Metric Ambiguity [Part 1][RETRACTED]

A simple example of how General Relativity isn't nearly as well understood as we think it is.

This is all wrong, see

Metric Ambiguity [Part 2]

for more details. Long story short, equation 4 is actually a constraint on what fully covariant rank-2 symmetric tensors could qualify as the local metric tensor. It doesn’t mean that the parallel transport of all fully covariant rank-2 symmetric tensors is path-independent. Idiotic mistake on my part!

General Relativity(GR), Einstein's theory of gravity, is often overshadowed by its more enigmatic counterpart, Quantum Mechanics(QM). QM is viewed as counterintuitive due to features like entanglement and Bell’s inequality being violated(which requires nonlocality, nondeterminism, or Superdeterminism), whereas GR is often seen as a more “normal” theory, grounded in classical physics. However, this perception is flawed. GR is a far more bizarre theory than most physicists realize. I will be showcasing GR’s peculiar properties in this and future blog posts. As my first physics post, I will demonstrate a feature of GR which is mathematically trivial to prove, at least to anyone with a passing familiarity to the math of differential geometry. It is my hope that this post will wake people up to the truly bizarre nature of GR, and make them more open to hearing other findings of mine.

The topic of today is something which I call Metric Ambiguity(MA). It is well known(among GR physicists) that one can flip the sign/signature of the metric tensor with no physical effect. This is analogous to space and time trading roles. In case you’re wondering, I prefer the -+++ signature over the +--- signature, but this feature of GR makes it purely a matter of convention. Another well-known aspect is that one can multiply the metric tensor field by any scalar they wish. This is essentially rescaling spacetime to be bigger or smaller. If a metric tensor field is a vacuum solution, all rescaled variants are also vacuum solutions. This property of GR is known as scale invariance. Put together, these two examples are equivalent to multiplying the metric tensor field by any real number, and this doesn’t count as a coordinate transformation. The reason we can do this with no physical effect is because this operation has no effect on the affine connection(to my knowledge, this is the term for the connection geodesics and parallel transport use in differential geometry. In standard GR, this is the same as the Levi-Civita connection). As the formulas for the Ricci and Reimann curvature tensors only require an affine connection, if the affine connection isn’t changed, neither will the curvature be. This is why the metric tensor field of a vacuum solution stays a vacuum solution even if the metric tensor field is multiplied by a scalar. Most physicists might be surprised that this property can be generalized to far more than just multiplying the metric tensor field with a scalar. Metric Ambiguity is far more general that that.

To start off, we will be given a torsion-free affine connection field and attempt to find its associated metric tensor field. We will start by assuming metricity(Q=0). This means that the covariant derivative of the metric tensor vanishes, I.E. the metric tensor is parallel transported. As you can see below, this allows us to prescribe the metric tensor at a point, and find its gradient at that point.

\(Q_{λμν}=\nabla_λg_{μν}=\partial_λg_{μν}-g_{νκ}\Gamma^κ_{λμ}-g_{μκ}\Gamma^κ_{λν}=0\rightarrow\partial_λg_{μν}=g_{νκ}\Gamma^κ_{λμ}+g_{μκ}\Gamma^κ_{λν}\qquad\left(1\right)\)

As you can see, the formula for the metric tensor’s gradient is linear, so the metric gradient of a linear combination starting metrics will be the same as the linear combination of said metrics’ gradients. The equations below explain this more coherently.

\(\begin{matrix}{\textcolor{magenta}{g_{μν}}=\textcolor{red}{g_{μν}}+\textcolor{blue}{g_{μν}}}\\{\partial_λ\textcolor{magenta}{g_{μν}}=\textcolor{magenta}{g_{νκ}}\Gamma^κ_{λμ}+\textcolor{magenta}{g_{μκ}}\Gamma^κ_{λν}=\textcolor{red}{g_{νκ}}\Gamma^κ_{λμ}+\textcolor{blue}{g_{νκ}}\Gamma^κ_{λμ}+\textcolor{red}{g_{μκ}}\Gamma^κ_{λν}+\textcolor{blue}{g_{μκ}}\Gamma^κ_{λν}=\partial_λ\left(\textcolor{red}{g_{μν}}+\textcolor{blue}{g_{μν}}\right)=\partial_λ\textcolor{red}{g_{μν}}+\partial_λ\textcolor{blue}{g_{μν}}}\\\end{matrix}\qquad\left(2\right)\)

This linear property is very important, as we’ll see later. If we plug this metric gradient into the Levi-Civita connection’s equation, we find that it’ll perfectly reproduce our given affine connection field.

