Block Time vs Q. Indeterminacy

[nitsuj]

With bob2c's last post I noticed another difference between block universe and the way I interpret spacetime diagrams. I think space is only 3 spatial dimensions which all share the quality of the same limited speed.

 

[stglyde]

With bob2c's last post I noticed another difference between block universe and the way I interpret spacetime diagrams. I think space is only 3 spatial dimensions which all share the quality of the same limited speed.

Don't ever forget that spacetime doesn't really exist like a particle or field do. When asked by reporters to summarize his theory, Einstein said: "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."

So it's just matter relationship with one another that we artifically called spacetime and model it as a 4th dimensional differential manifold which is not really there.

 

[bobc2]

Why do you keep insisting on a positive definite metric? The whole point of Minkowski spacetime is that the metric is *not* positive definite. That's what you need to describe a *spacetime* as opposed to a *space*.

It seems necessary for a 4-dimensional space-space (not space-time). Time should be nothing more than a parameter, playing exactly the same role as passage of time in the 3-D world. It seems elemental, given the different special relativity 3-D cross-sections that four spatial dimensions must be available for those views.

The Minkowski ict takes away the spatial equivalence of the 4th dimension in special relativity.

 

[nitsuj]

Don't ever forget that spacetime doesn't really exist like a particle or field do. When asked by reporters to summarize his theory, Einstein said: "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."

So it's just matter relationship with one another that we artifically called spacetime and model it as a 4th dimensional differential manifold which is not really there.

I never learned that spacetime doesn't exist?

It does exist, so I'm not sure in what sense you mean.

Who cares what Einstein told a reporter. (it seems like his point is regarding the role of calculating in defining, without objects to measure you can't determine spacetime's properties, from a physics perspective it's as if it doesn't exist. You need two to tango in other words ). I read that his theory implies spacetime is real and exists. It's properties effect everything.

 

[stglyde]

I never learned that spacetime doesn't exist?

It does exist, so I'm not sure in what sense you mean.

Who cares what Einstein told a reporter. I read that his theory implies spacetime is real and exists. It's properties effect everything.

Have you forgotten the thread "Spacetime doesn't really exist does it?"
https://www.physicsforums.com/showthread.php?t=487794&highlight=spacetime+doesn't+exist where you also participated? Someone there said:

"GR doesn't really describe gravity in terms of a field. GR deals with tensors, and the gravitational field is not a tensor. GR describes a gravitational wave as an oscillating curvature of spacetime."

Look spacetime is in same company as wavefunction, they are just mathematical abstraction.

Spacetime is just a 4 dimensional differential manifold.. curvature and other geometrical aspects are what producing "gravity". This means things like gravitational field as just PUN some love to say.

Now about Matter. Wave function is just a mathematical abstraction. It is all math. Large scale "things" are just results of decoherence of the mathematical wave functions.

Also Particles are just energy and momentum of the fields... or as Rovelli suggested.. "QFT should not be interpreted in terms of particle states, but rather in terms of eigenstates of local
operators."

Even virtual particles are just multi variate integrals, just purely mathematical artifacts as Neumaier emphasized.

btw... "don't exist" means no spacetime substrate.. for example... no "aether" substrate or "pilot wave" substrate... spacetime "aether" concept has problems with event horizons and pilot wave has problems with lorentz symmetry.. but if they can solve it.. then aether and pilot wave can be the substrate and make spacetime/matter "exist" in the sense there is a "material substrate".. but for now.. they are less likely...

In short, if you believe spacetime exist. Then you believe in spacetime aether? But it doesn't tally with the observations in event horizons in black holes.. so it's not likely.

 

[PeterDonis]

It seems necessary for a 4-dimensional space-space (not space-time).

I think you're missing the point. The 4-dimensional manifold we live in is a space-time, *not* a space-space. It has one timelike dimension and three spacelike dimensions. It has a metric that is not positive definite. These statements are invariant statements of physics; they do not depend on describing the timelike dimension using a "time" parameter. (For example, we can describe spacetime using null coordinates, which replace the timelike coordinate and one spacelike coordinate with a pair of null coordinates. But physically spacetime is still the same, just described differently.) So the 4-D manifold we live in is different, *physically*, than a 4-D space-space with a 4-D positive definite metric. You can't model the former with the latter.

Time should be nothing more than a parameter, playing exactly the same role as passage of time in the 3-D world.

