Extending Space-time with Property

Paul Stack, University of Tasmania
Robert Delbourgo, University of Tasmania

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Outline

Spent the last 4 or so years working on a PhD project at the University of Tasmania - An attempt to unify gravity with the other forces.

Currently we have two separate theories, general relativity and the standard model of particle physics. They don’t play nicely together and there are a few problems.

To attempt to solve these issues we introduce “property coordinates”, which describe the “what” that accompanies the when and where of space-time.

We consider the effects these have on General Relativity, with curvature in super-space.

What we find is that we can unify gravity with the other forces, and we get a cosmological constant!

This was all done using Mathematica, at the end I will discuss some of the techniques we used.

The Standard Model of Particle Physics

Excellent description of all short range particle interactions we observe, for instance at the LHC. However it has a few issues:

20+ constants

No explanation for particle generations

No candidate for dark matter

Doesn’t explain dark energy (cosmological constant)

Describes all known forces, except gravity

“Of these four forces, there's one we don't really understand." “Is it the weak force or the strong-- " “It's gravity."

Property

Space-time provides a solid framework to describe the “when” and “where” of particles and events. 

The “what” is missing. Normally this is done by assigning various fields and quantum numbers and then applying conservation laws to them. We however take a slightly different approach, what if we could include the “what” in our space-time framework?

To do so we introduce “property” coordinates, that describe the particles present. We label these property coordinates ζ and have them act as follows:

They are Complex, so we have both ζ and (OverBar[ζ]). This produces antiparticles.

They are Anti-commuting, this prevents an infinite number of particles being produced.

They are Lorentz scalar, this simplifies the mathematics, and separates our scheme from super symmetry.

We then assign quantum numbers to these and can produce all the particles of the standard model, plus some extra. Now the focus of this talk is to see what happens when these coordinates are introduced into general relativity.

Super space

The introduction of these property coordinates changes our space-time to space-time-property.

(x^m)=(ct,x,y,z)  becomes  (X^M)=((x^m),(ζ^μ),(ζ^(OverBar[μ])))

We adopt the following conventions:

Upper case roman letters: M, N, L, etc refer to space-time-property indices

Lower case roman letters: m, n, l, etc refer to space-time indices

Greek letters: μ, ν, λ, etc refer to property indices

We also introduce a grading function: [M], where [m] = 0 for space-time indices, and [μ] = 1 for property indices.

This allows us to express how our super space variables commute: (X^M)(X^N)=(((-1))^([M][N]))(X^N)(X^M). Factors like (((-1))^([M][N])) we call “sign factors”.

Super space - before integration over property

Integration over property

We don’t see any property coordinates in our current equations of physics, so we have one last step, which is to integrate over them.

It turns out that integrating anti-commuting coordinates is effectively the same as taking their derivative:

∫ζ dζ = 1 ,               ∫  dζ = 0,                  ∫f(x)(OverBar[ζ]) ζ dζ d(OverBar[ζ]) = f(x)

So whenever we work with property coordinates, the last step is to integrate over them to eliminate them from our equations. This also acts as conservation of property in interactions.

Super space - after integration over property

General Relativity

The process with general relativity is that you first specify a metric tensor, this is essentially a matrix that allows you to compare vectors, get their length and also measure distances. For example in flat Minkowski space-time the metric is:
(Subscript[η, mn])=(({{-1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}))  . The inner product between two vectors (v^m) and (u^n) is then given by v·u = (Subscript[η, mn])(v^m)(u^n)= - (v^0) (u^0)+(v^1)(u^1)+(v^2)(u^2)+(v^3)(u^3). 
Note we adopt Einstein summation notation, where the m and n are summed since they appear repeated in an up then down position.

