Dimensionless complex Minkowski coordinates u are given as: u=t+i*(x,y,z);
The Minkowski coordinates in legth units are given as:
U=t*cT0+i*R*(x,y,z);
R is a very large length, T0 is an arbitrary time unit, c is the speed of light, i is the imaginary unit.
The transformation to a quaternion q is given as: q=exp(u) with
q=exp(t)*exp(i*(x,y,z)) and
exp(i*(x,y,z))=(cos(|(x,y,z)|),sin((|x,y,z|)/|(x,y,z)|*(i*x+j*y+k*z))
The inverse transformation is then u=ln(q).
Most mathematical implementations are described in
"Quaternions and Hilbert spaces" By J.A.J. van Leunen http://vixra.org/pdf/1411.0178v6.pdf
The most important physical implication is a finite universe with size R.
The "new cosmology" then evaluates all other physical implementations, beginning with the Maxwell equations in quaternionic notation.
No it doesn't. The properties of the Lorentz group are known as are its relation to the group of quaternions, SU(2). The Lorentz group contains the group of quaternions as a subgroup.
A fixed length or time isn't consistent with global Lorentz invariance, incidentally, since only the spacetime interval is invariant under global Lorentz transformations. Cosmology requires invariance under local Lorentz transformations and it's known how to implement them consistently, since de Sitter and Friedmann.
Hi Stam,
thank you for your answer. But the described issue does not rise any doubt in the consistent implementations of Friedmann and De Sitter and does not address or require fixed length or time.
The only remarkable property is, that the invertible transformation to quaternions consistently maps vector addition in the Minkowskii space to multiplication in the quaternion space and this multiplication is finite in space. (Space is described by unit quaternions.)
The multiplication operator describes an isotropic space with a kind of rotational symmetry.
For any "size" of the universe, the transformation is mathematically possible and leads to a finite geometry.
Yes it does, because the statement about a unit of length and of a unit of time isn't invariant under Lorentz transformations. Lorentz transformations don't leave lengths and times individually invariant, but a specific combination of them, that isn't of definite sign.
The quaternions describe rotations, transformations that leave the sphere invariant. The sphere is a compact manifold and its radius can be set to any finite, non-negative, value. And Lorentz transformations do map spheres to spheres. But Lorentz transformations include boosts and these describe a non-compact manifold. This isn't a finite geometry. Just do the calculation. In more precise technical terms: SL(2,C), the (double cover of the) Lorentz group, isn't isomorphic to SU(2), the group of quaternions. It's wrong to identify the Lorentz group with the group of quaternions.
The Lorentz boosts don't map time-like directions to time-like directions and space-like directions to space-like directions; they map the whole spacetime to itself, simply preserving the individual number of time-like and space-like directions.
For cosmology what is relevant are local Lorentz transformations, in any case. And these include boosts. Rotations aren't enough.
The size of the Universe isn't an invariant quantity, so it doesn't make sense using it as a unit. The reason is that, even if it weren't expanding, it would be described by Minkowski spacetime, that doesn't allow the definition of any fixed size; however it is expanding-and the rate at which it is doing so is increasing, in a way described by Friedmann and de Sitter. Its metric is a Friedmann-Robertson-Lemaître-Walker metric with positive cosmological constant.
A different math notation should not give a different physics, but it usually does, causing concern that the reults might not be describing something physical.
S. N.: "It's wrong to identify the Lorentz group with the group of quaternions."
The mapping of complex Minkowski coordinates u= (t,i(x,y,z)) to quaternions q with q=exp(u) does not affect the Lorentz group. The only point is, that a vector addition in Minkowski space u3=u1+u2 can be evaluated with u3=ln(q1*q2).
The time in displacement u2 describes the velocity and is given by the Lorentz Transformation. But the transformation to and from quaternions has no effect on the velocity. The result t3 is as expected t3=t1+t2.
S.N.: "Roatations are not enough".
The mapping "exponentiation" with the inverse "natural logarithm" extended to "complex four vectors" and "quaternions" is a kind of mapping which leaves all group properties invariant.
The transformation of the complex Minkowski space to and from quaternations is surely allowed and does not imply any restriction. But it suggests a rotational symmetric isotropic Topology.
If as you suggest, something is wrong, it only can be the "complex" interpretation of the Minkowski space.
The (real) four-dimensional orthogonal group is related to Hamilton’s quaternion algebra consisting of elements q = q0σ0 + q1σ1 + q2σ2 + q3σ3 where the qi are real, σ02 = 1 and σ12 = σ22 = σ32 = σ1σ2σ3 = −1.
The Lorentz group is similarly related to the “Pauli spin algebra” defined by
σ02 = σ12 = σ22 = σ32 = σ1σ2σ3 = −1.
See P. Rastall, “Quaternions in Relativity,” Rev. Mod. Phys. 2, 820 (1964).
Dear Wolfgang,
There is a language problem in your question. The algebra equivalent to Lorentz group SO(3,1) is the one of the bi-quaternions and not to the quaternions one. That is to say, the one expanded by Dirac matrices 4x4 and no the 2x2 Pauli's ones.