I'm not sure I understand what you mean by this.

It seems elemental, given the different special relativity 3-D cross-sections that four spatial dimensions must be available for those views.

No, this is wrong. What the different SR cross-sections show us is that four *spacetime* dimensions must be available--three spacelike and one timelike (or, alternatively, as I said above, two spacelike and two null, which is physically the same thing). The different SR cross sections are *inconsistent* with there being four *spatial* dimensions, because the SR cross sections require a Minkowski metric, i.e., not positive definite. This is a genuine physical difference; you can't handwave it away.

 

[PeterDonis]

Have you forgotten the thread "Spacetime doesn't really exist does it?"
https://www.physicsforums.com/showthread.php?t=487794&highlight=spacetime+doesn't+exist where you also participated? Someone there said:

"GR doesn't really describe gravity in terms of a field. GR deals with tensors, and the gravitational field is not a tensor. GR describes a gravitational wave as an oscillating curvature of spacetime."

You should be careful interpreting this statement. If I'm wrong, bcrowell can jump in here and correct me, but I strongly suspect that by "gravitational field" in that statement he meant something like the "acceleration due to gravity", which is not a tensor. However, the "curvature of spacetime" *is* a tensor, the Riemann curvature tensor, and it's oscillations in that tensor that describe gravitational waves. I seriously doubt that bcrowell meant to imply that gravity, in the sense of spacetime curvature, and spacetime itself were not "real".

In short, if you believe spacetime exist. Then you believe in spacetime aether?

This is a non sequitur. Believing that spacetime "exists" does not require believing that there is some non-dynamical "background" portion of spacetime that does not interact with matter.

 

[stglyde]

You should be careful interpreting this statement. If I'm wrong, bcrowell can jump in here and correct me, but I strongly suspect that by "gravitational field" in that statement he meant something like the "acceleration due to gravity", which is not a tensor. However, the "curvature of spacetime" *is* a tensor, the Riemann curvature tensor, and it's oscillations in that tensor that describe gravitational waves. I seriously doubt that bcrowell meant to imply that gravity, in the sense of spacetime curvature, and spacetime itself were not "real".



This is a non sequitur. Believing that spacetime "exists" does not require believing that there is some non-dynamical "background" portion of spacetime that does not interact with matter.

but spacetime aether is dynamical.. see:
http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.1547v2.pdf

My understanding is that if spacetime exist.. it should be a "thing".. but if it doesn't.. meaning it is just mathematical abstraction like wave function.. then it isn't a thing.. unless you meant spacetime is like a particle that has independent existence? but if spacetime points are real... then diffeomorphism invariance (general covariance) is wrong.. so how can spacetime be anything other than mathematical abstraction?

 

[PeterDonis]

but spacetime aether is dynamical.. see:
http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.1547v2.pdf

Ah, ok, that clarifies what you meant by "ether" (the term has a lot of meanings). Then my response is, believing that spacetime exists does not require believing that there is a preferred "rest frame" at the quantum gravity level.

if spacetime points are real... then diffeomorphism invariance (general covariance) is wrong..

I don't see how this follows. General covariance just means that we can describe the spacetime manifold using any coordinate chart we like; in other words, that the physics of spacetime is independent of how we label the points in the spacetime with coordinates. It says nothing at all about whether the manifold itself, or the points in it, are "real". The latter question depends on what you mean by "real", but since we can measure the curvature of the spacetime manifold (as tidal gravity), it seems unproblematic to me to say that the manifold is real.

 

[bobc2]

...This is a non sequitur. Believing that spacetime "exists" does not require believing that there is some non-dynamical "background" portion of spacetime that does not interact with matter.

The post evoking this response was not mine. I'm not sure how it came out as my post.

 

[stglyde]

Ah, ok, that clarifies what you meant by "ether" (the term has a lot of meanings). Then my response is, believing that spacetime exists does not require believing that there is a preferred "rest frame" at the quantum gravity level.



I don't see how this follows. General covariance just means that we can describe the spacetime manifold using any coordinate chart we like; in other words, that the physics of spacetime is independent of how we label the points in the spacetime with coordinates. It says nothing at all about whether the manifold itself, or the points in it, are "real". The latter question depends on what you mean by "real", but since we can measure the curvature of the spacetime manifold (as tidal gravity), it seems unproblematic to me to say that the manifold is real.

