# Riemann Geometry and Physics

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Killtech
In terms of diff geo it seems like an obvious fact, that a manifold can be equipped with quite a variety of different Riemann metrics. But when it comes to physics (relativity theory in particular) it seems there is a very specific metric singled out. Now i do not entirely understand the connection between physics its mathematical formulation, but that is what i intend to find out.

So my understanding of Riemann geometry is that whether a surface has curvature or not depends entirely on the metric the manifold is equipped with. Furthermore there are many other metrics a manifold can have and the identify function should act as a pullback between them - since those are Riemann metrics i don't see how they could have incompatible topology amongst each other to to break the smoothness of ##id##.

So given a problem (e.g. an equation) on a specific Riemann manifold there should be no issue to pull it back to any other geometry (same manifold, different Riemann metric) and have an equivalent description of the problem there. My question is now if there is anything that distinguishes a specific and unique metric as in "the real depiction of the surface e.g. how it's really curved". Until now my understanding was that one metric is as good as another and if a distinction is made it's just for pragmatic reasons.

So far my naive view was that curvature was merely something resulting from how we chose to measure distances and therefore by itself not inherently a physical entity, which is dictated by nature. On the other hand i am very well aware that any information contained in the curvature of a physical problem is crucial, but i wonder if that's the only way to depict it.

Staff Emeritus
If I understand what you're asking, you're right, that a metric has no physical significance until you relate it to physically meaningful quantities.

The spatial component of the metric is used for computing lengths of objects. That has no physical significance unless you add the physical assumption that, in many situations, the length of a rigid object is unchanged as a function of time, and is unchanged by rotations and translations. Those properties of a rigid object are important for there to be such a thing as a "meterstick". If sticks were not rigid, but were instead stretchable like putty, then you couldn't use them to measure.

But given that there are such things as (approximately; no object is rigid in all possible circumstances) rigid objects, then the metric, and curvature (which is computed from the metric) is a physical property.

The metric also defines what is known as "geodesics", which is the analog of "straight lines" in curved space (actually, to define geodesics, you don't necessarily have to have a metric, but if you do have a metric, then geodesics can be defined in terms of the metric). Geodesics have very little physical relevance until you can connect them to the behavior of physical objects. The connection is through Newton's first law (which generalizes to curved space): An object in motion continues motion in a straight line unless acted upon by an external force. So you throw a rock, and assuming that electrical forces, friction forces, etc., are all negligible, then the path of the rock will be a geodesic.

(A difference between Newtonian physics and General Relativity is that for Newton, gravity was an external force, causing a rock's path to veer away from straight-line motion. For General Relativity, it is assumed that gravity is a manifestation of curved spacetime, so the path of a thrown rock is (approximately) geodesic even when there is gravity.)

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Furthermore there are many other metrics a manifold can have and the identify function should act as a pullback between them
I'm not really sure what you are saying here? Are you saying that the identity function from the manifold to itself should let you pull one metric back to become another metric? If so that is not the case, the identity function on a manifold pulls a metric back to the same metric. You can relate some metrics to each other by considering the pullback of a metric tensor under an automorphism of the manifold, which will generally relate the metrics simply by a coordinate transformation. However, this is not necessarily the case for arbitrary metrics. For example, the pullback will preserve things like curvature scalars etc so you cannot pull back a zero curvature metric and get a metric corresponding to non-zero curvature. In this sense, you cannot change the geometry of the Riemannian manifold using the pullback of an automorphism.

A very important concept is also that of Killing fields, which generate symmetry transformations of the manifold, i.e., the pullback of the metric under the symmetry transformation is the metric itself.

So far my naive view was that curvature was merely something resulting from how we chose to measure distances and therefore by itself not inherently a physical entity, which is dictated by nature. On the other hand i am very well aware that any information contained in the curvature of a physical problem is crucial, but i wonder if that's the only way to depict it.

On the contrary, curvature and more generally the geometry of a Riemannian manifold is something tangible that cannot be changed just through an automorphism's pullback.

PeroK
Killtech
If I understand what you're asking, you're right, that a metric has no physical significance until you relate it to physically meaningful quantities.