Usually, going to the continous groups of transformations, it can be constructed a surjective homomorphism from the special linear SL(2,C) as a double covering of the SO(3,1) which defines a (double) spinor map. In fact you have SL(2,C) has a subgroup SU(2)xSU(2) (Dirac's spinors)where SU(2) is a quaternion group (Pauli spinors) .
Even if you where able to represent the two forms of rotations contained in the 6-dimensional Lorentz group, this would not be enough for taking into account the gravitational Einstein's equations because your metric would be always flat (Minkowski' instead of Riemann's). Thus the cosmology would be very limited using the mathematical model that you purpose.
Dear Daniel,
With the language problem, you are perfectly right. The mapping I have considered comprises four dimensional coordinates (t,x,y,z) mapped to quaternions (e0,ie1,je2,ke3). I fully admit to your concern that a four dimensional consideration is not sufficient to address the full scope of cosmology.
But it may be sufficient for the following topological considerations:
The transformation describes a curved space. Space curvature has gravitational impact. The curvature described by the mapping is isotropic, and homogenous. (Presumably it is simply constant) Therefore it cannot lead to a gravitational force.
But it may be possible, that it induces a distance dependent gravitational potential difference between any two points.
I think four dimensions are sufficient to verify or falsify the possibility that such kind of gravitational potential can exist.
Dear Wolfgang,
I didn't tell you that four dimensions are not enough for presenting new cosmology. In fact I think that these are the dimensions necessary for every realistic cosmology ,within the present experimental knowledge that we have.
What I have said is that if you have a flat Minkowskian spacetime this is not going to give you enough information for considering its associated gravitation via Einstein's equations. On the other hand, these equations don't distinguish on Clifford algebrae or other representations of the Lorentz group. If you have a curvature this needs to be translate into Riemann curvatures.
On the other hand, if you try to find topological solutions instead of local geometrical ones, this doesn't means that you can avoid the curvature tensors because they are the main factor used by characteristic classes as Chern, Pontryaguin, Euler, Whitney, etc... Thus if we consider gravitation the fundamental interaction for cosmology it is clear that the flat Minkowski's metric is not enough, not for giving a new Cosmology, but for explaining even our present one.
Dear Daniel
Do you have an idea what kind of consequences a non zero isotropic constant curvature tensor would have?
Dear Wolfgang,
For sure there are many, locally and globally. The problem of your formalism is exactly that you haven't it. That is the main problem.
Dear Daniel,
I don't think that having "it", the mathematical item of a "constant isotropic Riemannian curvature tensor" can be a problem. Lets assume "it" exists and would make sense. What are the consequences?
My personal cababilities performing differential geometrical calculations are got kind of rusty but I think those skills would be helpful evaluating the mentioned consequences.
Dear Wolfgang,
I think that you need to be sure that the spacetime curvature exists (thus a non constant metric must be provided, never a Minkowskian's one) and this must be associated to the energy-momentum tensor. I cannot think in something which is not defined.
Einstein's equations don't depend of the rotations, which is the information given with the quaternios or biquaternions, therefore it is very difficult to imagine how can you provide such a link with the simple mathematical idea that you have presented. I cannot follow but perhaps you could try to say what is your aim or the new characteristics of the cosmology that you would find.
Hi Daniel,
The aim is to find a mathematical structure which represents a homogenous, isotropic an finite geometry and to evaluate possible physical consequences.
The idea is now that such a geometry is given by a solution of a partial differential equation, defined by setting some appropriate components of the Riemannian curvature Tensor to constant non zero values. Values which concern different spatial directions must be equal.
Dear Wolfgang,
FLRW metric does it. That is known for a long time and cosmologist employ it everyday.
Dear Daniel,
Then it is real!
If we assume a space with a constant non zero Ricci curvature Ric(X,X) in any direction X, then any dislocation in that space has an influence on the density integral of the volume per volume unit. (This is conformal to the concept of the "Ricci Flow") According to general relativity, expressed by the FLRW metric, moving between regions of different relative volume densities, is the same as moving across a gravitational potential difference.
With other words, in a space with constant non zero isotropic Ricci curvature, there is a gravitational potential difference between any two points depending on the separation of the points. But because of the isotropic nature of the effect, this potential difference does not generate a force, it only can be observed by something (e.g. a photon) which physically crosses the distance between the points.
This is not related with the quaternion algebra as you were asking.
Hi Daniel,
the relation is as follows:
The quaternion transformation of the Minkowski space describes a homogenous, isotropic, rotational, four dimensional geometry. Constant non zero curvature is the only possibility to get a homogeous and isotropic solution for the structure of that geometry.
Now it seems that volume dilatation on dislocation occurs in such a geometry with non zero curvature, which besides is equivalent to crossing a gravitational potential difference.
The possible existence of a gravitational potential, affecting any dislocation, without generating a static force, is indeed a new cosmology!
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