Are you familiar with the Hole Argument?

http://en.wikipedia.org/wiki/Hole_argument

"Einstein believed that the hole argument implies that the only meaningful definition of location and time is through matter. A point in spacetime is meaningless in itself, because the label which one gives to such a point is undetermined. Spacetime points only acquire their physical significance because matter is moving through them."

In other words. Spacetime points shouldn't be made of substance or General Invariance won't work. Now if Spacetime points are not substantive but just merely mathematical abstraction, then it may not have any independent existence. Without any matter/energy/stress.. do you think there would be spacetime?

 

[PeterDonis]

Are you familiar with the Hole Argument?

http://en.wikipedia.org/wiki/Hole_argument

Did you read this in the introduction to that same Wikipedia article:

It [the hole argument] is incorrectly interpreted by some philosophers as an argument against manifold substantialism, a doctrine that the manifold of events in spacetime are a "substance" which exists independently of the matter within it. Physicists disagree with this interpretation, and view the argument as a confusion about gauge invariance and gauge fixing instead.

In other words, the hole argument does not show that general covariance is inconsistent with spacetime being a "real thing". All it shows is that GR is a gauge theory.

Without any matter/energy/stress.. do you think there would be spacetime?

Minkowski spacetime, with zero stress-energy tensor everywhere, is a solution of the Einstein Field Equation, so yes, there is at least one possible "spacetime" without any matter/energy/stress. Einstein, when he made that statement to reporters that you quoted, either didn't think of that or conveniently ignored it because he didn't want to try to go into subtleties in that context (for which I can't blame him).

In other writings, IIRC, Einstein argued that Minkowski spacetime was not a counterexample to claims of the sort he made in your quote, because it required asymptotic flatness as a boundary condition, which was an extra physical assumption over and above the EFE, since the EFE doesn't explain how the boundary condition "at infinity" comes into being. In essence, he argued that the only way to truly satisfy the conditions he described in what you quoted was for the universe as a whole to be closed, so the question of boundary conditions "at infinity" would not arise. Hawking seems to believe something similar with his "no boundary" proposal for quantum cosmology.

Bottom line, I think the question you posed in what I quoted above is a physical question on which the jury is still out. GR, as it stands, is compatible with either alternative, since it has both closed solutions and solutions with boundary conditions at infinity. (More precisely, whichever solution we end up adopting at the quantum gravity level, there will be a GR model that will work as the classical limit of that solution.)

 

[stglyde]

Did you read this in the introduction to that same Wikipedia article:



In other words, the hole argument does not show that general covariance is inconsistent with spacetime being a "real thing". All it shows is that GR is a gauge theory.



Minkowski spacetime, with zero stress-energy tensor everywhere, is a solution of the Einstein Field Equation, so yes, there is at least one possible "spacetime" without any matter/energy/stress. Einstein, when he made that statement to reporters that you quoted, either didn't think of that or conveniently ignored it because he didn't want to try to go into subtleties in that context (for which I can't blame him).

In other writings, IIRC, Einstein argued that Minkowski spacetime was not a counterexample to claims of the sort he made in your quote, because it required asymptotic flatness as a boundary condition, which was an extra physical assumption over and above the EFE, since the EFE doesn't explain how the boundary condition "at infinity" comes into being. In essence, he argued that the only way to truly satisfy the conditions he described in what you quoted was for the universe as a whole to be closed, so the question of boundary conditions "at infinity" would not arise. Hawking seems to believe something similar with his "no boundary" proposal for quantum cosmology.

Bottom line, I think the question you posed in what I quoted above is a physical question on which the jury is still out. GR, as it stands, is compatible with either alternative, since it has both closed solutions and solutions with boundary conditions at infinity. (More precisely, whichever solution we end up adopting at the quantum gravity level, there will be a GR model that will work as the classical limit of that solution.)

To clarify my point. Is the wave function real in the sense that there are really physical (or whatever) waves that interfere before say the double slit electron reach the detector? There isn't. So is Spacetime. Nothing is curving there in space or time. It is just a math model like wave function. Spacetime is not the territory but just a map.. and they say not to mistake map for territory.

We don't know how such complex mathematical abstraction like wave function and spacetime connect to our world and physicists don't care. Physics is just studying the math models with a a huge disconnect in correponding to our reality. Agree?

 

[PeterDonis]

To clarify my point. Is the wave function real in the sense that there are really physical (or whatever) waves that interfere before say the double slit electron reach the detector? There isn't.

