Can We Flatten Space with York Decomposition?

[DBailen]

TL;DR Summary

Using York Decomposition can we move all the distortions typically associated with space curvature, into distortions in time flow.

Using York decomposition as a foundation, can we reformulate General Relativity to treat space as flat ($R_{ij} = 0$), redistributing all curvature into a vector field for time ($\vec{t}_i$)?

The York decomposition:
$$\tilde{h}_{ij} = \phi^{-4} h_{ij}$$
$$\tilde{K}_{ij} = \phi^2 K_{ij}$$

shows that geometric information can be fully decoupled from space. This suggests the possibility of maintaining flat space while allowing time to function as a vector field carrying the geometric information typically attributed to spacetime curvature.

Specifically:
1. Can York decomposition mathematically support this redistribution?
2. What constraints would apply to the time vector field?
3. Do the Einstein equations remain self-consistent in this framework, particularly:
$$R_{ab} - \frac{1}{2}Rg_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4}T_{ab}$$

Looking for rigorous mathematical analysis of this possibility.

 

[PeterDonis]

York decomposition

A reference (textbook or peer-reviewed paper) for where you are getting your understanding of the York Decomposition would be very helpful.

 

[PeterDonis]

Using York decomposition as a foundation, can we reformulate General Relativity

That's easy: no. What you are doing is choosing a particular coordinate chart. GR allows you to choose any coordinate chart you want. The Einstein Field Equation, as a tensor equation, is the same in every coordinate chart. So you are not "reformulating" GR. You are just doing what GR allows you to do already.

geometric information can be fully decoupled from space.

In the sense that you are choosing a coordinate chart in which "space" (3-slices of constant "time") is flat, yes.

But note that this definition of "space" might have no meaningful physical relevance, and the corresponding definition of "time" might not either. It might not correspond to the proper time of any relevant observers, and the "space" might not correspond to the natural "space" of any relevant observers. In certain special cases, it will, but those cases will not generalize.

Also, it's not clear that doing what you describe is even possible in a general spacetime; see below.

while allowing time to function as a vector field carrying the geometric information typically attributed to spacetime curvature.

This cannot be possible in the general case, because a vector field in 4-d spacetime only contains 4 independent components at each spacetime point, and the Riemann curvature tensor in 4-d spacetime contains 20 independent components.

A key reason for wanting a reference from you about the York Decomposition is to see what that reference will say about the conditions under which the decomposition is applicable.

 

[DBailen]

That's easy: no. What you are doing is choosing a particular coordinate chart. GR allows you to choose any coordinate chart you want. The Einstein Field Equation, as a tensor equation, is the same in every coordinate chart. So you are not "reformulating" GR. You are just doing what GR allows you to do already.


In the sense that you are choosing a coordinate chart in which "space" (3-slices of constant "time") is flat, yes.

But note that this definition of "space" might have no meaningful physical relevance, and the corresponding definition of "time" might not either. It might not correspond to the proper time of any relevant observers, and the "space" might not correspond to the natural "space" of any relevant observers. In certain special cases, it will, but those cases will not generalize.

Also, it's not clear that doing what you describe is even possible in a general spacetime; see below.


This cannot be possible in the general case, because a vector field in 4-d spacetime only contains 4 independent components at each spacetime point, and the Riemann curvature tensor in 4-d spacetime contains 20 independent components.

A key reason for wanting a reference from you about the York Decomposition is to see what that reference will say about the conditions under which the decomposition is applicable.

My thoughts were, if space is flattened, truly flattened and time were permitted to carry the distortion as a vector of change, (forgive my lingo problems) that spacetime would become a 3d vector field in many ways.

 

[PAllen]

That's easy: no. What you are doing is choosing a particular coordinate chart. GR allows you to choose any coordinate chart you want. The Einstein Field Equation, as a tensor equation, is the same in every coordinate chart. So you are not "reformulating" GR. You are just doing what GR allows you to do already.


In the sense that you are choosing a coordinate chart in which "space" (3-slices of constant "time") is flat, yes.

But note that this definition of "space" might have no meaningful physical relevance, and the corresponding definition of "time" might not either. It might not correspond to the proper time of any relevant observers, and the "space" might not correspond to the natural "space" of any relevant observers. In certain special cases, it will, but those cases will not generalize.

Also, it's not clear that doing what you describe is even possible in a general spacetime; see below.


This cannot be possible in the general case, because a vector field in 4-d spacetime only contains 4 independent components at each spacetime point, and the Riemann curvature tensor in 4-d spacetime contains 20 independent components.

A key reason for wanting a reference from you about the York Decomposition is to see what that reference will say about the conditions under which the decomposition is applicable.

I'll add another argument why I doubt this is possible in the general case.

