General Relativity and spacetime

[Peron]

It seems that GR claims to know that space and time exist. Not only that it claims to know the nature of space with out providing any proof that space exists in the first place. My reasoning goes something like this, if we ask the questions does space exist? we might answer it by saying that it does and we might give our description of what we think space is. classical physics will call it a volume. But let's say we remove all the matter in the universe are we still left with space? Some (I think Mach) have said that, no if matter is missing there is no space. Even if this is un true, whatever our concept of space is, does it not fall on GR to give us a working definition of space, a definition that works independent of matter?

[swansont]

It seems that GR claims to know that space and time exist. Not only that it claims to know the nature of space with out providing any proof that space exists in the first place.

 

It does?

 

My reasoning goes something like this, if we ask the questions does space exist? we might answer it by saying that it does and we might give our description of what we think space is. classical physics will call it a volume. But let's say we remove all the matter in the universe are we still left with space? Some (I think Mach) have said that, no if matter is missing there is no space. Even if this is un true, whatever our concept of space is, does it not fall on GR to give us a working definition of space, a definition that works independent of matter?

 

That's the realm of metaphysics.

[ajb]

The mathematical theory of general relativity assumes that space-time exists and is understood to be a manifold. The points in space-time have the physical interpretation as the collection of all possible events.

 

In general relativity space-time in not an emergent notion, it is inbuilt part of the theory. I don't know what one would mean by "proof" of the existence of space and time in this context.

[IM Egdall]

Here's what Einstein said about the existence of space:

 

"On the basis of the general theory of relativity . . . space as opposed to ‘what fills space’ . . . has no separate existence . . . If we imagine the gravitational field, i.e. the functions g_ab to be removed, there does not remain a (flat) space, but absolutely nothing . . . there is no such thing as an empty space, i.e. a space without a (gravitational) field.'

REF: Einstein (1952). As cited in John Stachel, Einstein from "B" to "Z", p. 297.

 

[Peron]

It does?

 

 

 

That's the realm of metaphysics.

 

Okay I understand that, yet what if metaphysicians come to the conclusion that space doesn't exist with our objects wouldn't that then invalidate GR? After all ajb states that there is a physical interpretation of spacetime. I take it to mean that space actually exists independent of objects, so it seems that GR is making a metaphysical claim.

[swansont]

Okay I understand that, yet what if metaphysicians come to the conclusion that space doesn't exist with our objects wouldn't that then invalidate GR? After all ajb states that there is a physical interpretation of spacetime. I take it to mean that space actually exists independent of objects, so it seems that GR is making a metaphysical claim.

 

I was mostly objecting to your claim that GR purports to "know the nature of space" which is separate from claiming it exists.

[Iggy]

I take it to mean that space actually exists independent of objects, so it seems that GR is making a metaphysical claim.

Einstein said the opposite, "People before me believed that if all the matter in the universe were removed, only space and time would exist. My theory proves that space and time would disappear along with matter."

 

Relativity doesn't insist that space exist independent of objects. Einstein believed it proved the opposite. He said in another quote that the "coordinates of the space-time continuum are entirely arbitrary choosable parameters, devoid of any independent physical meaning"

 

EDIT -- sorry, IM Egdall. I didn't mean to repeat your point without giving props

Edited September 23, 2011 by Iggy

[csmyth3025]

Okay I understand that, yet what if metaphysicians come to the conclusion that space doesn't exist without objects -wouldn't that then invalidate GR? After all ajb states that there is a physical interpretation of spacetime. I take it to mean that space actually exists independent of objects, so it seems that GR is making a metaphysical claim.

I think you might be confusing the curvature of space-time predicted by General Relativity with the more philosophical question: "Does space exist?"

 

I'm not in any way well educated about Einstein's General Theory of Relativity, so I hope other members here will correct any mis-statements I might make on the subject.

 

General Relativity is usually considered by the general public as theory that explains the phenomenon of gravity as the curvature of spacetime by objects such as planets, stars, galaxies, etc. It is this, and much more.

 

The following is presented as a reference and is not intended to confuse you. It's a simplified way of writing the Einstein Field Equations (EFE):

 

Using geometrized units where G = c = 1, this can be rewritten as

 

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how the curvature of spacetime is related to the matter/energy content of the universe.

(ref. http://en.wikipedia....thematical_form )

 

Note that the explanation given above says that what's on the left of the equals sign represents the curvature of spacetime and what's on the right side of the equals sign represents the matter/energy content. The term (8*pi) is a simple factor equal to about 25.13. The expression is the stress–energy–momentum tensor.