That's another physical question on which the jury is still out. There is certainly a large school of thought in quantum physics that believes this, but it is not established the way, say, the Earth going around the Sun is.

So is Spacetime. Nothing is curving there in space or time. It is just a math model like wave function.

Tidal gravity is not "just a math model". It's a physical observable. It is true that one is not *forced* to model tidal gravity using a curved spacetime; one could use another model. But in the context of that model, "spacetime curvature" is simply another name for "tidal gravity", so if tidal gravity is real (which it is), then spacetime curvature is real.

Spacetime is not the territory but just a map.. and they say not to mistake map for territory.

Yes, but that does not mean that nothing named on the map is real. The United States of America is also part of a map; there is nothing in the territory that carries intrisic labels saying "this belongs to the USA". Does that mean the USA is not real?

This kind of discussion can degenerate very quickly into philosophy instead of physics. The map-territory distinction is supposed to forestall such degeneration, not cause it. The map is not the territory, but the reason for having a map is to guide you through the territory. You can't make use of the map that way if you don't think of the things the map is labeling as real.

We don't know how such complex mathematical abstraction like wave function and spacetime connect to our world...

Certainly we do. We use these complex mathematical abstractions to make predictions that are confirmed to many decimal places. That requires a deep knowledge of how they connect to our world.

...and physicists don't care. Physics is just studying the math models with a a huge disconnect in correponding to our reality. Agree?

No. This kind of statement needs a *huge* amount of support which you have not given, and which I don't see how you could give in view of the predictive accuracy of our models which I have just referred to.

 

[stglyde]

That's another physical question on which the jury is still out. There is certainly a large school of thought in quantum physics that believes this, but it is not established the way, say, the Earth going around the Sun is.



Tidal gravity is not "just a math model". It's a physical observable. It is true that one is not *forced* to model tidal gravity using a curved spacetime; one could use another model. But in the context of that model, "spacetime curvature" is simply another name for "tidal gravity", so if tidal gravity is real (which it is), then spacetime curvature is real.



Yes, but that does not mean that nothing named on the map is real. The United States of America is also part of a map; there is nothing in the territory that carries intrisic labels saying "this belongs to the USA". Does that mean the USA is not real?

This kind of discussion can degenerate very quickly into philosophy instead of physics. The map-territory distinction is supposed to forestall such degeneration, not cause it. The map is not the territory, but the reason for having a map is to guide you through the territory. You can't make use of the map that way if you don't think of the things the map is labeling as real.



Certainly we do. We use these complex mathematical abstractions to make predictions that are confirmed to many decimal places. That requires a deep knowledge of how they connect to our world.



No. This kind of statement needs a *huge* amount of support which you have not given, and which I don't see how you could give in view of the predictive accuracy of our models which I have just referred to.

Hmm...

Anyway back to Block Spacetime. Is it a consensus that the past no longer exist or is it only your belief? I just saw Michio Kaku in TV describing how to go back in time.. how come they keep saying this when the past no longer exist? Any idea of how many percentage of physicists share your view, etc.?

 

[PeterDonis]

Is it a consensus that the past no longer exist or is it only your belief? I just saw Michio Kaku in TV describing how to go back in time.. how come they keep saying this when the past no longer exist? Any idea of how many percentage of physicists share your view, etc.?

With regard to what physicists believe, as far as I know, all of this talk about going back in time is speculative (certainly the stuff I usually see Michio Kaku saying when he does TV specials is speculative); no physicist claims that we *can* time travel to our past, and no physicist claims to know for certain that it's impossible. It's just speculation.

 

[stglyde]

With regard to what physicists believe, as far as I know, all of this talk about going back in time is speculative (certainly the stuff I usually see Michio Kaku saying when he does TV specials is speculative); no physicist claims that we *can* time travel to our past, and no physicist claims to know for certain that it's impossible. It's just speculation.

About general invariance.. it is synonymous to diffeomorphism invariance isn't it.. although other physicists use diffeo morph to mean background independence (no prior geometry). Which do you use?

So far have we actually confirmed general invariance (diffeo morph) in experiments? What experiments actually do that?

I'm interested in local lorentz invariance in quantum gravity studies. General Relativity is not compatible with QM so GR may just be lower limit of a more primary theory. What is your favorite primary theory and Why?

