) + (x,y,z) (alpha T - U)
P = -a (X,Y,Z)
H = aT + bU
M = a (alpha T - U) = mu + alpha H
mu = -c U

where

a = root(1 + zeta s^2 - lambda r^2)
b = kappa r^2/(a + c)
c = root(1 + zeta s^2) = a + alpha b

r^2 = x^2 + y^2 + z^2 + 2 beta t u + gamma u^2
s = t + alpha u

It is linear if kappa = 0 (with a = 1, b = 0, c = 1), non-linear
otherwise. The non-linear case reflects an underlying curved geometry.

The dimensions of the coordinates are x,y,z: L, t: T, u: LV.

Both a and c play the role of "extra" coordinates in the associated
conic sections:

a^2
+ lambda (x^2 + y^2 + z^2) - zeta t^2 = 1
c^2 - zeta s^2 = 1

For the Static/Carroll groups (alpha = 0, beta = 0), one has:

a = 1, b = kappa r^2/2, c = 1, r^2 = x^2 + y^2 + z^2, s^2 = t^2.

From this, you can integrate to find the action of the symmetry
transforms on the coordinates. The transform (psi) corresponding to (mu)
will have a non-trivial effect.

To determine these - which I won't do here - you will need to calculuate
the action of (J,K,P,H,M,mu) on (a,b,c); and these can be found from the following actions of (T,U,X,Y,Z) on (a,b,c):

a (T,U,X,Y,Z) a = kappa (beta t, 0, -alpha x, -alpha y, -alpha z)
ac (T,U,X,Y,Z) b = kappa (beta (au-bt), beta as, cx, cy, cz)
c (T,U,X,Y,Z) c = kappa (beta s, gamma s, 0, 0, 0)

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