Curvatures in space-time: actual reality or mathematical concept?

[tris_d]

According to GR is space actually supposed to be bending around mass, or is the concept of "curvature" just abstract and purely mathematical idea?

 

[haushofer]

You could ask that question for basically every physical concept. Physics just desribes reality as we perceive it and beyond, without referring to the "actual reality" (whatever that means).

Perhaps you like this story of Feynman:

In the Graduate College dining room at Princeton everybody used to sit with his own group. I sat with the physicists, but after a bit I thought: It would be nice to see what the rest of the world is doing, so I’ll sit for a week or two in each of the other groups.

When I sat with the philosophers I listened to them discuss very seriously a book called Process and Reality by Whitehead. They were using words in a funny way, and I couldn’t quite understand what they were saying. Now I didn’t want to interrupt them in their own conversation and keep asking them to explain something, and on the few occasions that I did, they’d try to explain it to me, but I still didn’t get it. Finally they invited me to come to their seminar.

They had a seminar that was like, a class. It had been meeting once a week to discuss a new chapter out of Process and Reality –some guy would give a report on it and then there would be a discussion. I went to this seminar promising myself to keep my mouth shut, reminding myself that I didn’t know anything about the subject, and I was going there just to watch.

What happened there was typical–so typical that it was unbelievable, but true. First of all, I sat there without saying anything, which is almost unbelievable, but also true. A student gave a report on the chapter to be studied that week. In it Whitehead kept using the words “essential object” in a particular technical way that presumably he had defined, but that I didn’t understand.

After some discussion as to what “essential object” meant, the professor leading the seminar said something meant to clarify things and drew something that looked like lightning bolts on the blackboard. “Mr. Feynman,” he said, “would you say an electron is an ‘essential object’?”

Well, now I was in trouble. I admitted that I hadn’t read the book, so I had no idea of what Whitehead meant by the phrase; I had only come to watch. “But,” I said, “I’ll try to answer the professor’s question if you will first answer a question from me, so I can have a better idea of what ‘essential object’ means. Is a brick an essential object?”

What I had intended to do was to find out whether they thought theoretical constructs were essential objects. The electron is a theory that we use; it is so useful in understanding the way nature works that we can almost call it real. I wanted to make the idea of a theory clear by analogy. In the case of the brick, my next question was going to be, “What about the inside of the brick?”–and I would then point out that no one has ever seen the inside of a brick. Every time you break the brick, you only see the surface. That the brick has an inside is a simple theory which helps us understand things better. The theory of electrons is analogous. So I began by asking, “Is a brick an essential object?”

Then the answers came out. One man stood up and said, “A brick as an individual, specific brick. That is what Whitehead means by an essential object.”

Another man said, “No, it isn’t the individual brick that is an essential object; it’s the general character that all bricks have in common–their ‘brickiness’–that is the essential object.”

Another guy got up and said, “No, it’s not in the bricks themselves. ‘Essential object’ means the idea in the mind that you get when you think of bricks.”

Another guy got up, and another, and I tell you I have never heard such ingenious different ways of looking at a brick before. And, just like it should in all stories about philosophers, it ended up in complete chaos. In all their previous discussions they hadn’t even asked themselves whether such a simple object as a brick, much less an electron, is an “essential object.”

 

[pervect]

It's space-time that's curved, not just space. But with the usual slicing of space-time into space + time, the spatail slices are curved as well.

As far as it being "real" or not - I think of it as "real". But I'm not sure what sort of experiment (if any), would convince you that "curvature" or anything else for that matter was "real" or "not real".

Typically, "reality" is addressed by philosophy and isn't experimentally testable.

 

[tris_d]

You could ask that question for basically every physical concept. Physics just desribes reality as we perceive it and beyond, without referring to the "actual reality" (whatever that means).

Perhaps you like this story of Feynman:

Can you rephrase that as a specific statement?

a.) "curvatures" are only abstract mathematical idea

b.) "curvatures" are abstract idea with a possibility to be actual reality

c.) "curvatures" are ...?


I'm not sure I agree with what you're saying. When physics say there is such thing as electron, do we consider it actual reality or just abstract idea? And when physics say there is such thing as electric or magnetic field, do we consider it actual reality or just abstract idea as well?

 

[tris_d]

It's space-time that's curved, not just space. But with the usual slicing of space-time into space + time, the spatail slices are curved as well.