Once you specify your general curved space-time metric (Subscript[g, mn]), you can then use this to calculate Christoffel symbols: ((Subscript[Γ, mn])^l)=(1/2)(g^(l s))((Subscript[g, s m,n])+(Subscript[g, s n, m])-(Subscript[g, m n,s])),
 
The Riemann curvature tensor (Subscript[R, j k l m])=(Subscript[(((Subscript[Γ, k m])^s)), ,l])-(Subscript[(((Subscript[Γ, k l])^s)), ,m])+((Subscript[Γ, k m])^r)((Subscript[Γ, r l])^s)-((Subscript[Γ, k l])^r)((Subscript[Γ, r m])^s)
 
and finally the Einstein field equations: (Subscript[R, m n])-(1/2)R (Subscript[g, m n])+ Λ (Subscript[g, m n])=κ (Subscript[T, m n])

Extended General Relativity

We move to extended GR by promoting our metric (Subscript[g, m n]) to (Subscript[G, M N]). We then use this to calculate our super Christoffel symbols ((Subscript[Γ, M N])^L), which are given by the same formula as the space-time versions, now ((Subscript[(Γ^([g])), mn])^l), except with sign factors introduced: ((Subscript[Γ, M N])^L)=(1/2)[(((-1))^([M][N]+[S][M]+[S]))(Subscript[G, S M,N])+(((-1))^([S][N]+[S]))(Subscript[G, S N, M])-(((-1))^([S][M]+[S][N]+[S]))(Subscript[G, M N,S])](G^(S L))

These are then used to get the Ricci tensor and Ricci scalar, which give us our Lagrangian and field equations. 

For N property coordinates, our general metric is as follows: (Subscript[G, MN])= (({{(Subscript[g, mn])C+(1/2)(l^2)(OverBar[ζ])((Subscript[A, m])(Subscript[A, n])+(Subscript[A, n])(Subscript[A, m]))ζ C', -(1/2) i (l^2)ζ (Subscript[A, m]) C', -(1/2) i (l^2)(OverBar[ζ]) (Subscript[A, m])C'}, {- (1/2)i (l^2)ζ (Subscript[A, n]) C', 0, (1/2)(l^2)(δ^(ν(OverBar[μ])))C'}, {- (1/2)i (l^2)(OverBar[ζ]) (Subscript[A, n])C', -(1/2)(l^2)(δ^(μ(OverBar[ν])))C', 0}})).
Where C and C’ are invariants of our group, i.e. for 2 coordinates they are C = 1+(Subscript[c, 1])(OverBar[ζ])ζ  +(((Subscript[c, 2])((OverBar[ζ]) ζ))^2) and  C’ = 1+(Subscript[c, 3])(OverBar[ζ])ζ  +(((Subscript[c, 4])((OverBar[ζ]) ζ))^2), where (Subscript[c, i]) are constants.
(Subscript[A, m]) is the gauge field that has to be introduced to allow the metric to transform correctly and l is the length scale for our additional coordinates.

Plugging this into our equations for extended general relativity produces thousands of terms. This is where we used Mathematica to perform the algebra that was virtually impossible to do by hand.

Results

We have applied this technique to several models, for N = 1, 2 and 3. I will outline here some of the results for the N = 2 case, but the general structure is quite similar for all 3 cases.
The Einstein field equations in vacuum are: (R^mn) - (1/2)(G^mn) = 0.
Evaluating this for our super space the field equations come out to be: ((R^([g]))^mn)-(1/2) (g^mn)(R^([g]))+Λ (g^mn)= κ Tr( (T^mn))
Where Λ is the cosmological constant, (R^([g])) is the space time version of the Ricci tensor and scalar and (T^mn) is the stress energy tensor for our gauge field. 

What does this mean? Well the resulting equation is exactly what describes gravity, with a cosmological constant and a gauge field, and this all came out of the super space geometry. We have effectively unified gravity with the force associated with the gauge field, and also produced a cosmological constant as expected!

