What's being curved, when mass bends the spacetime continuum?

[Muphrid]

I disagree. To me, GR suggests there is no gravitational field--there is simply the geometry of spacetime, which is warped and curved and which, in turn, affects the motions of test particles. Gravity as a field theory on a flat background doesn't require the conceptual middle man that is the spacetime geometry--there is a gravitational field, and it affects trajectories directly.

 

[Kraflyn]

Hi.

Yes, one may say that metric curvature can be equally well expressed as a field and vice versa. However, how equally well exactly? The difference between field description and curvature description is much like the difference between heliocentric and geocentric picture. In heliocentric picture, orbits are simple ellipses or second order curves, and in geocentric picture orbits are same those orbits plus epicycles. So heliocentric picture is more simple. In GR, gravity is not a force at all, there is no external field. Gravity of GR is a pseudo-force: it depends on Your choice of coordinates. Much like centrifugal force. Some systems have it - some may not. The field complication, much like geocentric epicycles, brings complications. Unlike innocent epicycles, field corrections tend to produce ghosts... Mythical particles that can do ... anything at all really. So field vs. curvature does not necessarily end with a draw. Fundamentally, every theory is wrong, so I'm not pro nor am I contra. I'm just having a conversation :D

Cheers.

 

[Kraflyn]

Hi.

What is exactly this mythical creature called "flat background"? Where did it come from? And why would it be more popular than, say, curved metric?

Yes, there is always a background, it's called vier-bein or tetrad, meaning four-frame. How come we prefer flat? Some of us like curvy a bit more, maybe?

Cheers.

 

[harrylin]

I disagree. To me, GR suggests there is no gravitational field [...]

Thus, as already became apparent, you disagree with Einstein's GR. That is an obvious disagreement of interpretation; the mathematics is necessarily the same.

Hi.

Yes, one may say that metric curvature can be equally well expressed as a field and vice versa. However, how equally well exactly? The difference between field description and curvature description is much like the difference between heliocentric and geocentric picture. [..] The field complication, much like geocentric epicycles, brings complications. Unlike innocent epicycles, field corrections tend to produce ghosts... Mythical particles that can do ... anything at all really. [..]

Hi, I am not aware of that kind of complication due to the field concept, in particular in view of the use of GR math tools to go with it. Does anyone have a concrete example?

 

[Kraflyn]

Hi.

Yes, examples, sorry.

OK, in order to explain precession of Mercury perihelion, field theory on flat background should invoke a force of the form F=GMm/r^2+A/r^3. There should be higher order corrections introduced. And yet, GR solves it elegantly with just V=-1/r.

Let's pay attention to more complicated system now. For example, near a black hole GR predicts an event horizon from just knowing the Newtonean approximation V=-1/r. Now imagine field theory on flat background try produce event horizon at r=2m. Field theory in zeroth approximation believes the only singularity is at origin. In order for field theory to produce another spherical singularity, field theory should begin to introduce singular corrections for no apparent reason. Singular fields should be introduced ad hoc, and You know what happens when we arrive at singularities: we can't handle them. And all that GR had to do is to say: "ah, yes, Newtonean approximation is V=-1/r and hence Schwarzschild radius is exactly at r=2m".

So this is not hard to grasp.

Now, there is something similar happening in quantum field theory. It is defined on a flat background and first ghost became known to it as "Landau ghost". The phenomenon has a standard name now: Faddeev-Popov ghost field. See, for example, http://en.wikipedia.org/wiki/Ghost_fields. Now I'm not saying ghost fields would be absent if flat background was replaced by a suitable metric. Well, ghosts would disappear if we could find such suitable metric: but we can't yet. We don't know how to do it yet. There are too many unknown details about particle interactions. The point is, rather, that there are huge difference depending on the approach one assumes. Flat background or curved geometry?

I hope this explains it a bit.

Cheers.

 

[A.T.]

I disagree. To me, GR suggests there is no gravitational field--there is simply the geometry of spacetime

But the geometry of spacetime is also described by a field. Newtons gravitation is a vector field. Einsteins gravitation is a tensor field:
http://en.wikipedia.org/wiki/Field_(physics )

 

[harrylin]

Hi.

Yes, examples, sorry.

OK, in order to explain precession of Mercury perihelion, field theory on flat background should invoke a force of the form F=GMm/r^2+A/r^3. There should be higher order corrections introduced. And yet, GR solves it elegantly with just V=-1/r.