I think "curvatures" can not really exist in some 3D spatial volume as actual geometrical feature without reference frame against which it would be curved against. Actual curves would require actual reference frame and "empty space" contains nothing, so for it to contain some actual topology seem to be direct contradiction.

As far as it being "real" or not - I think of it as "real". But I'm not sure what sort of experiment (if any), would convince you that "curvature" or anything else for that matter was "real" or "not real".

Typically, "reality" is addressed by philosophy and isn't experimentally testable.

If you can explain some phenomena by supposing space is actually curved that I can not explain with the concept of potentials, gradients or fields, then I will submit space is actually curved as the best explanation.

 

[Dale]

If you can explain some phenomena by supposing space is actually curved that I can not explain with the concept of potentials, gradients or fields, then I will submit space is actually curved as the best explanation.

That is one of the best criteria I have seen for "actual reality". Most people asking similar questions simply avoid trying to define what they mean by it. I congratulate you for making the effort.

However, if you have even the slightest mathematical background then you should be aware that it is always possible to write the same equation multiple ways. That is basically what algebra is all about. Therefore, it is always possible to come up with a different but mathematically equivalent expression. Meaning that there isn't anything in physics which meets this standard.

 

[A.T.]

When physics say there is such thing as electron, do we consider it actual reality or just abstract idea? And when physics say there is such thing as electric or magnetic field, do we consider it actual reality or just abstract idea as well?

Particles and fields are human ideas. They are useful in predicting phenomena in the observed reality.

 

[WannabeNewton]

I think "curvatures" can not really exist in some 3D spatial volume as actual geometrical feature without reference frame against which it would be curved against. Actual curves would require actual reference frame and "empty space" contains nothing, so for it to contain some actual topology seem to be direct contradiction.

You seem to be saying that in order to detect \ define curvature in any possible way you would need your hyper - surface to be embedded in some ambient space. This isn't true; even when dealing with regular surfaces in R^3 you will see that Gauss curvature is an intrinsic property of the regular surface. Also, on manifolds endowed with a riemannian metric, the curvature endomorphism is given in terms of the levi - civita connection which make sense without reference to some embedding. I don't understand your statement about "actual" topologies: you don't need to refer to curvature or the likes to even endow a set with a topology.

 

[Dickfore]

Perhaps you like this story of Feynman:

That's a great story! Where is it from?

 

[ZikZak]

If you can explain some phenomena by supposing space is actually curved that I can not explain with the concept of potentials, gradients or fields, then I will submit space is actually curved as the best explanation.

On what basis do you claim that curvature is less-than-real, while fields and gradients are really-real? What does really-real mean anyway? How do you determine when something is really-real?

 

[tris_d]

Sure. There are alternates to GR (and even reformulations of GR) that explain GW's as propagation of a real field against a flat spacetime backdrop. String theory after all demands existence of gravitons as fundamental string excitation that acts as a bona-fide force carrying 'particle'. What I was getting at with GW's is that 'spacetime curving' is a mystical concept without physical content if spacetime is purely a relational entity involving nothing but clocks and rulers. Get's down to what 'physical' implies. There are some that will probably argue that energy and momentum are just mental constructs.

I agree. The thing is people in cosmology field take the idea seriously and so we have these 'shape of the universe' models where the space is actually curved and can be closed and finite. I thought that was not logical assumption and is unreasonable because it could not be proven even if it was true, but people I spoke to felt differently and were firmly believing the space actually bends and curves and pointed me to 'Wilkinson Microwave Anisotropy Probe' measurements for the curvature of the whole universe. They measured zero curvature, as I naturally expected, but apparently NASA and our astronomer friends are far less agnostic about the whole "curved-space" idea.

 

[tris_d]

That is one of the best criteria I have seen for "actual reality". Most people asking similar questions simply avoid trying to define what they mean by it. I congratulate you for making the effort.

However, if you have even the slightest mathematical background then you should be aware that it is always possible to write the same equation multiple ways. That is basically what algebra is all about. Therefore, it is always possible to come up with a different but mathematically equivalent expression. Meaning that there isn't anything in physics which meets this standard.

I would assume that, but I would not claim it without trying it out myself. I think there is a difference when you call abstract things abstract names like "field" or "potential gradient", because even if they are curvatures in reality they would still conform to this abstract definition, relations to distance and mass would stay the same. But if you give abstract things specific names, then you kind of insist that what it actually is. So I understand how there are people who consider 'curvatures' are actual reality rather than abstract idea, I just wondered if that's a bad thing or whether I lack some understanding.

