Spacetime Curvature: Exploring General Relativity

[SinghRP]

General relativity has it that the spacetime continuum is curved. The physics of continuum is dealt with [stress] tensors.

My questions:
(1) The presence of a mass creates the curvature in spacetime. By how?
(2) If the curvature due to matter is positive, is the curvature due to antimatter negative?
(3) If the gravitational mass of matter is positive, is antimatter gravitational mass negative?
(3) Do other fundamental interactions (the strong, the weak, and electromagnetic) create curvatures in their respective fields?

I am working on the source of gravity, which I cannot reveal due to the Forums rules.

 

[Dale]

(1) The presence of a mass creates the curvature in spacetime. By how?

It is not just mass, it is the entire stress-energy tensor. For a brief overview see http://en.wikipedia.org/wiki/Stress-energy_tensor

(2) If the curvature due to matter is positive, is the curvature due to antimatter negative?

No. Antimatter has a positive mass/energy density.

(3) If the gravitational mass of matter is positive, is antimatter gravitational mass negative?

No. See (2).

(3) Do other fundamental interactions (the strong, the weak, and electromagnetic) create curvatures in their respective fields?

Yes. Any energy will curve spacetime see the stress-energy tensor.

I am working on the source of gravity, which I cannot reveal due to the Forums rules.

Thank you for abiding by the rules.

 

[SinghRP]

(1) Not a pertinent answer. How is the stress tensor created?
I thought the curvatures were in the spaceime continuum (x-y-z-t).

(2) and (3): I agree.

(4) I believe these forces are mediated by exchange of respective bosons. The strong - gluons. The weak: W and Z vector bosons. Electromagnetic - photons. I know the Standard Model well, but I never thought of stress tensors in those fundaamental interactions.

 

[Mentz114]

(1) Not a pertinent answer.

It is the best answer you'll get because GR does not tell us the 'by how' answer.

How is the stress tensor created?

Not sure what you mean. Did you look at the Wiki article ?

 

[Dale]

(1) Not a pertinent answer. How is the stress tensor created?

The same way that energy, momentum, pressure, etc are normally created: By the presence of matter or fields.

I thought the curvatures were in the spaceime continuum (x-y-z-t).

Yes.

 

[karlzr]

(1) Not a pertinent answer. How is the stress tensor created?
I thought the curvatures were in the spaceime continuum (x-y-z-t).

(2) and (3): I agree.

(4) I believe these forces are mediated by exchange of respective bosons. The strong - gluons. The weak: W and Z vector bosons. Electromagnetic - photons. I know the Standard Model well, but I never thought of stress tensors in those fundaamental interactions.

I don't know what you mean by how the stress tensor is created. In my opinion, the stress tensor include all the information we need to create spacetime curvature, such as the energy density, momentum density and pressure.
For SM lagrangian, the stress energy-momentum can be defined as the variation of the action with respect to the metric. Anyway, it's just a semi-classical theory, because we do not quantize the metric field.

 

[SinghRP]

Sorry, I did not say it right: how is the stress tensor created. I will be responding to you all who kindly answered my questions. I learned from you. Thank you.

Here is the situation. There is a piece of matter with mass in a spacetime continuum. GR has it that the spacetime continuum is a field of space-time geometry and that the mass creates warp/curvature in that field of spacetime geometry. (The spacetime geometry is a complex geometry with x, y, z, ict. So, whatever GR is doing, it is the projection of that on the said complex space.)

A force or a torque creates stress in a deformable continuous medium. The mathematical properties of that stress are represented by stress tensor. (Stress tensor does not create deformations in the medium.)

A force imparts a mass momentum and energy. (Pressure is a form of many momenta, temperature is due to translational motions). Forces are mediated/created. For examples, the strong force is mediated by gluons, the weak by W and Z bosons, and electromagnetic by photons. It is conjectured that gravity is mediated by gravitons. Thus, according to GR and quantum mechanics, gravitons are the quanta of the field of spacetime geometry.