The spatial component of the metric is used for computing lengths of objects. That has no physical significance unless you add the physical assumption that, in many situations, the length of a rigid object is unchanged as a function of time, and is unchanged by rotations and translations. Those properties of a rigid object are important for there to be such a thing as a "meterstick". If sticks were not rigid, but were instead stretchable like putty, then you couldn't use them to measure.

But given that there are such things as (approximately; no object is rigid in all possible circumstances) rigid objects, then the metric, and curvature (which is computed from the metric) is a physical property.
Yeah, this is an aspect i was thinking a lot about lately. In particular about the assumption that a rigid object, i.e. the "meterstick" length is unchanged. The structure of Riemann geometry makes me think this postulate is not physical in nature, given the circular definition of any real length measure. I mean how would you know that a meterstick doesn't change length? you have to measure it and every measurement requires comparing your sample vs a reference measure i.e. another meterstick.. so I'm not sure it's logically undecidable if it really does change and it would seem that at some point some meterstick must be simply defined as constant. Poincaré pointed out this problem in a letter from 1904.

But looking at the math i instead got the idea that it doesn't really matter because it's totally sufficient to think of all length measurements as in units of the local length of the meterstick rather then an invariant absolute value. A length measure defined via such a physical meterstick would then only require to suffice the properties of a metric, so even if we consider a stick not being rigid but having a well defined length at each point and direction of space time (Come to think of it this may be a very physical depiction of the metric tensor) it should suffice to define a Riemann metric - at least its purely spatial components (strictly speaking this is not true for the original meter bar in Paris, because it's length at a point of space time would depend on the way it got there. That's a bigger problem then it not being rigid). That at least would fit well with the picture that a manifold has many Riemann metrics to chose from and none is naturally highlighted as the real on in terms of mathematics.

Using different physical objects as reference for length that might change their relative length compared to each other would therefore merely represent different Riemann metrics physics could be represented with. As such there would be no absolute way to say whether a body is rigid in general, as this would depend on the choice of metric.

The metric also defines what is known as "geodesics", which is the analog of "straight lines" in curved space (actually, to define geodesics, you don't necessarily have to have a metric, but if you do have a metric, then geodesics can be defined in terms of the metric). Geodesics have very little physical relevance until you can connect them to the behavior of physical objects. The connection is through Newton's first law (which generalizes to curved space): An object in motion continues motion in a straight line unless acted upon by an external force. So you throw a rock, and assuming that electrical forces, friction forces, etc., are all negligible, then the path of the rock will be a geodesic.

(A difference between Newtonian physics and General Relativity is that for Newton, gravity was an external force, causing a rock's path to veer away from straight-line motion. For General Relativity, it is assumed that gravity is a manifestation of curved spacetime, so the path of a thrown rock is (approximately) geodesic even when there is gravity.)
About Newton and GRT. I wondered if a core difference between those theories could be interpreted that Newtown actually uses a different metric? I try to sketch how i get there: consider Newton and Einstein describing the identical scenario of a spherical mass (think of the Schwarzschild solution for a Earth proxy). Consider both building rings at different radiuses and measuring their length. in GRT space curvature will make it deviate from the ##2 \pi r## formula, so Newtowns case he will find his Euclidean geometry mismatches the measured data. Couldn't he therefore conclude that the 1-SI meter(stick) must change its length depending on its space location (in particular the gravity potential)? So he would have to presume his theory measures lengths differently then the measurement devices do but using a pullback to their metric could remedy it, no?

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Killtech
I'm not really sure what you are saying here? Are you saying that the identity function from the manifold to itself should let you pull one metric back to become another metric? If so that is not the case, the identity function on a manifold pulls a metric back to the same metric. You can relate some metrics to each other by considering the pullback of a metric tensor under an automorphism of the manifold, which will generally relate the metrics simply by a coordinate transformation. However, this is not necessarily the case for arbitrary metrics. For example, the pullback will preserve things like curvature scalars etc so you cannot pull back a zero curvature metric and get a metric corresponding to non-zero curvature. In this sense, you cannot change the geometry of the Riemannian manifold using the pullback of an automorphism.
Hmm, okay. Well, there are different types of manifolds i guess and we are not talking about the same here. Consider the identity function as acting only on the bare set of ##M## and forget about it knowing about any other structures defined on it maybe.