\(\begin{aligned}\Gamma^λ_{μν}=&\frac{1}{2}g^{λα}\left(\partial_νg_{αμ}+\partial_μg_{αν}-\partial_αg_{μν}\right)\\=&\frac{1}{2}g^{λα}\left(g_{μκ}\Gamma^κ_{να}+g_{ακ}\Gamma^κ_{νμ}+g_{νκ}\Gamma^κ_{μα}+g_{ακ}\Gamma^κ_{μν}-g_{νκ}\Gamma^κ_{αμ}-g_{μκ}\Gamma^κ_{αν}\right)\\=&\frac{1}{2}g^{λα}\left(2g_{ακ}\Gamma^κ_{μν}+\left(1-1\right)g_{μκ}\Gamma^κ_{αν}+\left(1-1\right)g_{νκ}\Gamma^κ_{αμ}\right)=g^{λα}g_{ακ}\Gamma^κ_{μν}\\=&\Gamma^λ_{μν}\end{aligned}\qquad\left(3\right)\)

This relies on the affine connection being torsion-free, as this proof requires switching the covariant indices in the affine connection. Another thing which is important is that the parallel-transport of the metric tensor is path-independent so long as the spacetime has no branch cuts. This can be seen in the equation below, which is used to find the path-dependence of the parallel transport of fully covariant rank-2 tensors. You can find more information about the path-dependence of the parallel-transport of arbitrary tensors here.

\(\left(\left[\nabla_ρ,\nabla_σ\right]+T^λ{}_{ρσ}\nabla_λ\right)g_{μν}=-R^λ{}_{μρσ}g_{λν}-R^λ{}_{νρσ}g_{μλ}=-\left(R_{μνρσ}+ R_{νμρσ}\right)=0\qquad\left(4\right)\)

Now that we have these four equations, we can now show the following. We have our given affine connection field and we specify the metric tensor at a point(let’s say the origin for simplicity). Equation 4 shows that we can parallel transport the metric tensor from the origin to every other point without issue(so long as spacetime has no branch cuts). We now have our metric tensor field. Equation 3 shows that if we calculate the Levi-Civita connection using this metric tensor field, we’ll perfectly reproduce the given affine connection field. Neither of these steps have imposed any limitations the metric at the origin. The linearity showcased by equations 1 and 2 shows that if we were to take two different starting metric tensors, calculate their metric tensor fields, and added them together, the result would be the same as if you had initially added them together to create a third starting metric tensor and found the metric tensor field of that.

This has several major implications. The first is that if you’re given a torsion-free affine connection field, there is no unique metric tensor field which can be reversed engineered from it. For nD, the space of possible metric tensor fields is equivalent to the space of initial starting metric tensors, ½n(n+1)D. So the metric tensor is actually very ambiguous.

The second implication is as follows. Imagine you have a system comprised of Kaluza-Klein Solitons(KKS) which performs some time dependent process. Maybe it’s some kind of machine, maybe it’s a cloud of KKS diffusing in some box, or maybe it’s a bunch or gravitational waves rippling across spacetime. Whatever it is, the metric tensor you’d initially use to model the system has one timelike extended dimension, and the system evolves as the time coordinate increases. Let’s say the metric tensor at the origin is the Minkowski metric. Let’s now take the affine connection field and use it to create several new metric tensor fields, starting at the origin. Our first metric tensor field had one time dimension which was aligned with the ct coordinate, but Metric Ambiguity shows we can get this same affine connection field with very different configurations. For instance, we can change the number of time dimensions from 1 to 0, 2, 3, or 4! Moreover, we can make the ct direction spacelike instead! Peculiarly, the dynamics of the system are as unaffected by these changes as they would be if the metric were rescaled instead. The machine(which could be the brain of an observer) cannot tell differentiate between these cases. Neither can the gas of diffusing KKS.

This makes it apparent that what we call “time” is likely a far more complicated phenomena than just the metric tensor having a negative eigenvalue. I’ve only recently discovered Metric Ambiguity, so I myself have no clue what “time” is. I suspect it arises from some Complex Systems phenomena. Given that distant points in our universe are able to agree on the number of time dimensions, it seems that the vacuum has nontrivial structure. Were it as simple as Minkowski spacetime, distant observers wouldn’t be capable of agreeing which directions were spacelike and which were timelike. I speculate that there’s a sort of omnipresent quantum foam(but classical as Dimensional Collapsar Theory(DCT) only has vacuum GR) which provides this level of consistency. It’s possible this foam could make the vacuum look different to matter and antimatter, which could explain our universe’s baryon asymmetry.

This is far from the weirdest property that I’ve discovered GR possesses, but it’s certainly the most comprehensible. I hope it’s enough to make people question just how well understood GR really is. I’ll post more of my findings here as time(whatever that is) goes on. That is all for now though, good day.