Sorry this must be a misunderstanding - it was solved by Einstein using the GR toolbox. Thus I asked how his gravitational field concept supposedly hindered him of doing just that, or how it must have been complicating for him. See my posts #41, #50.

Let's pay attention to more complicated system now. For example, near a black hole GR predicts an event horizon from just knowing the Newtonean approximation V=-1/r. Now imagine field theory on flat background try produce event horizon at r=2m. [...]

I did not fully understand the part that I do not cite here, but the misunderstanding may be in the phrase "flat background". I think that Einstein's GR proposes just the contrary, as I cited.

[..] Now, there is something similar happening in quantum field theory. It is defined on a flat background and first ghost became known to it as "Landau ghost". The phenomenon has a standard name now: Faddeev-Popov ghost field. See, for example, http://en.wikipedia.org/wiki/Ghost_fields. Now I'm not saying ghost fields would be absent if flat background was replaced by a suitable metric. Well, ghosts would disappear if we could find such suitable metric: but we can't yet. We don't know how to do it yet. There are too many unknown details about particle interactions. The point is, rather, that there are huge difference depending on the approach one assumes. Flat background or curved geometry?
I hope this explains it a bit.
Cheers.

It does! Thanks.

 

[Kraflyn]

Hi.

You're welcome. Thank You

Cheers.

 

[harrylin]

Just a little addition: perhaps the confusion of terms can be reduced by stating that Einstein's GR models the gravitational field as a "non-flat" background.

And in this context, "non-flat" does not mean that something is literally "curved", but that, as Einstein put it, the metrical qualities of space-time are partly conditioned by nearby matter.

 

[Kraflyn]

Hi.

Yes, terminology is a bummer with GR. For instance, consider an exact solid example: Schwarzschield metric element

\displaystyle\delta s^2 = \left( 1-\frac{2m}{r}\right) \delta t^2 - \frac{\delta r^2}{1-\frac{2m}{r}} - r^2 \delta \Omega^2

Term -1/r is a scalar potential. It is defined at every point of \mathbb{R}^4. As such, it is a scalar field. So, technically speaking, curvature of space-time, i. e. geometry, is defined through fields. All world events happen on curved background. Yet curved background is curved in reference to flat background. Connection between curved coordinates and flat ones being expressed through - fields. This flat background is actually necessarily of higher dimension than the actual space-time, so this makes it even more funky. The flat background never appears in equations because we never want to refer to it. We introduced connections called Christoffel Symbols - these are not tensor, by the way - and once we introduce those, we never ever have to refer to background flat space-time ever again. But that's the second flat background associated with GR we mentioned so far...

So, yes, one should exhibit extreme tolerance and care when trying to decipher the actual and true meaning of each sentence about GR... True story

Cheers.

 

[Kraflyn]

Hi.

Oh, we have very good definition for that. All of 64 Riemann curvature tensor components vanish simultaneously? Then space is flat at that point. Some of Riemann curvature tensor components are different than zero? Sorry, not flat. Einstein was poetic that day. Too much gulash, I guess. Heh, gulash is cooked with - wine!

Cheers.

 

[Muphrid]

But the geometry of spacetime is also described by a field. Newtons gravitation is a vector field. Einsteins gravitation is a tensor field:
http://en.wikipedia.org/wiki/Field_(physics )

Which tensor field in GR would you call the gravitational field?

 

[Kraflyn]

Hi.

Heh, it depends on nomenclature... For instance, in metric \delta s^2 = \left(1+2V(r)\right)\delta t^2-\frac{\delta r^2}{1+2V(r)}-r^2 \delta \Omega^2 metric tensor components are fields describing gravity... Strictly speaking, there is no exterior field for gravity. This is the very condition for gravity: E_{\mu \nu}=0. Nothing on the right-hand side. On the other hand... What is 1+2V(r) then?... It's some field, right? So... I'm becoming a bit bored of this now.

Cheers.

 

[StephenofCaly]

My simplistic understanding is that to get from Hither to Yon, you have to go a certain way. The rule for light is it has to get there the fastest way, not the straightest path. Since almost always they are the same, it's confusing when they diverge.

Another is like saying - how come the way to Mecca is off to the northeast for local Muslim folks? Mecca's actually at a slightly lower latitude to where I live, but not much. Ought to be SOUTHeast, by my guess. What is the thing that causes the Qibla (the way to Mecca) to curve? That's a fallacious question.