 

[Q-reeus]

I agree. The thing is people in cosmology field take the idea seriously and so we have these 'shape of the universe' models where the space is actually curved and can be closed and finite. I thought that was not logical assumption and is unreasonable because it could not be proven even if it was true, but people I spoke to felt differently and were firmly believing the space actually bends and curves and pointed me to 'Wilkinson Microwave Anisotropy Probe' measurements for the curvature of the whole universe. They measured zero curvature, as I naturally expected, but apparently NASA and our astronomer friends are far less agnostic about the whole "curved-space" idea.

tris_d - just to be clear on this matter, GR's geometrical curved spacetime basis does provide a very successful predictive model - which may however fail at a certain level yet to be tested. Current measurements suggests perfectly flat or very close to perfectly flat overall spatial curvature but this is a prediction of inflationary model of cosmology rather than GR itself. Real problem imo is this conceptual dichotomy where in GR 'gravitational field' has totally different connotation to say electric field. One glaring problem for me is that gravitational field is allowed to have an ambiguously defined energy content but not allowed to act as it's own further source, despite the insistence that all other forms of stress-energy must contribute. Ask an expert why and good luck getting a sensible answer.

The fact of gravitational waves as per binary pulsar data imo screams out one of two things - curvature of an actual physical medium on geometrical formulation of gravity, or physical field propagation through flat spacetime on field formulation of gravity. But definitely not just curvature of spatial and temporal relationships.

 

[haushofer]

Can you rephrase that as a specific statement?

a.) "curvatures" are only abstract mathematical idea

b.) "curvatures" are abstract idea with a possibility to be actual reality

c.) "curvatures" are ...?I'm not sure I agree with what you're saying. When physics say there is such thing as electron, do we consider it actual reality or just abstract idea? And when physics say there is such thing as electric or magnetic field, do we consider it actual reality or just abstract idea as well?

Well, in Newton's days one could say that gravity is actually some sort of invisible arm pulling masses together instantaneously. Einstein says it's because energy curves spacetime. Someone like Verlinde claims gravity is some sort of entropic force. From a string point of view it's because energy is exchanging closed strings with vibrational modes we call a "graviton". Some of these ideas overlap, some don't.

What is "actual reality"? That's rather an abstract notion for a physicist, I think. Perhaps it's some fixed point in "theory-space" of which we hope our theories are converging to. Perhaps it's not fixed at all. Perhaps there are more points. I would say that every theory which describes nature as accurately, consistently and completely as is possible in a certain paradigm is as close as you can get. Personally I don't like concepts as "actual reality" or "absolute truth". It makes you forget that physics is all about description and perception. A wise man once said that description and perception come with deception.

Perhaps I've missed it, but I didn't see a definition of "actual reality".

 

[haushofer]

That's a great story! Where is it from?

I'm sure it is in one of its books, but I've forgotten which one and google isn't helping :P

 

[Dale]

So I understand how there are people who consider 'curvatures' are actual reality rather than abstract idea, I just wondered if that's a bad thing or whether I lack some understanding.

That is fine, as long as they understand that EVERYTHING in physics is, according to your classification, an "abstract idea" and not "actual reality". Force, energy, mass, time, potential, momentum, and everything else all fail to meet your criterion for being "actual reality", so it shouldn't be surprising that curvature does also.

 

[jtbell]

I'm sure it is in one of its books, but I've forgotten which one and google isn't helping :P

A Google search for

feynman "essential object"

led me to Surely You're Joking, Mr. Feynman!, which fits because I remember reading that story myself a long time ago, and I think that book was his first "autobiographical" one.

http://books.google.com/books?id=7p...=onepage&q=feynman "essential object"&f=false

 

[PeterDonis]

Real problem imo is this conceptual dichotomy where in GR 'gravitational field' has totally different connotation to say electric field.

This is because gravity is different from electromagnetism. (Specifically, gravity obeys the equivalence principle while electromagnetism does not.) Why is that a problem? The two things are physically different, so the theories that best describe them are different as well.

If it's the word "field" that you're getting hung up on, then drop it; or at least, don't apply it to gravity. "Field" is just a word.