 

[bobc2]

I think you're missing the point. The 4-dimensional manifold we live in is a space-time, *not* a space-space. It has one timelike dimension and three spacelike dimensions. It has a metric that is not positive definite.

No. The R4 manifold we live in is space-space and positive definite. I thought you would show us, based on this, the rigorous mathematical procedure for arriving at the coordinate transformations that apply to a special relativity approach to describing our physical experience in the manifold. The physical objects present on the manifold are independent of the manifold and its metric space.

I can lay out a big sheet of paper on the table with a cartesian chart having an X1 horzontal axis and a X2 axis. Then, I arbitrarily place pencils, rulers, sticks, etc. on the paper with random orientations. It would be difficult to find an easy mathematical representation of the object locations, orientations and their relatedness.

On the other hand, I can place objects on the grid in very special thought-out patterns that are ordered in a special way. It may be difficult to describe the positions and orientations of the objects using the cartesian coordinates. However, given the symmetries associated with the object placement, I can perhaps find a special system of coordinates,
X1' and X2', X1'' and X2'', etc., for mathematically expressing the positions and orientations of the objects--along with relatedness among the objects. If the objects are all long, and if they are all generally oriented with their long directions at less than 45 degrees with respect to the cartesian X2 direction, the new X2' and X2'', etc., coordinates would be special as compared to the X1, X1', X1'', etc.

The 4-dimensional positive definite manifold with 4-D objects populating the manifold, lying along 4-D world lines should be viewed the same way.

These statements are invariant statements of physics; they do not depend on describing the timelike dimension using a "time" parameter. (For example, we can describe spacetime using null coordinates, which replace the timelike coordinate and one spacelike coordinate with a pair of null coordinates. But physically spacetime is still the same, just described differently.)

But, they were arbitrarily chosen. And without time as a parameter, along world lines, you are left without a physically understood picture of reality. It's easy to talk about time as the 4th dimension (or some ill-defined mixture of space and time), but in reality no one has the foggiest idea what that means physically. However, a 4-dimensional space with time passing as a parameter as some aspect of the observer moves along the 4th spatial dimension at the speed of light--is a concept that can be envisioned. Without 4 spatial dimensions, you really don't know what you've got for reality--except as an abstract mathematical description.

So, the 4-D manifold we live in is different, *physically*, than a 4-D space-space with a 4-D positive definite metric.

No. I agree that you can model physics with the Minkowski metric. But, that seems arbitrary and contrived. And it gives us a physical picture that is really not comprehensible (on one really knows what it means to have time as a physical dimension). So, in that sense the Minkowski manifold is different from what we live in.

We don't need an R4 manifold other than positive definite with a Euclidean orthonormal basis chart. The coordinate space we live in can be obtained through coordinate transformations.

You can't model the former with the latter.

Why can't you do it with coordinate transformations? We do it with curves on a sheet of paper all of the time. We start with cartesian coordinates and then draw in hyperbolic curves and affine coordinates, etc.

I'm not sure I understand what you mean by this.

We use parametric equations all the time--for example, to describe motion of projectiles in 3-D space, i.e., Y(t) and X(t). Time is just a parameter along the world lines in exactly the same sense.

No, this is wrong. What the different SR cross-sections show us is that four *spacetime* dimensions must be available--three spacelike and one timelike (or, alternatively, as I said above, two spacelike and two null, which is physically the same thing). The different SR cross sections are *inconsistent* with there being four *spatial* dimensions, because the SR cross sections require a Minkowski metric, i.e., not positive definite. This is a genuine physical difference; you can't handwave it away.

Time is not needed at all to describe the 4-dimensional universe populated with 4-dimensional objects. Although, it is useful in computing distances along world lines, since obervers are "moving" along world lines at c. But, from the "birds eye view" you alluded to in an earlier post, you can remove consciousness from the observers (which are, from the "birds" view, after all, just 4-D objects) and the super hyperspace "bird" just sees a static 4-dimensional structure. It would never occur to the "bird" that he would need anything other than an R4 manifold with an orthonormal basis set along with appropriate transformations to describe what he is viewing.

 

[PatrickPowers]

We can describe spacetime using null coordinates, which replace the timelike coordinate and one spacelike coordinate with a pair of null coordinates.

I find this most intriguing. Might you offer a reference?

 

[PeterDonis]

About general invariance.. it is synonymous to diffeomorphism invariance isn't it.. although other physicists use diffeo morph to mean background independence (no prior geometry). Which do you use?