So, I come back to “How a mass creates warp/curvature in the spacetime continuum.” (By the way, the great astrophysicist E. A. Milne asked this question a quarter century ago. We owe him an answer.)

Sooner or later, GR must accommodate Mach’s principle: “The inertial mass of a particle is determined by the gravitational effect of all the other matter in the universe.”

We are physicist first, mathematician second. We must at least try to determine how Nature in a given situation works.

I will bring in the sign of matter or antimatter gravitational mass later, maybe under another topic. This has implications for GR.

 

[Mentz114]

So, I come back to “How a mass creates warp/curvature in the spacetime continuum.”

I repeat, GR does not tell us how this happens. What we have is a set of equations that relate curvature to the SET of the source, G_ab=kT_ab. Or in the case of vacuum solutions, the RHS is zero !

Sooner or later, GR must accommodate Mach’s principle: “The inertial mass of a particle is determined by the gravitational effect of all the other matter in the universe.”

I think you'll find this view is much disputed. I myself regard it as complete hogwash. If we assume the equivalence principal it becomes a tautolgy without meaning.

[ In the words (nearly) of the writer A.A Milne - "A physicists life is terrible hard, says Alice" ]

 

[Naty1]

I'm not at all sure a number of statements in post #7 are correct...nor what their point may be...
but I do like: " (By the way, the great astrophysicist E. A. Milne asked this question a quarter century ago. We owe him an answer.)"

Many here will disagree regarding "why" questions in physics...


In post #8, I'd not go so far as does Mentz, but I agree that the equivalence principle seems to supersede Mach's ideas because it leads more directly to concrete insights and predictions.

Singh:

Sooner or later, GR must accommodate Mach’s principle: “The inertial mass of a particle is determined by the gravitational effect of all the other matter in the universe.”

I myself regard it as complete hogwash.

Instead, I'd say that general relativity DOES incorporate those portions of Mach's principle which are quantifiable...or one might say some of his ideas are expressed withinin GR and others are not believed correct...

See here: http://en.wikipedia.org/wiki/Mach's_principle and also note the Lense Thirring effect which IS an example of Mach's ideas within GR. (See the excerpts of Einstein's letter to Mach on this subject...)

 

[Mentz114]

If express Mach's principle as

“The inertial mass of a particle is determined by the gravitational effect of all the other matter in the universe.”

and combine with

"inertial and gravitational mass are the same thing", we get

“The gravitational mass of a particle is determined by the gravitational effect of all the other matter in the universe.”

The problem is the Mach's principle is not clearly stated. In a cosmological solution, the overall curvature is the result of all the matter/energy in the universe, but if this is to affect an individual body's inertia, there would have to be instantaneous effects.

 

[Passionflower]

So, I come back to “How a mass creates warp/curvature in the spacetime continuum.” (By the way, the great astrophysicist E. A. Milne asked this question a quarter century ago. We owe him an answer.)

I think this is more a philosophical than a science question but just to spend a few words on it.

The question implies that mass (among other things) is a cause of the curvature of spacetime. We do not know if this is true, one of three things could be the case:

- Mass is the cause of spacetime curvature
- Spacetime curvature is the cause of mass
- They are exactly the same thing
​

Unfortunately, the answers to philosophical questions tend to generate yet even more questions.

 

[Naty1]

The question implies that mass (among other things) is a cause of the curvature of spacetime. We do not know if this is true

I was thinking the same thing when posting my last comments...I do not think there is any experimental evidence to confirm actual curvature of spacetime...??

I think its because other aspects of relativity like, orbtial precession, the fixed speed of light, bending of light by gravity and time dilation as a result of speed and gravitational potential, so far seem to substantiate relativty...so we take curvature to be correct as well...it would indeed be odd to have all that apparently experimentally verified and then find spacetime doesn't curve...Is that possible?

