# What is Space made of?

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5 hours ago, studiot said:

Wouldn't it just be different?

It wouldn’t be spacetime, because there would be no concept of distance in space or separation in time.

5 hours ago, studiot said:

Extension (distance, interval) is now fundamental; and the location of an object is a computational result

I haven’t read Eddington, but I agree with this quote. This is what I meant when I said that a collection of events without any additional structure could not manifest as spacetime in the way we experience it. So in that sense, relationships between events are more fundamental (in terms of physics) than the events themselves.

4 hours ago, studiot said:

A very emphatic no I'm afraid. Dropping the coordinate idea of contours or isolines (t = a constant) is the most important idea both Marcus and Eddington stress.

The idea of t = a constant is dangerously close to leading towards an absolute coordinate system - an anathema to relativity.

Actually, it is possible to describe spacetime as an ordered set (called a foliation) of spacelike hyperslices, where t=const for each slice. The result is somewhat like the pages in a book - each page represents a snapshot of 3D space, and is labelled by a number, which plays the role of time. There is a well defined sequence of page numbers, corresponding to the arrow of time. Or you could think of it as the frames in a movie. This is called the ADM formalism, and allows you to write GR in terms of Hamiltonian dynamics. Both the (non-constant) separation between hyperslices, as well as the spatial geometry of the slices themselves, make up the curvature of spacetime.

The ADM formalism is very useful in numerical GR, as well as in the mathematics of some models of quantum gravity.

Edited by Markus Hanke

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3 hours ago, geordief said:

These invariant intervals ,can it be  usefully said that they are not subject to any uncertainty in their measurements?

You would deal with the position and momentum uncertainty with quantum particles however relativity itself is a classical theory which it's mathematics doesn't incorporate probabilities or harmonic oscillators for the uncertainty principle. Those get incorporated when you deal with theories such as QFT.

However freefall paths via principle of least action (Langrangian) does involve uncertainty in the chosen path that the particle will take at each infinisimal. (GR itself doesn't get too much into the Langrangian) so when studying GR I wouldn't worry about uncertainty in freefall paths.

That's would be far too distracting until you get really comfortable with GR

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23 hours ago, Markus Hanke said:

Actually, it is possible to describe spacetime as an ordered set (called a foliation) of spacelike hyperslices, where t=const for each slice. The result is somewhat like the pages in a book - each page represents a snapshot of 3D space, and is labelled by a number, which plays the role of time. There is a well defined sequence of page numbers, corresponding to the arrow of time. Or you could think of it as the frames in a movie. This is called the ADM formalism, and allows you to write GR in terms of Hamiltonian dynamics. Both the (non-constant) separation between hyperslices, as well as the spatial geometry of the slices themselves, make up the curvature of spacetime.

The ADM formalism is very useful in numerical GR, as well as in the mathematics of some models of quantum gravity.

So have the boffins got this to work yet, with or without "auxiliary fields"  ?

Quote

Quote Wikipedia

ADM energy is a special way to define the energy in general relativity, which is only applicable to some special geometries of spacetime that asymptotically approach a well-defined metric tensor at infinity – for example a spacetime that asymptotically approaches Minkowski space. The ADM energy in these cases is defined as a function of the deviation of the metric tensor from its prescribed asymptotic form. In other words, the ADM energy is computed as the strength of the gravitational field at infinity.

If the required asymptotic form is time-independent (such as the Minkowski space itself), then it respects the time-translational symmetry. Noether's theorem then implies that the ADM energy is conserved. According to general relativity, the conservation law for the total energy does not hold in more general, time-dependent backgrounds – for example, it is completely violated in physical cosmology. Cosmic inflation in particular is able to produce energy (and mass) from "nothing" because the vacuum energy density is roughly constant, but the volume of the Universe grows exponentially.