 

Most people think that this stress–energy–momentum tensor includes energy and matter, but this isn't necessarily so. Remember that E=mc², so mass (matter, as most people think of it) doesn't have to be present. This equation is perfectly valid if represents only energy.

 

The result would be a universe in which no matter exists, but which has spacetime curvature that is perfectly described by the EFE. Such a universe was first postulated in 1917 by Willem de Sitter and is appropriately termed a de Sitter universe.

 

I mention all of this because of your OP comment that: "...It seems that GR claims to know that space and time exist. Not only that it claims to know the nature of space with out providing any proof that space exists in the first place..."

 

General Relativity doesn't "claim to know that space and time exist", it just assumes that it does - as we all do in everyday life. Nor does General Relativity claim to know "the nature of space". It predicts how space, time, energy and (by extension through the equivalence formula E=mc²) matter interact.

 

Don't misunderstand the quote from Einstein that IM Egdall provided. "What fills space doesn't have to be matter. It can be - and in some models it is - just energy.

 

The short answer to your question: "...What if metaphysicians come to the conclusion that space doesn't exist without objects -wouldn't that then invalidate GR?" would be "no".

 

Chris

[ajb]

I take it to mean that space actually exists independent of objects, so it seems that GR is making a metaphysical claim.

 

General Relativity gives us a mathematical model in which to describe gravitational phenomena. Part of the mathematical structure is space-time. I don't think there is any real metaphysical claim here. We have a good model for gravity and part of that is space-time.

 

You then go away and observe the Universe and see that everything agrees with your predictions. You could then suggest that space-time is real, at least on the scales you have probed. All your experiments suggest that space-time is a good notion.

 

Generally I find it can be really difficult to understand what is mathematics and what physical. I also do not think that physicists have been the best at separating these. I certainly include myself here.

 

In my mind the only true real things are those that can be measured or observed. Things that cannot be observed are auxiliary and mathematical. But this may not be the best definition.

 

So can we observe an empty small piece of space-time? I claim not as physically we know the Universe is filled with many different fields and that theses fluctuate on the smallest scales. There is no true empty vacuum and this will make disentangling an empty piece of space-time from the mix impossible.

 

So is space-time real and independent of stuff on it? I am not really sure how to answer that. But for sure out notions of space and time will be remoulded once we have a better understanding of the quantum nature of space-time.

 

Einstein said the opposite, "People before me believed that if all the matter in the universe were removed, only space and time would exist. My theory proves that space and time would disappear along with matter."

 

The only thing I find perplexing here is that we can write down solution to the field equations in the absence of any sources. For example Minkowski space-time is a solution. However, without any test particles one could not probe any geometry, and maybe you could argue that this means no geometry.

 

 

Relativity doesn't insist that space exist independent of objects.

 

I am not sure it really says much either way. You just assume you have a space-time with a signature Lorentzian metric. The interpretation is that the points are the collection of all possible events. I have never been clear on what possible means here.

 

Must we assume that something has happened or eventually will happen at a given arbitrary point in space-time?

 

This is not really how I read it. I think it just means space-time can be covered with coordinate charts in which we could use to make measurements of any arbitrary hypothetical event. The events need not be physical in any true sense.

 

Einstein believed it proved the opposite. He said in another quote that the "coordinates of the space-time continuum are entirely arbitrary choosable parameters, devoid of any independent physical meaning

 

This is right and gets at the heart of diffeomorphism invariance of physics. One is free to chose whatever coorinates you like. This is not in any way special to Einsteinian relativity, it holds true for all physics. Any theory worth anything will have this property. Newtonian physics has this property for example, you are free to use non-inertial frames if you want even for simple physics.

 

There is a related notion which is quite special to general relativity and really is non-trivial. This is also often called diffeomorphism invariance. The important statement is that there are no preferred coordinates systems in general relativity. This means that there is no priori geometry as the metric is a dynamical variable.

 

In special relativity and Newtonian physics we have the inertial frames as preferred frames, though the two notions here are not the same.

[Iggy]

The only thing I find perplexing here is that we can write down solution to the field equations in the absence of any sources. For example Minkowski space-time is a solution. However, without any test particles one could not probe any geometry, and maybe you could argue that this means no geometry.

or at least that the geometry itself doesn't have physical meaning without the test particle.

 

I am not sure it really says much either way. You just assume you have a space-time with a signature Lorentzian metric. The interpretation is that the points are the collection of all possible events. I have never been clear on what possible means here.

 

Must we assume that something has happened or eventually will happen at a given arbitrary point in space-time?

 

This is not really how I read it. I think it just means space-time can be covered with coordinate charts in which we could use to make measurements of any arbitrary hypothetical event. The events need not be physical in any true sense.