Perhaps i better make a concrete example: consider a smooth 2 dimensional manifold ##M## and a coordinate chart ##\phi##. Now let's suppose that someone lost the information about what surface ##M## actually was and we don't have its metric anymore - i.e. we only know how it looks via its coordinates. So we have to look at the possibilities: it could have been just flat surface or maybe it was part of a sphere with a curvature? We can equip ##M## with the possible Riemann metrics for either of those cases, but the set ##M## itself stays identical and we could go from one option to the other by using the ##id## function on the set of ##M##. So ##id## actually maps between two different Riemann manifolds - which is a bit confusing i admit. Now ##id## will be bijective in any case but smoothness requires the topologies on both sides to agree. Anyhow i don't see how ##id## would remain isometric for that case.

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Staff Emeritus
Yeah, this is an aspect i was thinking a lot about lately. In particular about the assumption that a rigid object, i.e. the "meterstick" length is unchanged. The structure of Riemann geometry makes me think this postulate is not physical in nature, given the circular definition of any real length measure. I mean how would you know that a meterstick doesn't change length?

Well, you would certainly know if different metersticks changed size in different ways. Then two metersticks that started out the same size would end up different sizes.

If you're asking whether it's possible that everything in the whole universe changed size in exactly the same ratio, so comparisons never changed, well, that wouldn't be observable. In that case, you could just define the length of something to be the length relative to a standard. Then by definition, if the ratio didn't change, then the lengths would be constant.

There is a certain flexibility or tradeoff between "laws of physics" and "geometry". If you wanted to define the surface of the Earth to be a flat plane, you could do so, but then you would find all sorts of effects, such as the lengths of objects changing as you move them around.

The only thing that you can physically test is the combination of particular laws of physics with particular assumptions about geometry. You can't test the two completely separately (I don't think). For example, in a rotating coordinate system, there is an apparent "centrifugal force" seeming to fling objects away from the center of rotation. Properly understood, this is not an actual force, but is a manifestation of geometry.

You could, however, attribute it to a physical force, instead of geometry. That is, you could assume that your coordinates are not rotating, but that there is some strange "anti-gravity" emanating from the center, pushing things out. That's certainly a possible interpretation, but it would lead to a strange theory, because there is no source to this force field. There's no actual object in the center that could be the source of some kind of anti-gravity.

So the choice between what's geometry and what's physical forces is not completely determined, but we choose the geometry that makes the laws of physics as simple as possible.

But looking at the math i instead got the idea that it doesn't really matter because it's totally sufficient to think of all length measurements as in units of the local length of the meterstick rather then an invariant absolute value.

Right. I guess I should have read this before responding with the above.

About Newton and GRT. I wondered if a core difference between those theories could be interpreted that Newtown actually uses a different metric? I try to sketch how i get there: consider Newton and Einstein describing the identical scenario of a spherical mass (think of the Schwarzschild solution for a Earth proxy). Consider both building rings at different radiuses and measuring their length. in GRT space curvature will make it deviate from the ##2 \pi r## formula, so Newtowns case he will find his Euclidean geometry mismatches the measured data. Couldn't he therefore conclude that the 1-SI meter(stick) must change its length depending on its space location (in particular the gravity potential)? So he would have to presume his theory measures lengths differently then the measurement devices do but using a pullback to their metric could remedy it, no?

Yes. There is no such thing (as far as I know) as an effect that absolute cannot be explained using Newton's physics, with flat space and absolute time, as long as you're creative enough about making up new, weird forces. Before Einstein, people did attempt to explain things such as the null result of the Michelson-Morley experiment by proposing forces that physically affected the lengths of rods and the timing of clocks as they move around. It can be made to work, but it's a lot more complicated than Einstein's approach, which makes use of a different spacetime geometry than Newton used.

Killtech
Killtech
If you're asking whether it's possible that everything in the whole universe changed size in exactly the same ratio, so comparisons never changed, well, that wouldn't be observable. In that case, you could just define the length of something to be the length relative to a standard. Then by definition, if the ratio didn't change, then the lengths would be constant.
i don't exactly understand your premise here. You can define various physical objects or length which do change relative to the SI-meter depending on where they are (and in a well defined way). My thinking was that one could equally set these things to have constant length instead thus rendering the SI-meter to not being constant.