Also significantly is that there are no other terms, the thousands of terms simplify down to this (relatively) simple expression. It is a little more complicated if we show what Λ and κ are:
κ = (((Subscript[c, 3])(l^2))/(4(2(Subscript[c, 4])-3((Subscript[c, 3])^2)+2(Subscript[c, 1])(Subscript[c, 3])-(Subscript[c, 2])))), Λ =((24 (Subscript[c, 1])(Subscript[c, 2])-38 ((Subscript[c, 1])^2)(Subscript[c, 3])-40 (Subscript[c, 2])(Subscript[c, 3])+110 (Subscript[c, 1])((Subscript[c, 3])^2)-75 ((Subscript[c, 3])^3)-40 (Subscript[c, 1])(Subscript[c, 4])+60 (Subscript[c, 3])(Subscript[c, 4]))/((l^2)(2 (Subscript[c, 4])-3 ((Subscript[c, 3])^2)+2 (Subscript[c, 1])(Subscript[c, 3])-(Subscript[c, 2])))).

Results

Results

Mathematica Implementation

For our Mathematica implementation we wanted:

To be easily able to verify our results by hand against Mathematica at each stage

A complete representation of the expressions

An output format that was easily readable

Knowledge of how our code was operating

The last of these points was achieved by writing our own code to do almost everything. This includes our own version of Non Commutative Multiplication, conversion to TeX and some simplification routines.

Non Commutative Multiplication

We wanted our non commutative multiplication to work like a calculation by hand would. Early in the project it was important to be able to easily check that our results were correct. We initially tried using the package NCAlgebra, which worked, but we wanted more control. We used NonCommutativeMultiply, and extended it with rules for our property coordinates. This allowed us to have anti-commutation automatically calculated for us. 

For example:

ζ[1]**ζ[2] evaluates to -ζ[2]**ζ[1]
ζ[1]**ζ[1]evaluates to 0
ζ[(Subscript[α, 3])]**ζb[(Subscript[α, 2])]**ζ[(Subscript[α, 1])] evaluates to  ζb[(Subscript[α, 2])]**ζ[(Subscript[α, 1])]**ζ[(Subscript[α, 3])]

We then just had to write an function to deal with a mix of non commutative and commutative parts, to allow Mathematica’s inbuilt simplification routines then to take over. For example:

(ζ[1]+ζ[2])**(2 ζ[1]+(Subscript[c, 3])ζ[2]) becomes  -ζ[2]**ζ[1] (-2+(Subscript[c, 3]))

Representation

Take for instance the metric tensor (Subscript[g, mn]), how do we represent this accurately inside Mathematica? After a few attempts to use subscripts, we found the best way was as follows: g[d[m],d[n]]

While ugly to read, thus allowed us to specify the index order, and whether they were up or down precisely. Now to get an output format that is easily readable, what we did was to build upon Mathematica’s ToString[“”,TeXForm] function, by wrapping it inside another function. When this acted on say g[d[m],d[n]] it would produce the following string: “g{}_{m}{}_{n}”, which when parsed with LaTeX gives us (Subscript[g, mn]).

This two stage process allows us to very precisely represent our equations, but then also produce exactly what we want in terms of readable output. It also made writing papers easy since Physics Journals use LaTeX.

Representation - example

As an example here is the output in Mathematica after calculating the Ricci scalar and applying several simplifications:

-((384 (Subscript[c, 1]) (Subscript[c, 2]))/(l^6))+((608 (Subscript[c, 1]^2) (Subscript[c, 3]))/(l^6))+((640 (Subscript[c, 2]) (Subscript[c, 3]))/(l^6))-((1760 (Subscript[c, 1]) (Subscript[c, 3]^2))/(l^6))+((1200 (Subscript[c, 3]^3))/(l^6))+((640 (Subscript[c, 1]) (Subscript[c, 4]))/(l^6))-((960 (Subscript[c, 3]) (Subscript[c, 4]))/(l^6))-(1/(l^2))2 g[u[(Subscript[s, 1])],u[(Subscript[s, 2])]] g[u[(Subscript[s, 3])],u[(Subscript[s, 4])]] (Subscript[c, 3]) tr[I matA[{deriv[A[d[(Subscript[s, 1])]],(Subscript[s, 3])],Acomm[d[(Subscript[s, 2])],d[(Subscript[s, 4])]]}]-matA[{deriv[A[d[(Subscript[s, 1])]],(Subscript[s, 3])],F[d[(Subscript[s, 2])],d[(Subscript[s, 4])]]}]-I matA[{A[d[(Subscript[s, 1])]],A[d[(Subscript[s, 3])]],F[d[(Subscript[s, 2])],d[(Subscript[s, 4])]]}]+matA[{A[d[(Subscript[s, 1])]],Acomm[d[(Subscript[s, 2])],d[(Subscript[s, 3])]],A[d[(Subscript[s, 4])]]}]]

