Metric Ambiguity [Part 2]

It’s been disproven, good riddance!

Metric Ambiguity has been thoroughly disproven! I mentioned it to Dr. Beetle, a GR professor at my school, and they pointed out a flaw in my proof. The error lies in equation 4

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

where I simply assumed that a fully covariant symmetric rank-2 tensor could always be parallel transported in a small closed loop and always return to its original value, I.E. the tensor’s parallel transport would always be path independent. This is always the case for the metric tensor, so I sort of assumed that this would work for any tensor with a similar enough structure. Dr. Beetle pointed out that this was very much not the case. After extensive testing, I’ve determined that if a Levi-Civita connection field is created with a metric tensor field, then in general, only scalar multiples of that metric tensor field will be capable of reproducing said Levi-Civita connection field. As such, the ambiguity in the metric tensor is only one dimensional. Furthermore, I’ve learned that in general, Levi-Civita connection(by which I mean the symmetric connection geodesics use(might be botching terminology)) fields will not be metric compatible. If you use them to create a mixed-variance Riemannian curvature tensor, then it’ll always satisfy the first Bianchi identity and have antisymmetric 3^rd and 4^th indices. Generally speaking though, the contraction of the first two indices will be nonzero. If the contraction of the Levi-Civita connection field is irrotational though, then the aforementioned contraction of the first two indices will be zero, as it’s supposed to be. Even if this is the case, it’s no guarantee that the connection is metric-compatible. If it is metric-compatible, then there should exist a metric tensor at that point where if the metric tensor is used to lower the Riemannian curvature tensor’s contravariant index, the first two indices of the result should be antisymmetric. In general though, this is not the case, even if we have a vacuum solution.

If you guys want examples of mixed-variance Riemannian curvature tensors, here’s a metric-compatible Riemannian curvature tensor and its metric tensor:

g={{2,-1,7},{-1,-4,2},{7,2,-4}}

Riem={{{{0,98,-542},{-98,0,-256},{542,256,0}},{{0,-284,-192},{284,0,-1128},{192,1128,0}},{{0,64,324},{-64,0,552},{-324,-552,0}}},{{{0,-147,71},{147,0,636},{-71,-636,0}},{{0,90,36},{-90,0,628},{-36,-628,0}},{{0,-600,-122},{600,0,-2676},{122,2676,0}}},{{{0,-49,165},{49,0,164},{-165,-164,0}},{{0,38,-24},{-38,0,692},{24,-692,0}},{{0,-188,506},{188,0,-372},{-506,372,0}}}}

And here’s a metric-incompatible Riemannian curvature tensor:

Riem={{{{0,-25420,-86740},{25420,0,55820},{86740,-55820,0}},{{0,49620,89480},{-49620,0,110950},{-89480,-110950,0}},{{0,33660,1720},{-33660,0,-261560},{-1720,261560,0}}},{{{0,-59486,-88297},{59486,0,-54763},{88297,54763,0}},{{0,41850,82092},{-41850,0,223316},{-82092,-223316,0}},{{0,136855,37948},{-136855,0,66924},{-37948,-66924,0}}},{{{0,128844,-79502},{-128844,0,-23878},{79502,23878,0}},{{0,3180,-40308},{-3180,0,-399944},{40308,399944,0}},{{0,-16430,4648},{16430,0,-279136},{-4648,279136,0}}}}

I do not believe that our universe’s connection has been tested for metric compatibility before, suggesting that this could allow for a new test of GR. I’m not exactly sure how one might compute the metric incompatibility of a given Levi-Civita connection field though. The best I can currently do is test if a given mixed-variance Riemannian curvature tensor has an associated metric. In general, if two metric compatible mixed-variance Riemannian curvature tensors are added together, the result will respect the symmetries of the Riemannian curvature tensor but will no longer be metric compatible. This is the case even if the second tensor is weighted to be practically infinitesimal. We could measure the geodesics of particles in a coordinate system to reconstruct the Levi-Civita connection in a region, at least in theory. Any measurements we perform will be noisy and inaccurate though, and as such, all measured Levi-Civita connections will be metric incompatible, regardless of whether they actually are. Hence why a proper formula for the metric incompatibility of a Levi-Civita connection field would be required for such a task. Even if we did detect this metric incompatibility at significant levels, that still doesn’t necessarily mean GR is wrong. For all I know, Kaluza-Klein Theory(KKT) in its most general form(no cylindrical condition) might not result in a metric compatible 4D Levi-Civita connection field. As such, this might not actually be able to test GR.