## Application to modified gravity

By using the ADM decomposition and introducing extra auxiliary fields, in 2009 Deruelle et al. found a method to find the Gibbons–Hawking–York boundary term for modified gravity theories "whose Lagrangian is an arbitrary function of the Riemann tensor".[6]

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2 minutes ago, studiot said:

So have the boffins got this to work yet, with or without "auxiliary fields"  ?

I am not familiar with that particular work, so I can’t comment on it.

But as for the ADM formalism itself - yes, it works fine, it’s just a different way to formulate the same model (GR).

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4 minutes ago, Markus Hanke said:

yes, it works fine, it’s just a different way to formulate the same model (GR).

Despite the middle paragraph of the Wiki quote?

here is the full reference.

I also don't see how using a t axis like that is compatible with the "Principle of Relativity"

Edited by studiot

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28 minutes ago, studiot said:

Despite the middle paragraph of the Wiki quote?

ADM energy is just one of many different concepts of energy you find in GR; it applies only to some very specific types of spacetime, but can be quite useful in those cases, since it is relatively straightforward to calculate. What issue specifically do you see with this?

28 minutes ago, studiot said:

I also don't see how using a t axis like that is compatible with the "Principle of Relativity"

Foliating a region of spacetime into space-like hyperslices is not the same as postulating an “absolute time” axis, because there are infinitely many possible foliations. In practical terms, you can label the slices in whichever way is suitable for the given problem at hand, there is no physically preferred foliation scheme, so there is no issue with the principle of relativity. The overall model retains full diffeomorphism invariance.

Just to make this clear, the ADM formalism is just a different mathematical formalism of the same theory of GR - it has all the same symmetries, makes the same predictions, and has the same physical content. It’s simply a straightforward application of the Hamiltonian framework (a commonly used a very useful tool) to GR; so you just use a different set of dynamic variables to describe the exact same thing. It is particularly useful, and routinely used, in numerical GR.

P.S. If you are interested in the precise details of how this works, then Misner/Thorne/Wheeler “Gravitation” devotes an entire chapter to this formalism. Well worth a read.

Edited by Markus Hanke

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On 12/7/2019 at 12:00 PM, geordief said:

Looked at differently*, is it possible (in the model) to abstract space from spacetime by using   regions of spacetime where t equals a constant.?

On 12/7/2019 at 4:32 PM, Markus Hanke said:

Actually, it is possible to describe spacetime as an ordered set (called a foliation) of spacelike hyperslices, where t=const for each slice.

How is this possible without an underlying coordinate system ?

On 12/7/2019 at 4:32 PM, Markus Hanke said:

It wouldn’t be spacetime, because there would be no concept of distance in space or separation in time.

I can't agree with this since the sticks (intervals) have a clearly defined measure.

And clearly there exists a stick between each pair of events in the set.

Even if the set includes every number in  $\Re \otimes \Re \otimes \Re \otimes \Re$

Edited by studiot

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46 minutes ago, studiot said:

How is this possible without an underlying coordinate system ?

You do have to impose a coordinate system to define a foliation (which mathematically is just a set of functions of the metric), but you are free to choose whichever coordinate system works best for the problem at hand. There is no physically preferred one. So different observers are free to choose different foliations for the same scenario, but these will be related via diffeomorphisms, so they describe the same spacetime. This is the exact same situation as standard GR, just written differently.

46 minutes ago, studiot said:

I can't agree with this since the sticks (intervals) have a clearly defined measure.

I agree that we need ‘sticks to connect the events’ - that’s really what I was trying to say all along, just in different words.

You need to endow your manifold with a connection and a metric, before you can define a (quantifiable) notion of separation between events. Without that extra structure (connection & metric), you have a set of events, but no way to meaningfully define separations in time and space, nor indeed any kind of causal structure. So it wouldn’t be spacetime as we experience it, because it would lack any structure, geometry, or topology.

In GR, this is done by endowing the underlying manifold with the Levi-Civita connection, as well as the metric as dynamic variable constrained by the Einstein equations. That is why, when we perform actual calculations in GR to do with separations in time and/or space, these are always based on the metric.