Wiki says this:

All results are in agreement with general relativity.[55] However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.

http://en.wikipedia.org/wiki/Curved_space_time#Gravitational_time_dilation_and_frequency_shift

 

[Mentz114]

I do not think there is any experimental evidence to confirm actual curvature of spacetime

There never can be. How do you measure 'spacetime curvature' ?

(Radioshack is completely out of curvometers. ).

 

[SinghRP]

Thank you , parrticularly Metzi114, for confirming my line of thinking: "GR does not describe HOW matter creates warp/curvature in the field of space-time geometry."

Regarding gravitational mass of antimatter, GR and Mach's principle, Principle of equivalence (two meanings), etc., I think my statements are humbly compatible with my colleagues'.

I have a request to one of you who can implement my wish. I wish the Forums create a separate section, such as New Approaches or similar, where physicists may put their current line of work without going through peer review. The Forums need to put in a strong disclaimer though. Two criteria for acceptance would be: (1) internal consistency and (2) not contradictory to accepted physical observations. You never know there could be a gem in one of them. (Einstein's papers on the two relativities did not go through peer reviews -- thanks God!) I recall Niels Bohr: "The opposite of a true statement is a false statement. The opposite of profound truth may well be another profound truth."

To Metzi114: I love A.A.Milne's words. Is s/he related to the great E.A. Milne?

Kind regards.

 

[Mentz114]

There is a section of the Forums called 'Independent Research' here

https://www.physicsforums.com/forumdisplay.php?f=146[My A.A Milne 'quote' was frivolous. AA Milne and EA Milne are not related, to my knowledge]

 

[Dale]

The spacetime geometry is a complex geometry with x, y, z, ict.

That is one way of doing special relativity that was in fashion several decades ago, but I believe that it is not compatible with GR and so has fallen out of use.

Sooner or later, GR must accommodate Mach’s principle

I agree with Mentz's comments on this above. I have a great distrust of Mach's principle since it is so vaguely stated as to be experimentally meaningless, and it usually leads to absurd discussions about "otherwise empty" universes that have nothing to do with physics in this universe which is the only one that we can test.

We are physicist first, mathematician second. We must at least try to determine how Nature in a given situation works.

It seems to me like you are looking for a "bedtime story" about how gravity works. We have a theory whose experimental predictions agree with all of the observations made to date. That is all that science can do, test a theory's agreement with experiment. In science, all of the rest (beyond the mathematical framework and the experimental data) is just a story that we tell ourselves to make it easier to remember which variables to plug into the equations.

 

[Dale]

I do not think there is any experimental evidence to confirm actual curvature of spacetime...??

There never can be. How do you measure 'spacetime curvature' ?

I disagree with these comments. Some of the effects of GR, e.g. time dilation, can occur in an accelerated reference frame in flat spacetime (equivalence principle), but there are many effects that confirm curvature.

1) relative acceleration of inertial observers
2) deflection of light by the sun
3) precession of Mercury
4) redshifting of the CMB
etc.

 

[Mentz114]

I disagree with these comments. Some of the effects of GR, e.g. time dilation, can occur in an accelerated reference frame in flat spacetime (equivalence principle), but there are many effects that confirm curvature.

1) relative acceleration of inertial observers
2) deflection of light by the sun
3) precession of Mercury
4) redshifting of the CMB
etc.

Obviously, I disgree. You can't measure spacetime curvature any more than you can measure the vector potential of EM or the wave equation in QM.

What we measure is distance and time, and compare those with predictions. We are measuring gravitational effects, not spacetime curvature, which is only a model ( that works rather well).

There are theories that make the same predictions as GR that do not use curvature.

That is all that science can do, test a theory's agreement with experiment. In science, all of the rest (beyond the mathematical framework and the experimental data) is just a story that we tell ourselves to make it easier to remember which variables to plug into the equations.

 

[Dale]

You can't measure spacetime curvature any more than you can measure the vector potential of EM or the wave equation in QM.

What we measure is distance and time, and compare those with predictions. We are measuring gravitational effects, not spacetime curvature.

Curvature is simply a deviation of distances and times from what you would expect if the spacetime were flat. If you can measure distance and time then you can measure curvature.

