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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