One thing to note is that if spacetime were actually quantum and existed in a superposition, this might result in the Levi-Civita connection being truly metric incompatible. I personally don’t think this is the case, so I won’t advocate much for such a concept. GR is also an extremely successful theory, so the likelihood that this metric incompatibility is actually a thing is fairly unlikely IMO.

Now that I’m done with that explanation, I must say how happy I am that metric ambiguity has been disproven. In all honesty, it only caused me issues. For example, in Dimensional Collapsar Theory(DCT)(A theory of mine which is like KKT in the sense it’s literally just vacuum GR but explicitly with compact spaces), I had realized that the scale-invariance of vacuum GR could be broken by a compact space if the cylindrical condition failed(I’ll elaborate on that more in a later post). This would result in the size of the scaling compact space(the largest one) setting the scale of all particles/Kaluza-Klein Solitons(KKSs). Metric ambiguity threw a wrench in that idea though by absolutely destroying any and all coherent notions of relative size. Now that metric ambiguity has been disproven, I can go back to working on DCT’s dilaton halo aspects. I’ve already calculated the effective Einstein tensor of a dilaton halo(specifically that of the KKS version of a Schwarzschild blackhole)!

\(\begin{matrix}{ds}^2=-{dct}^2+\left(1-\frac{r_s}{r}\right)^{-1}{dr}^2+r^2\left({dr}^2+\sin^2{\theta}{d\theta}^2\right)&A_\mu=0&\phi=\sqrt{1-\frac{r_s}{r}}\\\end{matrix}\)

\(\mathrm{5D\ spacetime\ interval}:\ {d\widetilde{s}}^2={ds}^2+\left(1-\frac{r_s}{r}\right){d\phi}^2\)

\(\begin{matrix}R_{\mu\nu}=G_{\mu\nu}=\frac{r_s}{2r^3}\left[\begin{matrix}0&0&0&0\\0&-\frac{2}{\left(1-\frac{r_s}{r}\right)}&0&0\\0&0&r^2&0\\0&0&0&r^2\sin^2{\theta}\\\end{matrix}\right]\\G^\mu\:_\nu=\frac{r_s}{2r^3}\left[\begin{matrix}0&0&0&0\\0&-2&0&0\\0&0&1&0\\0&0&0&1\\\end{matrix}\right]\\G^{\mu\nu}=\frac{r_s}{2r^3}\left[\begin{matrix}0&0&0&0\\0&-2\left(1-\frac{r_s}{r}\right)&0&0\\0&0&r^{-2}&0\\0&0&0&r^{-2}\sin^{-2}{\theta}\\\end{matrix}\right]\\\end{matrix}\)

I was also able to determine that dilaton halos can’t gravitate non-solitonic waves, suggesting that our universe is devoid of non-solitonic waves. This would mean that if one were to reformulate GR as an Integrable System, then said Integrable System should have no continuous Scattering Data, only discrete Scattering Data. This is because the former corresponds to non-solitonic waves whereas the latter corresponds to proper solitons.

I talked about this more on ekkolápto’s YouTube channel in the video Physicist DISPROVES Metric Ambiguity(As a reminder, the physicist who did so was Dr. Beetle, not me. Thank you again Dr. Beetle!). ekkolápto also has a playlist of interviews of me talking about various ideas which I haven’t gotten around to posting here yet, such as Dimensional Collapsar Theory:

Sorry for any and all confusion I’ve caused with that mistake in equation 4. To be completely honest though, I think I hurt myself more than anyone else with that blunder. This is a good lesson in always being paranoid that you’re making a horrible mistake, and to double check your work. Have a good rest of your day.