By "general covariance" I mean, as I said in a previous post, that you can describe spacetime using any coordinates you like. Since transformations between different coordinate systems are diffeomorphisms, this definition of general covariance is equivalent to diffeomorphism invariance of physical laws; i.e., valid physical laws must be expressible in a form that is diffeomorphism invariant. GR meets this requirement since it expresses all physical laws in terms of tensors and other geometric objects, which are diffeomorphism invariant.

I don't know for sure whether the above is equivalent to background independence or not. A physics that included "prior geometry" might still be expressible in diffeomorphism invariant terms. For example, suppose I have a theory that says that the "preferred frame" of the universe is the "comoving" frame in the standard FRW model, so that, for example, the motion of the Earth relative to this frame appears in physical laws. I could still write such laws in terms of tensors and other geometric objects; for example, I could express the motion of the Earth relative to the preferred frame as the contraction of the Earth's 4-momentum at a given event with a 4-vector normal to the spacelike hypersurface of constant comoving time t that contains that event. Such a law would be diffeomorphism invariant but would still express the presence of a preferred frame.

So far have we actually confirmed general invariance (diffeo morph) in experiments? What experiments actually do that?

How would you confirm diffeomorphism invariance in particular by doing experiments? The physical laws of GR are certainly confirmed very well, and those are expressible in diffeomorphism invariant terms. Does that count?

I'm interested in local lorentz invariance in quantum gravity studies. General Relativity is not compatible with QM so GR may just be lower limit of a more primary theory. What is your favorite primary theory and Why?

This is kind of getting off topic, but I agree that we are likely to find that GR is the classical, low energy limit of some quantum gravity theory that may look quite different. I don't have a "favorite" here because we have no experimental data in the regime where such a theory would be expected to show itself (particle energies approaching the Planck energy), so there are no constraints on such a theory other than the need to have GR as a low energy classical limit, which isn't a very tight constraint.

 

[PeterDonis]

I find this most intriguing. Might you offer a reference?

Try this page for a start:

http://www.mathpages.com/rr/s1-09/1-09.htm

 

[PeterDonis]

The post evoking this response was not mine. I'm not sure how it came out as my post.

That's weird, I'm not sure either. The editing window was still open on my post so I went back and fixed the quote. Sorry for the mixup.

 

[PeterDonis]

No. The R4 manifold we live in is space-space and positive definite.

What is your basis for this claim? It seems obviously false to me since spacetime is locally Lorentz invariant, not Euclidean invariant, which is what your statement would imply. (Btw, by "basis" I mean "physical basis"--what physical experiments show you that we live in an R4 manifold with a positive definite metric?)

We use parametric equations all the time--for example, to describe motion of projectiles in 3-D space, i.e., Y(t) and X(t). Time is just a parameter along the world lines in exactly the same sense.

*Proper* time, yes. *Coordinate* time, no. In your terms, all four coordinates, including the "time" coordinate, are functions of the "time parameter" along a timelike curve, which I'll refer to as proper time since that's the standard term. More precisely, a parametrization of a timelike worldline in spacetime is a one-to-one mapping of proper time values to 4-tuples of coordinate values.

I did not mean to say that such a mapping was not possible or that it was not useful for understanding coordinate charts. It certainly is. But proper time should not be confused with coordinate time; they are two different things.

But, they were arbitrarily chosen. And without time as a parameter, along world lines, you are left without a physically understood picture of reality.

I agree with this to an extent. Coordinates, in general, are not physical observables; in some cases you can choose coordinate charts that match up well with certain symmetries of a spacetime and therefore can be more or less equated to certain physical observables, but those are special cases. Proper time, however, is an obvious physical observable: it can be directly read off clocks that follow a given worldline.

But proper time is not left out of the coordinate models I was describing; in fact, it's precisely the physical requirement of making sure the correct proper time is assigned to any given segment of a curve that makes the metric so important. And it's precisely the physical fact that not all curves are timelike that requires a non-positive definite metric; along a non-timelike curve, there is no proper time, and parametrizing such a curve cannot be done using a "time" parameter. You have to use a parameter that corresponds to something else, physically, besides proper time.

Why can't you do it with coordinate transformations? We do it with curves on a sheet of paper all of the time. We start with cartesian coordinates and then draw in hyperbolic curves and affine coordinates, etc.

