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Right just some more investigations I have made to add to the collection, actually, some of these idea's are based on some older ones I had when I was considering Gaussian curvature about a year ago.The mean curvature is And a mean curvature of regular surface in at some point is given in standard form asWhere is the shape operator.It is said that when the shape operator is seen in terms of tensor analysis, it will describe the curvature of a system - this is without saying, since that type of mathematics involves the Christoffel symbols and in our investigation, we are only concerned with sphere dynamics - we did show those dynamics in the previous post. It is also standard that the Gaussian curvature which is just and the mean curvature satisfies an inequalityThat's interesting because it has formal similarities to (this time the exact dimensional form) of an inequality I derived concerning a Geon particle modelAnd suggested this principle goes for all kinds of waves and there is even possibiities that it could have thermal consequences as well. When the mean curvature is equal to the Gaussian curvature, it means the curvature is the same in any direction. That is true for a 2-dimensional surface of course. The Gaussian curvature of a three dimensional hypersphere equal to the Planck wave isWhere is the formal definition of a Gaussian curvature involving the principles bundles. Motz notes that the three dimensional hypesphere of a radius, the Guassian curvature is so we also haveThis already implies dimensional possibilities of invoking the spacetime uncertainty on the LHS! The Gaussian curvature further can be represented in terms of general relativity asWe use this in the following form, where is the proper density. Relating this to a function of smooth Riemannian manifold with a boundary and is the Gaussian curvature of and is a geodesic curvature of the boundary. where is some differential change of area on the surface and is the metric of the boundary. Only if a boundary is ''piece-wise smooth'' then we can interpret the integral as corresponding to the sum of the angles by which they turn the corners of the boundary. The value of the Euler characteristic is simply for a sphere. This last equation is known as the Gauss-Bonnet Theorem, or at least, another form of it.
The Riemann tensor is defined already in literature as the commutator of two connections. We modify its interpretation in terms of commutivity between a spatial and a temporal set of connections in the gravitational field:where it is the non-commutation between space and time but has units that satisfy the deviation of curvature ie.The great thing is, working in 2 dimensions simplifies everything greatly and so we won't need to work with the more complicated form of the Riemann tensor for n-dimensions. It is said that the Riemann curvature tensor has which is 24 = 16 components in two-dimensional space. It is remarked that is confusing that the Ricci scalar, the Ricci tensor and the Riemann tensor all use the letter [R] and we can only distinguish them by the number of indices, i.e. 4 for the Riemann tensor and 2 for Ricci tensor and 0 for Ricci scalar - since the Ricci scalar is we may exclude this on n-dimenisonal purposes. Though the geometry of a sphere does satisfyIt does not take into account the required indices to describe the commutation. From this link http://www.physicspages.com/2014/10/21/ricci-tensor-and-curvature-scalar-for-a-sphere/ which deals with a Ricci tensor for the curvature of a sphere. however, it seems we do not want a contracted case but instead The non-vanished indices fit the approach it seems of looking at spacetime commutation https://arxiv.org/pdf/1305.2403v4.pdfThis probably means the real picture we seek is a non-contracted form of the Ricci curvature, with a non-vanishing indice for the space and one for the time.
Thermal energy is related to equipartition as
Quote from: Dubbelosix on 24/08/2017 14:38:32The Riemann tensor is defined already in literature as the commutator of two connections. We modify its interpretation in terms of commutivity between a spatial and a temporal set of connections in the gravitational field:where it is the non-commutation between space and time but has units that satisfy the deviation of curvature ie.The great thing is, working in 2 dimensions simplifies everything greatly and so we won't need to work with the more complicated form of the Riemann tensor for n-dimensions. It is said that the Riemann curvature tensor has which is 24 = 16 components in two-dimensional space. It is remarked that is confusing that the Ricci scalar, the Ricci tensor and the Riemann tensor all use the letter [R] and we can only distinguish them by the number of indices, i.e. 4 for the Riemann tensor and 2 for Ricci tensor and 0 for Ricci scalar - since the Ricci scalar is we may exclude this on n-dimenisonal purposes. Though the geometry of a sphere does satisfyIt does not take into account the required indices to describe the commutation. From this link http://www.physicspages.com/2014/10/21/ricci-tensor-and-curvature-scalar-for-a-sphere/ which deals with a Ricci tensor for the curvature of a sphere. however, it seems we do not want a contracted case but instead The non-vanished indices fit the approach it seems of looking at spacetime commutation https://arxiv.org/pdf/1305.2403v4.pdfThis probably means the real picture we seek is a non-contracted form of the Ricci curvature, with a non-vanishing indice for the space and one for the time.I don't know why I am juggling covariant and contravariant terms in the first and last post, but never mind that, ignore it. It hasn't really changed the physics we were looking for. Using the commutation relationship between the connections in terms of the possible non-trivial spacetime relationshipwe end up with something like:Where is the spacetime uncertainty recognized in form from string theory. The following is the reinterpretation of that commutation in terms of two connections of the fieldwhich is proportional to the curvature tensor of form . Andjust comes from the geometry of a sphere. We note that there is a change of length in there due to the application of uncertainty principle.
I've just noticed, there may be a metric term attached to the spacetime uncertainty, looking back on certain notes by Crowell. It hasn't devastated anything here, its just a matter of convention... and the metric is dimensionless so its not changed the dimensional structure in any way and would look like:http://www.scienceforums.com/topic30384-on-revised-work-concerning-a-qm-approach-to-wheelers-geons/#entry350501