One glaring problem for me is that gravitational field is allowed to have an ambiguously defined energy content but not allowed to act as it's own further source, despite the insistence that all other forms of stress-energy must contribute. Ask an expert why and good luck getting a sensible answer.

We've been here before, of course. You didn't like the answers you got before, but that doesn't mean they're not sensible (or correct, for that matter). But how about the answer I gave above: gravity is different from other things. Does that help?

I really don't understand why people think it's a big problem that gravity is different from other things. What's so hard about that? I agree it would make some theoretical tasks easier if gravity shared more properties with other things, but the fact is that it doesn't.

The fact of gravitational waves as per binary pulsar data imo screams out one of two things - curvature of an actual physical medium on geometrical formulation of gravity, or physical field propagation through flat spacetime on field formulation of gravity. But definitely not just curvature of spatial and temporal relationships.

What is the difference, in your view, between "curvature of an actual physical medium" and "curvature of spatial and temporal relationships"?

Btw, you are correct that, in the particular case of the binary pulsar data, you can also interpret the result as "physical field propagation", but not through a flat background spacetime--through a curved background spacetime. The propagating gravitational waves do not constitute the entire curvature; they only constitute the rapidly varying portion of the curvature.

However, this "physical field propagation" interpretation breaks down in other cases, such as cosmology; I'm not aware of any workable interpretation of the expansion of the universe in terms of "physical field propagation".

 

[tris_d]

Particles and fields are human ideas. They are useful in predicting phenomena in the observed reality.

Yes, everything is human idea, it's all just words. We give things names, and we name their properties as well. We then relate this information to other such things we know, and that's our knowledge, it's all relative. The problem here I see is the question whether this thing we call "curvature" is a property of "space", or property of "mass", or both, whatever those things actually are.

 

[WannabeNewton]

The problem here I see is the question whether this thing we call "curvature" is a property of "space", or property of "mass", or both, whatever those things actually are.

I don't see the problem there. For our purposes, the curvature of a pseudo riemannian 4 - manifold is well - defined and computed on local coordinate charts. The EFEs then give the relation between the curvature and the stress - energy tensor.

 

[harrylin]

According to GR is space actually supposed to be bending around mass, or is the concept of "curvature" just abstract and purely mathematical idea?

Curvature is here a mathematical characteristic of the "world of events" that is called space-time.

Compare: http://www.bartleby.com/173/17.html
http://en.wikipedia.org/wiki/Spacetime (see the intro + "Spacetime in general relativity")

[..] If you can explain some phenomena by [..] fields[..]

As a matter of fact, Einstein and others also used the expression "gravitational field". But "field" is a very vague, abstract concept; its original meaning is merely "area" or "zone".

 

[tris_d]

You seem to be saying that in order to detect \ define curvature in any possible way you would need your hyper - surface to be embedded in some ambient space. This isn't true; even when dealing with regular surfaces in R^3 you will see that Gauss curvature is an intrinsic property of the regular surface. Also, on manifolds endowed with a riemannian metric, the curvature endomorphism is given in terms of the levi - civita connection which make sense without reference to some embedding. I don't understand your statement about "actual" topologies: you don't need to refer to curvature or the likes to even endow a set with a topology.

By 'actual' I mean just "measurable". But as long as the same measurement can be explained with some other concept where the space can stay flat I will believe that explanation is better. Basically, if there is some experiment that strongly suggests gravity indeed curves space rather than being some field or something in flat space, I'd like to hear about it.

 

[Q-reeus]

Q-reeus: "One glaring problem for me is that gravitational field is allowed to have an ambiguously defined energy content but not allowed to act as it's own further source, despite the insistence that all other forms of stress-energy must contribute. Ask an expert why and good luck getting a sensible answer."
We've been here before, of course . You didn't like the answers you got before, but that doesn't mean they're not sensible (or correct, for that matter). But how about the answer I gave above: gravity is different from other things. Does that help?

Hello Peter - I just had this sixth-sense feeling we would meet up again about here . Sure the two (gravity vs EM) are different, and yes it's true I didn't like the answers given before because as I recollect there was no proper dealing with the question of gravity being excluded as source term in the SET. As I further recall you cited ADM definition as evidence gravity (but only as GW's) sort of contributed. But that begs the question - what is fundamentally different about static vs propagating field curvature that one can sort of contribute but absolutely not the other? This is what I mean about consistency issues in GR. There is no such divide with EM as source for instance - static or wave, they both carry energy and both act as contributors in SET. Hocus without the pocus?