Yes, and we interpret the lengths along the curves, physically, as lengths--*proper* lengths. But lengths are not times; they are physically different things. You can measure time in the same *units* as length, by using the speed of light as a conversion factor, but that does not make proper times the same, physically, as proper lengths. So if we want to use an R4 manifold to model the actual physical spacetime we live in, we cannot put a metric on it that only allows for one type of measure along a curve; there have to be three (timelike, spacelike, and null), and the measures for nearby curves have to be related in a way that preserves Lorentz invariance. A Euclidean, positive definite metric simply cannot model that.

Note, please, that I am not talking now about "time" or "space" as coordinates; I am talking about them as physical measures along curves. The physical measure along a timelike curve is proper time; the curve can be expressed, as I noted above, as a one-to-one mapping between proper time values and 4-tuples of coordinates, and we can label points on the curve by their proper time values and talk about them without ever using coordinates. But the physical measure along a spacelike curve is proper length, *not* proper time; it is a physically different thing. That is why our model needs to treat time and space differently: because the physical measure along timelike curves is fundamentally different than the physical measure along spacelike curves.

We don't need an R4 manifold other than positive definite with a Euclidean orthonormal basis chart. The coordinate space we live in can be obtained through coordinate transformations.

No, it can't, for the reasons given above.

Time is not needed at all to describe the 4-dimensional universe populated with 4-dimensional objects.

As a coordinate, no. As a measure along timelike curves, which is fundamentally different than the measure along spacelike curves, absolutely yes, it is. Otherwise the correspondence between the model and the real world, which you are so concerned about, is not there.

But, from the "birds eye view" you alluded to in an earlier post, you can remove consciousness from the observers (which are, from the "birds" view, after all, just 4-D objects) and the super hyperspace "bird" just sees a static 4-dimensional structure. It would never occur to the "bird" that he would need anything other than an R4 manifold with an orthonormal basis set along with appropriate transformations to describe what he is viewing.

It sure would, as soon as he tries to capture the physical difference between timelike and spacelike curves. That difference does not require conscious observers following the timelike curves.

 

[stglyde]

By "general covariance" I mean, as I said in a previous post, that you can describe spacetime using any coordinates you like. Since transformations between different coordinate systems are diffeomorphisms, this definition of general covariance is equivalent to diffeomorphism invariance of physical laws; i.e., valid physical laws must be expressible in a form that is diffeomorphism invariant. GR meets this requirement since it expresses all physical laws in terms of tensors and other geometric objects, which are diffeomorphism invariant.

I don't know for sure whether the above is equivalent to background independence or not. A physics that included "prior geometry" might still be expressible in diffeomorphism invariant terms. For example, suppose I have a theory that says that the "preferred frame" of the universe is the "comoving" frame in the standard FRW model, so that, for example, the motion of the Earth relative to this frame appears in physical laws. I could still write such laws in terms of tensors and other geometric objects; for example, I could express the motion of the Earth relative to the preferred frame as the contraction of the Earth's 4-momentum at a given event with a 4-vector normal to the spacelike hypersurface of constant comoving time t that contains that event. Such a law would be diffeomorphism invariant but would still express the presence of a preferred frame.

Are you saying "prior geometry" is the same as "preferred frame"? This is getting confusing. Let us define the terms well so we can understand one another. In the Jan 2004 Sci Am cover story "Loop Quantum Gravity". It is defined thus (according to Smolin):

"In particular we kept two key principles of general relativity at the heart of our calculations.

The first is known as background independence. This principle says that the geometry of spacetime is not fixed. Instead the geometry is an evolving, dynamical quantity. To find the geometry, one has to solve certain equations that include all the effects of matter and energy. Incidentally, string theory, as currently formulated, is not background independent; the equations describing the strings are set up in a predetermined classical (that is, nonquantum) spacetime.

The second principle, known by the imposing name of diffeomorphism invariance, is closely related to background independence. This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations. A point in spacetime is defined only by what physically happens at it, not by its location according to some special set of coordinates (no coordinates are special). Diffeomorphism Invariance is very powerful and is of fundamental importance in
General Relativity. "

You said diffeomorphism invariant can still be expressed in the presence of a preferred frame. Given Smolin input. You think diffeomorphism invariant can also be expressed in the presence of a background (background dependence or prior geometry)?