But kudos on very good use of my own words against me

 

[pervect]

We can compute the curvature given that we know the behavior of clocks and rulers. With SI standard clocks and SI standard rulers, one can't avoid concluding that space-time is curved from their observed behavior.

It may be possible to hypothesize a fundamentally flat underlying space-time in which the standard clocks and rulers are deformed / influenced by gravity. Some theorists have done this as an approach to quantum gravity, i.e. http://arxiv.org/abs/gr-qc/9512024. But this requires the introduction of different rulers and clocks than the ones we actually measure things with.

I'm not aware of any popular treatments that take this non-standard approach, though it has some promise as dealing with curvature seems to confuse a lot of people.

It's also unclear if/how this approach would deal with black holes or other situations where global topological issues arise, though it appears to give results equivalent to GR on a small scale.

 

[Mentz114]

Curvature is simply a deviation of distances and times from what you would expect if the spacetime were flat. If you can measure distance and time then you can measure curvature.

I must disagree, because you're not measuring curvature. You're fitting observations to a theory that uses curvature to model the effects of gravity. Sure, we can measure the curvature of a physical thing like a football or a piece of string. But what are you measuring the curvature of when it's 'spacetime' ?

Anyhow, I see it as an interpretational thing. Operationally it is not important. I don't think GR is the final theory of gravity and I don't think 'spacetime curvature' is a physical thing ( now we're getting into definitions so this is going nowhere).


Pervect, thanks for the reference, it looks well worth a read.

 

[Daverz]

I must disagree, because you're not measuring curvature.

That's how curvature is measured. You might want to call it "fabdoodle", but curvature has a well understood meaning.

 

[PAllen]

I must disagree, because you're not measuring curvature. You're fitting observations to a theory that uses curvature to model the effects of gravity. Sure, we can measure the curvature of a physical thing like a football or a piece of string. But what are you measuring the curvature of when it's 'spacetime' ?

Anyhow, I see it as an interpretational thing. Operationally it is not important. I don't think GR is the final theory of gravity and I don't think 'spacetime curvature' is a physical thing ( now we're getting into definitions so this is going nowhere).

Suppose we have no theory of gravity all, in the sense of relationship between curvature and stress/energy. In fact suppose a universe where there is no gravity. Suppose, instead, we have a fixed background spacetime, as in SR, that happens to have have curvatures. We detect this, for example, by finding that parallel laser beams intersect at great distances in certain direction. Would we not conclude, if we have a mathematical theory of intrinsic curvature, that our universe is curved? So, if we do similar experiments in the world we actually live in, and find curvature by similar criteria, why should we avoid concluding that we observe curvature? Yes, you can find ways to interpret not having curvature (in both cases; if one, but not the other, you are simply being inconsistent), but that becomes exactly equivalent the school of mathematicians (there is such a school) that denies the existence of intrinsic curvature at all, and treats all curvature as a manifestation of embedding. So I would say you can deny the existence of curvature in our universe only in the same manner as you can deny the existence of intrinsic curvature altogether.

 

[Dale]

I must disagree, because you're not measuring curvature. You're fitting observations to a theory that uses curvature to model the effects of gravity.

I don't understand the distinction you are trying to make here. If you have a theory that describes a quantity and tells you how to experimentally measure it then when you perform the experiment it seems clear (to me) that the observations are a measurement of the quantity.

It seems to me that you could say the same thing about any physical quantity. Refering e.g. to Newton's second law you could say: "You're not measuring force. You're fitting observations to a theory that uses force to model the effects of inertia."