All I am really trying to say here is that a collection of events alone does not constitute ‘spacetime’ - you need a connection and a metric structure as well (which would correspond to the ‘sticks’ you mentioned) to define meaningful relationships between these events. You need sticks to connect events, in your words. Without this, I’m pretty sure you wouldn’t even have a manifold in the mathematical sense, because there is no locally defined affine structure to the set (open to correction on this point, though).

It seems to me that we are actually in agreement on this point, we are just explaining it in different ways.

47 minutes ago, studiot said:

And clearly there exists a stick between each pair of events in the set.

Only if we have a manifold endowed with a connection and a metric, otherwise not. So we need that extra structure.

Edited by Markus Hanke

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On 12/8/2019 at 5:44 PM, Markus Hanke said:

You need to endow your manifold with a connection and a metric, before you can define a (quantifiable) notion of separation between events. Without that extra structure (connection & metric), you have a set of events, but no way to meaningfully define separations in time and space, nor indeed any kind of causal structure. So it wouldn’t be spacetime as we experience it, because it would lack any structure, geometry, or topology.

Having a metric is not an essential requirement for topological spaces.
If a topological space has a metric it is a metric topological space.
This is important because there is no requirement to measure the 'length' of the sticks in a topological network of connected sticks.
The connectivity is all important in determining precedence or causality.

So I maintain it would just be different, although topologically equivalent.

I do agree that if you restrict the use of 'spacetime' to Minkowski (who coined the word after all) then it would not necessarily be spacetime.

Note also that in the first millenium and a half before coordinate systems were invented Geometry functioned perfectly well.

In fact the introduction of coordinate systems, principally by Descartes, introduced extra information into Geometry which was not present before.
This extra information is that everything now has an orientation.
Before an equilateral triangle was the same whichever way up it presented.
Whereas the same triangle standing on a vertex or a base are considered to be different different.
The issue then becomes is this redundant or required information ?

There is a move in modern Geometry to return to the pre Descartes era.

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18 hours ago, studiot said:

Having a metric is not an essential requirement for topological spaces.

Indeed not, but it is an essential requirement in order for said topological space to be considered a spacetime manifold, i.e. a model we can extract quantifiable physical predictions from.

18 hours ago, studiot said:

The connectivity is all important in determining precedence or causality.

Yes, absolutely. In GR, the connectivity (i.e. relations between tangent spaces at different points) is given by the Levi-Civita connection, and the metric provides a way to define measurements.

18 hours ago, studiot said:

I do agree that if you restrict the use of 'spacetime' to Minkowski (who coined the word after all) then it would not necessarily be spacetime.

I am unsure whether we are talking about the same thing here now. In order for a given manifold to be a spacetime manifold in the sense of GR, it has to be endowed with both a connection and a metric, or else we are no longer doing GR. Of course, purely mathematically speaking, you can have manifolds without a metric, and these can be studied (ref differential topology), but then you can’t assign a consistent notion of length to curves on this manifold. This makes them rather useless, in terms of extracting physical predictions from them, other than general statements of topology.

18 hours ago, studiot said:

The issue then becomes is this redundant or required information ?

Well, I guess that depends on what it is you are trying to model with these manifolds. Within GR, we want to be able to study relationships between events, and quantify those in a consistent manner. For that purpose, you do need both a connection and a metric. For other purposes, a connection alone might be sufficient.

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On 5/30/2015 at 7:11 PM, Mordred said:

This question is amongst one of the most commonly asked questions in relativity.

I guess the answer is science doesn't know. Why guess?

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I guess the answer is science doesn't know. Why guess?

Theories are not guesses.  They are best estimates  at any specific time, that may or may not change, as technology advances and more data becomes available. Those same  theories grow in certainty as they continue to match the observational and experimental data: The theory  of  evolution of life is actually a fact...other theories such as the BB, SR/GR are overwhelmingly supported.

But didn't I explain this to you elsewhere earlier on?

Glad to be of further assistance anyway.

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