Consider the goal of foliating a spacetime such that for some finite region, spatial slices are flat. If this were possible, then it must be possible to coordinatize the flat slices with the Euclidean metric. This amounts to putting 6 coordinate conditions on the full metric. However, it is well known that you can put only 4 coordinate conditions on the metric without loosing generality of geometry.

 

[A.T.]

My thoughts were, if space is flattened, truly flattened and time were permitted to carry the distortion as a vector of change, (forgive my lingo problems) that spacetime would become a 3d vector field in many ways.

Consider a large static structure, build around a massive object, at rest relative to that object. According to GR the geometry of that structure is non-Euclidean, following the spatial curvature outside of the mass (Flamm's paraboloid). This is a physical, frame-invariant fact: you can count the number of truss elements of known proper lengths, measure their stresses etc. These parameters are time-invariant, because the structure is static in the rest frame of the mass. So time is irrelevant here and cannot account for the non-Euclidean geometry of the structure.

 

[PeterDonis]

if space is flattened, truly flattened

There is no such thing. "Space" is an artifact of your choice of coordinate systems. You aren't "flattening" anything in reality when you choose York Decomposition coordinates. It's a mathematical artifact.

spacetime would become a 3d vector field in many ways.

A 3d vector field would be even worse: only 3 independent components, as compared to 20 for the Riemann curvature tensor.

I don't think you have carefully considered what "the Riemann curvature tensor has 20 independent components" actually means.

 

[PeterDonis]

Consider a large static structure, build around a massive object, at rest relative to that object. According to GR the geometry of that structure is non-Euclidean

In the light of my post #7 just now, I want to clarify: the non-Euclidean geometry in this case is not the geometry of "space" due to some choice of coordinates: it is the physical geometry of the static structure. The fact that the structure is static means that we can define an invariant notion of what that physical geometry is that does not depend on any choice of coordinates.

 

[DBailen]

Thank you all for your responses.

I understand your concerns about coordinate transformations and the Riemann tensor's 20 components vs. a vector field's 4 components.

However, I may not have been clear enough about what I'm proposing.

I'm not suggesting a simple coordinate transformation or trying to compress the Riemann tensor into fewer dimensions. Rather, I'm exploring whether the physical effects we attribute to spacetime curvature might emerge from pattern dynamics in flat space with vector time flow.

Regarding A.T.'s example of the static structure: In this framework, what appears as spatial curvature would emerge from pattern density effects in the perfect fluid medium. The structure's apparent non-Euclidean geometry would arise from how these patterns interact with time flow, even in a static configuration.

I acknowledge this is a significant departure from traditional geometric interpretations of GR. My question isn't about coordinate choices within existing GR, but about whether there might be a more fundamental way to understand these phenomena through pattern dynamics.

I am newbie, I apologize if I am off my mark, but I can't seem to stop seeing time as vector.

 

[DBailen]

While considering how to characterize space itself, it occurred to me that it behaves remarkably like a perfect inviscid fluid - offering no resistance to motion, maintaining perfect continuity, preserving patterns without loss. Not adding a substance to space (as in aether theory), but recognizing these intrinsic properties of space itself.

From this perspective, I became interested in whether York Decomposition might offer insights into how curvature effects could emerge from pattern dynamics in such a medium. My question about redistributing geometric information into time flow stems from this consideration of space's fluid-like behavior.

 

[PeterDonis]

I may not have been clear enough about what I'm proposing.

You shouldn't be "proposing" anything. That's personal speculation, which is off limits here. You should be giving us a reference on the York Decomposition that supports what you are saying. Do you have one?

 

[PeterDonis]

I'm not suggesting a simple coordinate transformation

Yes, you are. That's what the York Decomposition is: a choice of coordinates. Discussion of something else is not discussion of the York Decomposition, which is the topic of this thread.

 

[PeterDonis]

I'm exploring whether the physical effects we attribute to spacetime curvature might emerge from pattern dynamics in flat space with vector time flow.

This is personal speculation and is off limits here. It is also, as I have already pointed out, obviously impossible since a vector does not have enough independent components to represent the same things that spacetime curvature represents.

I am closing this thread for now. @DBailen if you have a reference on how the York Decomposition is used, you can PM it to me so it can be reviewed. In the absence of a reference, we do not have a valid basis for discussing what you appear to want to discuss.

 

[PeterDonis]

My question about redistributing geometric information into time flow stems from this consideration of space's fluid-like behavior.

As one further note: this kind of idea, of spacetime as some sort of fluid-like elastic medium, is not new. There were some papers published along these lines in, IIRC, the 1960s and 1970s, including one by Sahkarov, the Russian physicist. However, none of those ideas turned out to be workable. But in any case, they were not "redistributing geometric information into time flow". The physicists who proposed them knew perfectly well what I have stated in this thread, that a "time flow" vector does not have enough independent components to describe what needs to be described.