 

[Mentz114]

Suppose we have no theory of gravity all, in the sense of relationship between curvature and stress/energy. In fact suppose a universe where there is no gravity. Suppose, instead, we have a fixed background spacetime, as in SR, that happens to have have curvatures. We detect this, for example, by finding that laser beams intersect at great distances in certain direction. Would we not conclude, if we have a mathematical theory of intrinsic curvature, that our universe is curved? So, if we do similar experiments in the world we actually live in, and find curvature by similar criteria, why should we avoid concluding that we observe curvature? Yes, you can find ways to interpret not having curvature (in both cases; if one, but not the other, you are simply being inconsistent), but that becomes exactly equivalent the school of mathematicians (there is such a school) that denies the existence of intrinsic curvature at all, and treats all curvature as a manifestation of embedding. So I would say you can deny the existence of curvature in our universe only in the same manner as you can deny the existence of intrinsic curvature altogether.

This isn't science, it is sophistry. You haven't added anything to the argument except silky phrases. Your position requires that there is some substance that has intrinsic curvature, and you're welcome to hold that view.

In fact suppose a universe where there is no gravity.

No, I won't, it seems pointless.

Suppose, instead, we have a fixed background spacetime, as in SR, that happens to have have curvatures.

This does not make sense. Is this spacetime curved or not ?

Yes, you can find ways to interpret not having curvature (in both cases; if one, but not the other, you are simply being inconsistent),

How do you know what I might or might not do.

but that becomes exactly equivalent the school of mathematicians (there is such a school) that denies the existence of intrinsic curvature at all, and treats all curvature as a manifestation of embedding.

I don't deny the existence of intrinsic curvature ( see several recent posts of mine where I say intrinsic curvature is the key to understanding the maths of GR).

So I would say you can deny the existence of curvature in our universe only in the same manner as you can deny the existence of intrinsic curvature altogether.

I agree, you could say that. But it would be untrue. It doesn't follow from your earlier suppositions.

If I could produce a theory of gravity where

1. the action ( a Lagrangian density) did not contain the Riemann tensor
2. it could be shown that this action was equivalent to the Einstein-Hilbert action (up to a divergence)
3. in that it makes identical predictions to GR

what reason would I have for believing that curvature has anything to do with gravity ?

I don't know why you're so exercised by this. You should employ your considerable talents in a more productive way, than attacking my 'shut-up and calculate' philosophy that doesn't make unsustainable and unnecessary assumptions.

 

[Mentz114]

I don't understand the distinction you are trying to make here. If you have a theory that describes a quantity and tells you how to experimentally measure it then when you perform the experiment it seems clear (to me) that the observations are a measurement of the quantity.

Only because you've adopted a particular model ! That is an act of your volition - NOT a law of physics.

It seems to me that you could say the same thing about any physical quantity. Refering e.g. to Newton's second law you could say: "You're not measuring force. You're fitting observations to a theory that uses force to model the effects of inertia."

It's ironic that you say that when GR is all about force-free motion ! But isn't that exactly what you're doing ? Force cannot be measured directly but has to be calculated from times and distances ? (If force is such a 'physical' concept, why are there so many discussions about 'real' and 'fictitious' forces ? ). Isn't there a theory that is equivalent to Newtonian mechanics in the large limit - quantum mechanics, which does not have forces ? ( I could be wrong about this - all I can remember is unitary evolution of wave functions and interaction terms - which must be the force carrying bosons ).

I would like to add, for all my esteemed correspondents in this matter, that we are wasting time, because this is a philosophical discussion. You simply cannot produce a convincing physical argument to support your belief because there isn't one.

When I calculate curvatures, or SETs in curved spacetime, I get the same answers as someone who believes differently about curvature so it doesn't matter.

 

[Dale]

Only because you've adopted a particular model ! ... Force cannot be measured directly but has to be calculated from times and distances ?

I think you are confusing force with acceleration, but in any case the point is that the same can be said of any measurement of any physical quantity. Some models are more universally adopted than others, but to measure any physical quantity you have to adopt some model. I just don't understand what you think distinguishes curvature in this respect.

Let's try this, you think that measurements of curvature and force are just "fitting observations to a theory" and I assume that you think that other quantities are somehow "more"* than this. So what measured quantities would you place in the "fitting observations to a theory", what quantities would you place in the "more" category, and what do you think distinguishes measurements of the two?