What is the difference, in your view, between "curvature of an actual physical medium" and "curvature of spatial and temporal relationships"?

Gravitational waves for one!

Btw, you are correct that, in the particular case of the binary pulsar data, you can also interpret the result as "physical field propagation", but not through a flat background spacetime--through a curved background spacetime. The propagating gravitational waves do not constitute the entire curvature; they only constitute the rapidly varying portion of the curvature.

I understand that source - binary pulsars - as gravitating objects necessarily generates a static/quasi-static curvature through which GW's must propagate, but that's in geometric interpretation. You are well aware there exist other formulations of gravity where everything is owing to fields existing in an unobservable flat background spacetime. That's what I meant there.

However, this "physical field propagation" interpretation breaks down in other cases, such as cosmology; I'm not aware of any workable interpretation of the expansion of the universe in terms of "physical field propagation".

I'm no expert, but there are alternate descriptions/formulations using field concept - Yuri Baryshev comes to mind but there are others. In any case, sticking to geometric concept, how do you explain energy-momentum transport in GW's without it directly implying a physically 'stressed' medium of some sort?

 

[harrylin]

By 'actual' I mean just "measurable". But as long as the same measurement can be explained with some other concept where the space can stay flat I will believe that explanation is better. Basically, if there is some experiment that strongly suggests gravity indeed curves space rather than being some field or something in flat space, I'd like to hear about it.

As far as I know, experiment cannot distinguish such things. Note that also "flat space" is a mathematical term; it has no meaning that I know of in a physical sense (how can empty space be literally "flat"?; that's like saying that it is "green"!).

 

[Q-reeus]

As far as I know, experiment cannot distinguish such things. Note that also "flat space" is a mathematical term; it has no meaning that I know of in a physical sense (how can empty space be literally "flat"?; that's like saying that it is "green"!).

Sum of angles in a triangle! Greater than 180 degrees = +ve curvature, less than 180 degrees = -ve curvature. Note that this can be explained also using a non-geometric field theory = 'effective' curvature.

 

[tris_d]

On what basis do you claim that curvature is less-than-real, while fields and gradients are really-real? What does really-real mean anyway? How do you determine when something is really-real?

What's measurable is real enough for me. Only, we can explain measurements with all sorts of mathematical concepts, so the question is which one is better. For example, I see these equations that are supposed to be about curves are more about some tensors really, but I don't see anyone suggesting space is actually made of little bouncy springs. You see what I mean?

 

[harrylin]

Sum of angles in a triangle! Greater than 180 degrees = +ve curvature, less than 180 degrees = -ve curvature. Note that this can be explained also using a non-geometric field theory = 'effective' curvature.

Ah, then you actually mean with "flat space" something else, it's about "geometrical measurements with tools" - not the vacuum (nothingness?) itself to which tris seems to refer.

Compare: http://www.bartleby.com/173/23.html

The point is, as several people here mentioned, that one can map phenomena to a flat background - which is just as much the fruit of our imagination as a curved one. In a parallel thread we are now even discussing a locally flowing background!

 

[ZikZak]

What's measurable is real enough for me. Only, we can explain measurements with all sorts of mathematical concepts, so the question is which one is better. For example, I see these equations that are supposed to be about curves are more about some tensors really, but I don't see anyone suggesting space is actually made of little bouncy springs. You see what I mean?

No, because your preference for fields is at least as arbitrary. A field is a mathematical object which assigns a numerical value to each point in space, or space-time. There's nothing more mathematically abstracted from reality than that.

 

[Dale]

By 'actual' I mean just "measurable".

Curvature is definitely measurable. It is tidal effects. So I guess that curvature is "actual" by this definition but not "actual reality" by the previous definition.

 

[tris_d]

One glaring problem for me is that gravitational field is allowed to have an ambiguously defined energy content but not allowed to act as it's own further source, despite the insistence that all other forms of stress-energy must contribute. Ask an expert why and good luck getting a sensible answer.

I am not aware of it. What theory, what equation are you referring to?

The fact of gravitational waves as per binary pulsar data imo screams out one of two things - curvature of an actual physical medium on geometrical formulation of gravity, or physical field propagation through flat spacetime on field formulation of gravity. But definitely not just curvature of spatial and temporal relationships.

I don't know about that either. What did we measure and what was the reading?