So the choice between what's geometry and what's physical forces is not completely determined, but we choose the geometry that makes the laws of physics as simple as possible.
Yeah, that is exactly my thinking. But what the simples possible form laws of physics can take is always depending on the context of your problem, no? For me it looks like in this regard the metric/geometry resembles coordinate transformations in that you could switch between them to simplify your problem. But coordinates are chosen purely for practicality and there is no point of thinking of any choice as the actually "true" or "real" one.

Case in point, when dealing with acoustics, one could you just define a measure of length by using sonar location for distance measurement? due to the properties of the medium the speed of sound differs locally but is well defined to suffice as a metric in regular circumstance i think (ofc, not in a vacuum), so if you measure 1 SI-meter in different locations in terms of the two-way speed of sound the results will deviate. But using the special metric should simplify acoustic physics by rendering the speed of sound artificially constant - at the expense of adding curvature to a previously flat space.

Yes. There is no such thing (as far as I know) as an effect that absolute cannot be explained using Newton's physics, with flat space and absolute time, as long as you're creative enough about making up new, weird forces. Before Einstein, people did attempt to explain things such as the null result of the Michelson-Morley experiment by proposing forces that physically affected the lengths of rods and the timing of clocks as they move around. It can be made to work, but it's a lot more complicated than Einstein's approach, which makes use of a different spacetime geometry than Newton used.
Hmm, you arrive at the very same conclusions i have. I take it as a good sign that I'm not going nuts :). However don't think getting creative in finding ways to fix Newtons physics would be very productive. I would rather look for a way to identify the metrics of Newton and Einstrein and try to find a pullback between them in order to have a clearly defined way to translate all laws of physics from one geometry to another. I mean i would assume there could be some simplifying advantages of having a flat space time. And going back to the example of the 1-SI meter changing length in Newtown's grav potential - by the definition of the meter via the speed of light it implies it's not constant there as well. So Maxwells equations in vacuum would have to get something like a refractive index induced by Newtowns gravity - i.e. something resembling a medium. if a pullback to that geometry existed a Newtonian equivalent of Maxwell could be calculated from GRT right away, right? It would be interesting to know if the old aether theories (like the one originally used to explain Michelson-Morley result) could actually be equivalent to GRT but using a different metric.

Anyhow i don't want to discuss the details of such a approach here but rather trying find out if my way of thinking about this has any major logical flaws or oversights. So i welcome pointing out wherever the math does not warrant such interpretation.

Staff Emeritus
Yeah, that is exactly my thinking. But what the simples possible form laws of physics can take is always depending on the context of your problem, no?

No, I don’t think so. It’s true that geometric effects can look like physical forces. For example, the apparent centrifugal force. But that doesn’t mean that choosing what is forces and what id geometry is arbitrary. Geometric effects have certain universal qualities: They affect all objects in the same way, and they have a characteristic dependence on the velocity of the object. And, more to the point, the effect on test particles can always be locally eliminated by a coordinate change.

That’s not true for other physical effects. If you have an electric field, for example, it affects positively charged particles differently than negatively charged particles. This difference cannot be transformed away through a coordinate change.

Killtech
No, I don’t think so. It’s true that geometric effects can look like physical forces. For example, the apparent centrifugal force. But that doesn’t mean that choosing what is forces and what id geometry is arbitrary. Geometric effects have certain universal qualities: They affect all objects in the same way, and they have a characteristic dependence on the velocity of the object. And, more to the point, the effect on test particles can always be locally eliminated by a coordinate change.

That’s not true for other physical effects. If you have an electric field, for example, it affects positively charged particles differently than negatively charged particles. This difference cannot be transformed away through a coordinate change.
It is clear that all the information contained within the geometry cannot simply get lost via a transformation, so it has to transform into something else. My thinking was not about trying to choose what should be a force and what geometry - but merely chose a different metric and see what happens: If a map can be constructed between the original manifold and the new one arising by equipping the same space with the new metric (the map should therefore be based on ##id## of the underlying space) and that map is homeomorph - then pulling back laws of physical via that map cannot lose any physical information - therefore giving an equivalent description of the very same physics in the new metric. Therefore whatever metric you choose, gravity cannot ever vanish but rather must take some different form.

And about geometries affecting all objects in the same way - it is certainly true, yet the meaning of "same way" is perhaps misleading.