*or choose some other descriptive label

 

[Dale]

I just reviewed the conversation and I believe from your earlier comments that you would put distance and time in the "more" category. Where would you put angles? If angles are also in the "more" category then it is easy to show that curvature can be measured also. If not then what distinguishes angles from distances in your idea?

 

[yuiop]

I must disagree, because you're not measuring curvature. You're fitting observations to a theory that uses curvature to model the effects of gravity. Sure, we can measure the curvature of a physical thing like a football or a piece of string. But what are you measuring the curvature of when it's 'spacetime' ?

Your comments led me to ponder the difference between spatial curvature and spacetime curvature and I came up with this series of scenarios for discussion.

First a list of the terms I use in the discussion:

(1) Optical flatness.

A surface is optically flat if when any two spatially separated points on the surface are connected by a laser beam, then a point mid way along the laser beam will be on the surface. If the surface midpoint is below the light beam, then the surface will be said to have positive optical curvature.

(2) Gravitational flatness.

A surface is gravitationally flat if objects that are free to move stay where they are. If objects all tend to roll towards a common point then the surface will be said to be have positive gravitational curvature and if all objects tend to roll away from each other then the surface will be said to have negative gravitational curvature. Saddle points on a surface will have a mixture of positive and negative gravitational curvature that is directionally dependent but I don't think this is a major worry, because such a spatial saddle point will presumably fail all other flatness tests. I think almost by definition, spatially separated clocks on a gravitationally flat surface will run at the same rate.

(3)Euclidean flatness.

A surface is Euclidean flat if the internal angles of any traiangle on the surface add up to 180 degrees and if the ratio of the circumference to the radius of any circle on the surface is 2Pi, measured from a point at the centre of the circle but also on the surface. If the circle ratio is less than 2Pi or the triangle angles add up to more than 180 degrees, then the surface will be said to have positive Euclidean curvature.

(4) Spatial flatness.

For this definition I define a hypothetical "rigid" plane that is constructed out of Unobtainium, far away from any gravitational sources. At the time of construction the rigid plane is designed and tested to have optical and Euclidean flatness and is sufficiently rigid to resist being physically bent by gravitational forces. This rigid plane is meant to be a "spatial flatness ruler" that can be transported to where we want to measure flatness.

Now the question is which of the above definitions most closely describes spacetime curvature?
Here are some scenarios for possible discussion.

Consider a neutron star that conveniently has a radius equal to its photon orbit radius. The surface of this star would be optically and gravitationally flat and "look" flat to an observer standing on the surface of the star. It would not be spatially flat when compared to the rigid plane and it will have considerable positive Euclidean curvature.

Now imagine the rigid plane is placed close to this star so that the centre of the plane is closest to the star and the plane is orthogonal to a radial line from the centre of the star. To an observer on this "rigid spatially flat plane" the plane is not optically flat and will appear to an observer near the centre that he is standing at the lowest point of a bowl depression. The rigid spatial flat plane is also not gravitationally flat in this scenario and objects placed on it will tend to roll towards the centre in agreement with the observers optical perception that he is in a bowl. The rigid plane also fails the Euclidean flatness test. The rigid flat plane in this situation has positive optical curvature, positive gravitational curvature and positive Euclidean curvature.

Now consider an object with obvious spatial curvature but insignificant gravitation such as a tennis ball. The surface will not have Euclidean flatness because 3 points on the surface can have internal angles greater than 180 degrees. However, if we poke holes in the ball we can see that the 3 points can be connected by laser beams and although the surface has spatial curvature the 3 points are still embedded in flat space. We can still find a point (inside the ball) that is the centre of a circle passing through the 3 points and the radius to circumference ratio is Euclidean. So, although the ball surface is not flat, points on the surface of the ball can still be considered to be embedded in flat space. Now if we transport our ball to very close to an extreme gravitational source, the points on the ball will fail the Euclidean flatness test and so we might conclude there is something different about about the space the ball is embedded in near an extreme gravitational source. Can we call this difference spacetime curvature?