 

I find this really interesting and I want to see if I understand what you mean. The events, by my understanding, are physically meaningful by themselves but the space-time coordinates are not. The coordinate chart is arbitrary and could be changed so that it looks different, but the events in the new coordinate system would still interact in the same way. The same test particles, for example, would intersect.

 

This is right and gets at the heart of diffeomorphism invariance of physics. One is free to chose whatever coorinates you like. This is not in any way special to Einsteinian relativity, it holds true for all physics. Any theory worth anything will have this property. Newtonian physics has this property for example, you are free to use non-inertial frames if you want even for simple physics.

 

There is a related notion which is quite special to general relativity and really is non-trivial. This is also often called diffeomorphism invariance. The important statement is that there are no preferred coordinates systems in general relativity. This means that there is no priori geometry as the metric is a dynamical variable.

 

In special relativity and Newtonian physics we have the inertial frames as preferred frames, though the two notions here are not the same.

 

Absolutely, the wiki article where I pulled Einstein's quote talks about the same:

For the philosophically inclined, there is still some subtlety. If the metric components are considered the dynamical variables of General Relativity, the condition that the equations are coordinate invariant doesn't have any content by itself. All physical theories are invariant under coordinate transformations if formulated properly. It is possible to write down Maxwell's equations in any coordinate system, and predict the future in the same way.

 

But in order to formulate electromagnetism in an arbitrary coordinate system, one must introduce a description of the space-time geometry which is not tied down to a special coordinate system. This description is a metric tensor at every point, or a connection which defines which nearby vectors are parallel. The mathematical object introduced, the Minkowski metric, changes form from one coordinate system to another, but it isn't part of the dynamics, it doesn't obey equations of motion. No matter what happens to the electromagnetic field, it is always the same. It acts without being acted upon.

 

In General Relativity, every separate local quantity which is used to describe the geometry is itself a local dynamical field, with its own equation of motion. This produces severe restrictions, because the equation of motion has to be a sensible one. It must determine the future from initial conditions, it must not have runaway instabilities for small perturbations, it must define a positive definite energy for small deviations. If one takes the point of view that coordinate invariance is trivially true, the principle of coordinate invariance simply states that the metric itself is dynamical and its equation of motion does not involve a fixed background geometry.

 

--Meaning of coordinate invariance

[Peron]

Thanks all for the posts, I think I get it now.

[ajb]

I find this really interesting and I want to see if I understand what you mean. The events, by my understanding, are physically meaningful by themselves but the space-time coordinates are not.

 

The coordinate chart is arbitrary and could be changed so that it looks different, but the events in the new coordinate system would still interact in the same way. The same test particles, for example, would intersect.

 

 

You can have peculiarities in certain charts, for example the coordinate singularity at the Horizon of the Schwarzschild black hole. This is a pure artefact of the poor choice of coordinates.

 

But yes, the coordinates themselves have little or no physical meaning. This can cause trouble when interpreting physics on curve space-times. Our basic understanding is build on physics on flat space-time. So we have some intuition about energy conservation, conservation of momentum, angular coordinates and angular momentum etc... However one has to take great care in placing physical significance to things in general relativity and in particular things related to symmetries.

 

So yes the physics is independent of the coordinates chosen, but it is not always easy to understand the physics in an arbitrary coordinate system. There may well be coordinate systems suited to the problem you are studying, but this choice is yours and not imposed by any deep meaning in general relativity.

 

In special relativity things are different. Assuming no external forces nature singles out the class of inertial coorinates. It is true that no single coordinate system is preferred, but rather a whole class is selected by nature. The deep reason for this is the Poincare group.

[Schrödinger's hat]

You can have peculiarities in certain charts, for example the coordinate singularity at the Horizon of the Schwarzschild black hole. This is a pure artefact of the poor choice of coordinates.

 

One thing that really helped me realise this is looking at a Rindler chart, or other charts/coordinate systems built around objects (in flat space) undergoing hyperbolic motion. My understanding of GR is still rather lax so I don't know how long the analogy lasts, but looking at the universe using a Rindler chart is a lot like trying to use Schwarzschild coordinates near a black hole.

You have this big scary event horizon beside you, nothing ever crosses it, and light beams or other geodesics will always curve towards it. Even though you're accelerating hard you don't get any further away from it, and if your friend Bob turns his engine off he'll fall towards the event horizon, slowly red-shift away and never be heard from again.

If you change to a more suitable set of coordinates (such as Minkowski) you'll see that it was all in your head and it's actually safe to turn your engines off (except you may never be able to contact your friend Steve, who wouldn't listen to you and kept his engines on).

The event horizon around a black hole is a bit more objective, but if you change to some other coordinates such as Kruskal you see similar effects.