I'll try to make a simple purely mathematical model to sketch that thought. Consider the Minkowski space with only two fields for which the Maxwell equations ##m_1## holds (assuming charge and current always to ##0## for this example). Now let's pick some specific cartesian coordinates and add another two fields for which another Maxwell equations ##m_2## holds that however differs only by using having a different ##c_2## in that particular coordinates. Let's do a Lorentz boost: ##m_1## will obviously be invariant, but because ##m_2## adheres to a different Lorentz group associated with ##c_2## its transformation becomes quite messy. This can be remedied however by it pulling back to its natural Minkowski space arising from ##c_2##. Within that geometry ##m_2## becomes Lorentz invariant but this comes at the expense that ##m_1## won't be anymore. Now i don't think a geometry exists that can achieve both equations to have invariant transformation behavior at the same time, so in terms of Riemann geometry we can construct a case where we have two distinct native metrics for the same space where all physics (##m_1## and ##m_2##) take the simplest possible form.

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Killtech
I'm not really sure what you are saying here? Are you saying that the identity function from the manifold to itself should let you pull one metric back to become another metric? If so that is not the case, the identity function on a manifold pulls a metric back to the same metric.
The simples case of what i have in mind would be a Weyl transformation. This also uses the ##id## for the pullback. But in this scenario ##id## is neither an automorphism (no self mapping) nor an isometry - as it maps between two actually different metric spaces. It should only be self-homeomorphism in the topological sense because it is a mapping to the very same topological space.

However i don't want the restriction that limits the transformation to remain within the same conformal class.

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It is not clear to me what you are trying to do. An induced metric is the pullback of the metric in the target space of the embedding to the embedded space. The embedded space does not a priori have a metric - the induced metric becomes what it becomes. If you take a manifold M and it can be made a Riemannian manifold (M,g) by the introduction of a metric g, the pullback of g using the identity mapping (i.e., embedding M in itself using the identity map) is going to be g itself - not a Weyl transform of g.

Killtech
It is not clear to me what you are trying to do. An induced metric is the pullback of the metric in the target space of the embedding to the embedded space. The embedded space does not a priori have a metric - the induced metric becomes what it becomes. If you take a manifold M and it can be made a Riemannian manifold (M,g) by the introduction of a metric g, the pullback of g using the identity mapping (i.e., embedding M in itself using the identity map) is going to be g itself - not a Weyl transform of g.
Oh, no no no. I'm definitively not talking about an induced metric. Okay maybe i call my map ##\phi## instead of ##id## to point out it maps between different metric spaces. Furthermore i don't want to pullback the original metric, therefore the image of the map is not an embedded space.

Think of the two different Riemann manifolds ##(M,g)## and ##(N,h)##, except that ##M## and ##N## is actually the same topological space... so ##\phi: (M,\tau) \rightarrow (N,\tau) = (M,\tau)## (where ##\tau## ist supposed to be the topology). Only in that sense ##\phi## is the identity function. You could still pullback ##g## to ##N## but if you do that the induced metric will be exactly ##g## again - as you said yourself. However i explicitly wish to change the metric by not using an isometric map. So in this case if ##g## and ##h## disagree ##\phi## is no isometry, right?

And technically if ##g = f h## for some function ##f>0##, then ##\phi: (M,g) \rightarrow (M,h)## would be a conformal map i.e. a Weyl trafo. but again, i don't really want to restrict to that special case.

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But that is not a pullback then as you wanted in your OP. It is just a map between different geometries. Different geometries will have different physical characteristics - curves will geerally be of different lengths, angles will be different etc. The geometry is part of the physical description and so the physics are different.

Staff Emeritus
But that is not a pullback then as you wanted in your OP. It is just a map between different geometries. Different geometries will have different physical characteristics - curves will geerally be of different lengths, angles will be different etc. The geometry is part of the physical description and so the physics are different.

I think that @Killtech is saying that the same physics can be described by different geometries, if you also adjust the forces (and the laws for how matter and energy affect geometry).