Now consider widely spaced clocks on the surface of the Earth. They are all at sea level and running at the same rate, so they are on a gravitationally flat surface. They are obviously not on an optically flat surface, because sufficiently far points are over the horizon and we cannot connect 3 clocks in a line using a laser. So the Earth's surface is gravitationally flat but not spatially flat, optically flat or Euclidean flat. Is there any evidence of spacetime curvature here? Let us slice a piece off the Earth that is spatially flat (as measured by our rigid plane) and place 3 clocks on this cut surface. Will we measure slight positive Euclidean curvature on this "flat surface"? I think we will, as well as slight positive gravitational and optical curvature. While in a destructive mood let us cut off another slice using a powerful laser so that the new cut surface is optically flat. This new cut slice is not spatially, gravitationally or Euclidean flat.

After considering the above scenarios, it would seem one operational definition of spacetime flatness is to define an optically flat surface in the region under consideration and then check for Euclidean flatness on that optically flat surface. If the optically flat surface is not Euclidean flat then it seems there is evidence that there is intrinsic spacetime curvature. Euclidean curvature of a surface does not necessarily prove that the spacetime the surface is embedded in is intrinsically curved and optical flatness of a surface by itself does not demonstrate that the surface is embedded in flat spacetime. It seems we need to perform both tests.

So which if any of the above scenarios best illuminates spacetime curvature rather than just space curvature?

Perhaps the answer is "none of the above". I think Dalespam gave a good example of intrinsic spacetime curvature with the example of the changing spatial separation of two free falling observers due to tidal effects. While clocks in an accelerating rocket show Doppler shift with height as in a gravitational field, sequentially dropped objects in the rocket will maintain constant mutual spatial separation, demonstrating that the presence or absence of intrinsic curvature of the background can be detected even in an accelerating rocket. However, is defining tidal effects as intrinsic spacetime curvature just semantics? Newton would have been very aware of tidal separation of falling particles and it never for a moment led him to consider that this was evidence of time and space changing in some mysterious way.

P.S. I am not taking any strong position on anything I say above and I am just kicking some ideas around so that maybe someone can lead me to a clearer understanding in intuitive terms. Some of my definitions of what is positive or negative curvature may contradict already established formal conventions, so corrections are welcome

 

[PAllen]

(Ignoring ad hominem comments)
I am curious about your position. Posing questions helps clarify.

This does not make sense. Is this spacetime curved or not ?

I said it was curved. Say, a Swarzchild geometry not associated with any matter source. Just like SR has a very specific geometry, fixed geometry. Just like a plane is a specific geometry, not absence of geometry.

I don't deny the existence of intrinsic curvature ( see several recent posts of mine where I say intrinsic curvature is the key to understanding the maths of GR).

You say you can measure curvature on a football, and you think intrinsic curvature exists. What I am getting at, is "Is there a method by which you think it is meaningful to talk about curvature of the spacetime we occupy, in principle (like flat being in a 2-sphere)?" If the answer to this is no, then that is the nub of the philosophic difference. It would seem you admit actual curvature of physical objects in our universe, but do not accept, in principle, the concept of curvature of our universe. Then, there is no getting around this fundamental difference. If, instead, you accept that it could be meaningful to say our universe has curvature, but we don't know that about our actual universe, then I would be curious to know what procedure could detect the former but would not detect curvature in our actual universe.

 

[Passionflower]

While clocks in an accelerating rocket show Doppler shift with height as in a gravitational field, sequentially dropped objects in the rocket will maintain constant mutual spatial separation, demonstrating that the presence or absence of intrinsic curvature of the background can be detected even in an accelerating rocket.

I am not sure I understand this.

In a linearly accelerating rocket if we drop one object at v(t) and another one at v(t+1) then the dropped objects will have a relative velocity wrt each other right? In other words their separation will grow over time.

 

[yuiop]

I am not sure I understand this.