Evaluating totex on this results in the following string:

In LaTeX this comes to be :

Simplification

One of the larger challenges we encountered was simplifying expressions like: (g^mn)(g^kl)(Subscript[A, m,l])(Subscript[A, k,n])-(g^kn)(g^lm)(Subscript[A, k,n])(Subscript[A, m,l]).
Now this is a fairly simple example, some terms involved 7 or more repeated summation indices. The question is, do these cancel out? Or are they different? It is easy to see here that they are different, but when there are roughly a thousand terms with many more summation pairs it becomes a lot more difficult.

Our solution was to first relabel all summation indices by numbers: 
(g^((Subscript[s, 1])(Subscript[s, 2])))(g^((Subscript[s, 3])(Subscript[s, 4])))(Subscript[A, (Subscript[s, 1]),(Subscript[s, 4])])(Subscript[A, (Subscript[s, 3]),(Subscript[s, 2])])-(g^((Subscript[s, 3])(Subscript[s, 2])))(g^((Subscript[s, 4])(Subscript[s, 1])))(Subscript[A, (Subscript[s, 3]),(Subscript[s, 2])])(Subscript[A, (Subscript[s, 1]),(Subscript[s, 4])]).

The next step is to relabel them in case some were removed, for instance by contracting (A^(Subscript[s, 1]))(Subscript[g, (Subscript[s, 1])(Subscript[s, 2])])(g^((Subscript[s, 2])(Subscript[s, 3])))=(A^(Subscript[s, 1]))((Subscript[δ, (Subscript[s, 1])])^(Subscript[s, 3]))=(A^(Subscript[s, 3])), we lose two indices.

Next we reorder the indices by two methods, first by simply going through each term and extracting how each of the (Subscript[s, i]) appear, and just swapping them into numerical order. This is fairly light computationally, but does not always completely simplify expressions because of the commutation between the g’s and A’s.

The other method we used involved converting each term into an array, and then swapping the indices into every permutation to find the “least” value. This was a lot more computationally expensive (O(n!) in the number of summation pairs), but guaranteed to match up terms that have the same structure.

The result of all this was matching up of terms, and simplification from 1000+ terms down to ~20 or so, which could be handled by several other smaller routines.

Simplification - example

As another example, one step of the calculation for the two coordinate case involves an expression with 4174 terms. Taking just 6 of these and performing the LaTeX output gives:

Performing our first simplification method on this expression reduces the whole thing to 381 terms, and the 6 terms here reduce to:

Summary

In an attempt to unify gravity with the other forces, we included additional property coordinates in space-time.

We extended General Relativity to include these coordinates, and worked out the Einstein Field equations for our super space.

What we found was that the we had the forces and a cosmological constant arise out of the curvature in the super space, effectively unifying gravity with the other forces.

This was done using Mathematica, and I explained some of the techniques we used.

Thank you for listening!

References

We have several papers published covering this topic, with more upcoming:

The Relativity of Chiral Property. Robert Delbourgo and Paul D. Stack. IJMPA, 2015.
http://arxiv.org/abs/1504.01494

The General Relativity of two properties. Paul D. Stack and Robert Delbourgo. IJMPA, 2015. 
http://arxiv.org/abs/1411.5483

Where-When-What: the general relativity of space-time-property. Robert Delbourgo and Paul D. Stack. IJMPA, 2014. 
http://arxiv.org/abs/1401.1238