 

[PatrickPowers]

Try this page for a start:

http://www.mathpages.com/rr/s1-09/1-09.htm

I notice that this the ninth section of a full-sized book. I have read the first four sections and this seems to be exactly what I've been looking for. It will take a while to work through.

 

[atyy]

Let us define the terms well so we can understand one another. In the Jan 2004 Sci Am cover story "Loop Quantum Gravity". It is defined thus (according to Smolin):

...

The second principle, known by the imposing name of diffeomorphism invariance, is closely related to background independence. This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations. A point in spacetime is defined only by what physically happens at it, not by its location according to some special set of coordinates (no coordinates are special). Diffeomorphism Invariance is very powerful and is of fundamental importance in
General Relativity. "

Smolin is wrong on a detail. Theories prior to general relativity can be expressed in diffeomorphism invariant form. As Andrade, Marolf and Deffayet say, "The point here is that any local theory (e.g., a single free scalar field) can be written in diffeomorphism-invariant form through a process known as parametrization."

 

[PeterDonis]

You said diffeomorphism invariant can still be expressed in the presence of a preferred frame.

Not quite. I said that a theory with a preferred frame *might* still be expressible in diffeomorphism invariant form, and I gave an example of how that *might* be possible. Basically, if it turns out that a theory with a preferred frame in it can still be expressed in any coordinates I choose, then it's still diffeomorphism invariant, even though it has a preferred frame.

Given Smolin input. You think diffeomorphism invariant can also be expressed in the presence of a background (background dependence or prior geometry)?

You'll note that Smolin does not say that background independence and diffeomorphism invariance are the same; he only says they're closely related. So it would appear that he would also admit the possibility that there could be a theory that was diffeomorphism invariant but not background independent. This is not to say that anyone actually has such a theory; just that the possibility means we should be careful about making dogmatic statements about what is "required" of a theory.

 

[atyy]

Not quite. I said that a theory with a preferred frame *might* still be expressible in diffeomorphism invariant form, and I gave an example of how that *might* be possible. Basically, if it turns out that a theory with a preferred frame in it can still be expressed in any coordinates I choose, then it's still diffeomorphism invariant, even though it has a preferred frame.
You'll note that Smolin does not say that background independence and diffeomorphism invariance are the same; he only says they're closely related. So it would appear that he would also admit the possibility that there could be a theory that was diffeomorphism invariant but not background independent. This is not to say that anyone actually has such a theory; just that the possibility means we should be careful about making dogmatic statements about what is "required" of a theory.

An explicit example of such a theory is given by Laddha and Varadarajan. Andrade et al refer to Torre who discusses such formulations at length.

Also, Newtonian and Nordstrom gravity both have formulations as geometric theories, even though they are more conventionally formulated non-geometrically.

 

[bobc2]

It sure would, as soon as he tries to capture the physical difference between timelike and spacelike curves. That difference does not require conscious observers following the timelike curves.

As usual you've done a comptetent job of representing the conensus view of special relativity. I understood your points quite well. I was never exposed to anything different throughout my graduate physics curriculum. Usually, whether it was a course in classical field theory, modern physics, special relativity, general relativity, cosmology, etc., we were given the Minkowski metric as a starting point, then went from there.

But I was never satisfied with that, because I don't believe it treats time correctly--and still have not resolved my concerns. My PhD advisor would not discuss the subject--he thought it was a waste of time. And at that stage as a student I knew he was right--that I needed to concentrate on learning physics before challenging established ideas.

At this point on the forum, I'm afraid I've pushed on this to the point of violating the agreed upon rules here. The forum advisors have given me more leeway than was probably justified. Hopefully, others have benefitted from the occasion to consider more carefully the fundamental mathematical machinery upon which special relativity is based (thanks primarily to PeterDonis's contributions).

 

[PeterDonis]

Usually, whether it was a course in classical field theory, modern physics, special relativity, general relativity, cosmology, etc., we were given the Minkowski metric as a starting point, then went from there.

But I was never satisfied with that, because I don't believe it treats time correctly--and still have not resolved my concerns.

This is where I get confused; the only "concerns" you have expressed that I can see are: (1) about time being a parameter, which I've agreed with and shown that relativity includes; and (b) that the metric of spacetime is not positive definite, which I've given a good physical reason for: timelike curves are physically different than spacelike curves (and null curves are different from both). So either I'm misunderstanding the point of these concerns, or there are other concerns that I'm not seeing. I don't think expressing them would violate forum rules.