It's sort of trivially true. Suppose we pick some coordinate system ##x^\mu##, and according to the "true" laws of physics (say, General Relativity plus some force law), the path of a particular particle is given by:

##\dfrac{d U^\mu}{d\tau} = F^\mu - \Gamma^\mu_{\nu \lambda} U^\nu U^\lambda##

where
• ##U^\mu = \dfrac{dx^\mu}{d\tau}##
• ##d\tau = \sqrt{g_{\mu \nu} dx^\mu dx^\nu}##
• ##g_{\mu \nu} = ## the metric tensor
• ##\Gamma^\mu_{\nu \lambda} = \dfrac{1}{2} g^{\mu \sigma} (\partial_\nu g_{\sigma \lambda} + \partial_\lambda g_{\nu \sigma} - \partial_\sigma g_{\nu \lambda})##
• ##g^{\mu \sigma} = ## the inverse of ##g_{\mu \sigma}##.

Now, let's let ##g'_{\mu \nu}## be any candidate alternative metric. Then we can in terms of ##g'_{\mu \nu}## compute an alternative proper time ##\tau'##, and an alternative connection ##\Gamma'^\mu_{\nu \lambda}## and an alternative 4-velocity ##U'^\mu##. Finally we can compute (using the "true" laws of physics) the expression

##F'^\mu = \dfrac{d U'^\mu}{d\tau'} + \Gamma'^\mu_{\nu \lambda} U'^\nu U'^\lambda##

Then ##F'^\mu## will in general be a messy combination of the original force vector ##F^\mu## and the original connection coefficients ##\Gamma^\mu_{\nu \lambda}## and the original 4-velocity ##U^\mu##. Nevertheless, presumably it can be expressed as a function of the coordinates ##x^\mu##, their derivatives with respect to the "false" proper time, ##\tau'## via ##U'^\mu = \dfrac{x^\mu}{d \tau'}##, and the false metric tensor ##g'_{\mu \nu}##.

Killtech
This seems very similar to the other question you asked.

Killtech
But that is not a pullback then as you wanted in your OP. It is just a map between different geometries. Different geometries will have different physical characteristics - curves will geerally be of different lengths, angles will be different etc. The geometry is part of the physical description and so the physics are different.
To be honest i am not entirely sure how this type of transformation of an exchange of metrics is to called properly and I am indeed not sure if applying a pullback via a non-isometric map achieves what i have in mind. Generally i am thinking of something akin to Weyl-transformations but to arbitrary metrics thus without the restriction of conformality. In any case @stevendaryl has written down exactly what i am looking for.

As for physics being different, is that really so if you interpret it right? I mean the mapping to another metric should be perfectly reversible so any solution to a physical problem calculated in the alternative metric can be transformed back again to the original metric - i.e. it is still a solution to the original laws of physics in the original metric. Therefore it seems like it is just a mathematical transformation that will still yield the identical predictions albeit in a different calculus.

But my reason to ask this in the first place is that i do not think that lengths "map" to the physical reality in either trivial nor absolute way. Understanding that connection is what in part motivated my question and the fact that math indicates that there are a lot of alternatives - so i am looking for an interpretation of what that means. I arrive at the view that a metric represents rather an actual real object then an absolute length measure: the metric tensor is kind of an infinitesimal representation of a real Parisian meterstick at the given location and each direction (for the imagination it's perhaps easier to use the old SI-definition here). In particular it seems to describe how much a difference in coordinate value is in term of multitudes the meterstick at that location and in the direction of the coordinate - and therefore not in terms of an abstract absolute length. This also makes it tangible as to what an exchange in the metric means.

Killtech
I think that @Killtech is saying that the same physics can be described by different geometries, if you also adjust the forces (and the laws for how matter and energy affect geometry).

It's sort of trivially true. Suppose we pick some coordinate system ##x^\mu##, and according to the "true" laws of physics (say, General Relativity plus some force law), the path of a particular particle is given by:
[...]
Thank you! this is exactly what i have in mind. But is there a general name for these type of transformations? As you can see i am still lacking the terminology to properly express my questions, hence i easily create misunderstands by using a wrong formulation. There is also the issue that i have trouble finding lectures and papers covering such specific topics to learn from so i know how to properly talk about that.

Well, the forums are kind of my last resort to get any help with that.

This seems very similar to the other question you asked.
Yeah, in part they are. But as you can read in the answer just above i have some trouble formulating the questions correctly, hence my questions usually gets lost in the discussion trying to clear up the misunderstandings. But i am learning and as you can see in this thread some are finally able to understand and answer some my questions.

I mean would really love to find some books or papers to read through that cover that specific stuff rather then having to ask this on a forum. But it is what it is.