In a linearly accelerating rocket if we drop one object at v(t) and another one at v(t+1) then the dropped objects will have a relative velocity wrt each other right? In other words their separation will grow over time.

Oops, your right. One will have constant velocity relative to the other. I wonder if the distinction is that in a real gravitational field with intrinsic curvature, one will be accelerating relative to the other?

 

[pervect]

Gravitatonal time dilation is a REALLY BIG clue that space-time is curved. If you think of gravity as just being a force, there's no reason it should affect how clocks operate. But we observe that it does. Recent experiments can detect the gravitational time dilation for a height difference as small as a foot.

When we draw the resulting space-time diagram, we have diagrams that look like they should be parallelograms, but the sides are of unequal length. (The "sides" of unequal length in this diagram are the clocks which run at different length because it's a space-time diagram).

This isn't possible in an Eulidean geometry, and suggests (as most textbooks will mention) that space-time is curved.

To actually calculate a curvature invariant and show that it is nonzero is a bit more involved, but it's certainly possible in principle. It involves measuring the metric accurately enough to get the second partial derivatives. This will rule out some special cases such as being in an elevator. But basically, the reason we know that space-time is curved is that clocks appear to run at different rates depending on their gravitational potential.

 

[yuiop]

Gravitatonal time dilation is a REALLY BIG clue that space-time is curved. If you think of gravity as just being a force, there's no reason it should affect how clocks operate. But we observe that it does. Recent experiments can detect the gravitational time dilation for a height difference as small as a foot.

Isn't there a problem with this, in that clocks lower down in an accelerating rocket will tick slower and red shift relative to clocks higher up in the rocket even in the rocket is accelerating in flat space? In other words, differential clock rates due differences in "height" does not seem in itself to be proof of intrinsic spacetime curvature.

 

[Passionflower]

Isn't there a problem with this, in that clocks lower down in an accelerating rocket will tick slower and red shift relative to clocks higher up in the rocket even in the rocket is accelerating in flat space? In other words, differential clock rates due differences in "height" does not seem in itself to be proof of intrinsic spacetime curvature.

We could claim that the clocks run at a different rate because of the difference in curvature (that is path curvature, not spacetime curvature) of the worldlines, just like in the case of an accelerating rocket.

However in curved spacetime there is of course an additional factor.

Consider the moment when a free falling observer and a stationary observer are both at location r=4 in a Schwarzschild solution where m=1/2, then:

The stationary observer measures a distance to the EH of

$$\sqrt {r \left( r-1 \right) }+\ln \left( \sqrt {r}+\sqrt {r-1} \right) = 4.781059513$$

The free falling observer measures a distance of 3 while the Fermi normal distance is: 3.488654305.

The local velocity between the stationary and free falling observer is:

$$\sqrt {{r}^{-1}} = 0.5$$

So if we (of course falsely) assume for a moment that spacetime is flat we would expect the distance to the EH for the free falling observer from the perspective of the stationary observer to be simply the Lorentz contracted distance. However this is not the case, as shown as follows:

The Lorentz factor becomes:

$${\frac {1}{\sqrt {1-{v}^{2}}}} = 1.154700538$$

And thus the (assuming flat spacetime) distance would be:

4.781059513 / 1.154700538 = 4.140518997

Which is different from 3 (or a Fermi normal distance of: 3.488654305) as measured by the free falling observer.

And from the perspective of the free falling observer when applying the reverse Lorentz factor he calculates a distance of 3.464101617 (or 4.028351006 from the perspective of a Fermi normal distance) as opposed to 4.781059513.

So in a Schwarzschild spacetime we see a kind of 'bifurcation' of spatial distances. This by the way also demonstrates that Lorentz covariance applies only locally in GR.

Here is a demonstration of this, the graphs are a little complicated, notice that the Y axis is Distance/Schwarzschild Coordinate -1:

[PLAIN]http://img258.imageshack.us/img258/7867/001bifurcation2.gif [PLAIN]http://img245.imageshack.us/img245/4461/001bifurcation1.gif