Exploring the Curvature of Space: Intrinsic vs Extrinsic and the Role of Mass"

[Kevin McHugh]

In the absence of mass, G=0. Does this imply R=0, and that space is flat in between massive objects? If space is always curved is that intrinsic or extrinsic curvature? Thanks in advance for your insight.

 

[Grinkle]

I think (and I may well be wrong) -

G is not zero in between massive objects
In a universe totally without particles it might be difficult to define space in a way that has any meaning for general relativity
In a universe totally without particles space would be flat, if there are no fundamental issues in defining space for such a universe

 

[Kevin McHugh]

According to MTW G=0 in the absence of mass.

 

[newjerseyrunner]

If I remember correctly, without mass, space is curved outwards and will rapidly expand.

 

[PAllen]

In vacuum regions in GR with no cosmological constant, the (two index) Ricci curvature is zero, as is the Ricci curvature scalar derived from contracting it. However, the full, 4 index curvature tensor is NOT zero in vacuum regions. A zero Einstein tensor in no way causes the full curvature tensor to vanish. Furthermore, the Kretschmann curvature scalar formed by the full contraction of the 4 index curvature tensor with its cotensor, is not zero in vacuum regions. The type of curvature that may exist with zero Ricci tensor is called Weyl curvature.

 

[PAllen]

If I remember correctly, without mass, space is curved outwards and will rapidly expand.

Not really. The zero mass limit of FLRW solutions is flat Minkowski space (full curvature tensor is everywhere vanishing) but with funny coordinates. In these coordinates, each constant time spatial slice has negative (hyperbolic) curvature, and the distance measured in these slices between Hubble flow world lines expands rapidly. However, each Hubble flow world line is just a timelike geodesic (inertial world line), and a coordinate transform can take you to ordinary Minkowski coordinates. This example shows the genuine difficulty in separating which features of standard cosmology description are invariants versus coordinate dependent. In particular, it shows that maximal recession rate is achieved in flat spacetime, thus is not even remotely a difference between SR and GR.

 

[Kevin McHugh]

In vacuum regions in GR with no cosmological constant, the (two index) Ricci curvature is zero, as is the Ricci curvature scalar derived from contracting it. However, the full, 4 index curvature tensor is NOT zero in vacuum regions. A zero Einstein tensor in no way causes the full curvature tensor to vanish. Furthermore, the Kretschmann curvature scalar formed by the full contraction of the 4 index curvature tensor with its cotensor, is not zero in vacuum regions. The type of curvature that may exist with zero Ricci tensor is called Weyl curvature.

So if I read you correctly, the Reimann is never zero, and spacetime always has curvature. True?

 

[PAllen]

So if I read you correctly, the Reimann is never zero, and spacetime always has curvature. True?

For any GR solution except the flat spacetime degenerate solution (i.e. no gravity or matter).

 

[pervect]

In the absence of mass, G=0. Does this imply R=0, and that space is flat in between massive objects? If space is always curved is that intrinsic or extrinsic curvature? Thanks in advance for your insight.

If the Einstein tensor $G_{ab}$ is zero, as other posters have noted the Ricci tensor $R_{ab}$ is zero, but this doesn't imply that "space is flat", because the fulll Riemann tensor $R_{abcd}$ is not necessarily zero.

When we talk about curvature in GR, we're pretty much always talking about intrinsic curvature, just because we are imagining that we live within the universe, not outside it. To have extrinsic curvature, we'd need a concept of what lay outside the universe. This isn't a part of most theories, mainly due to experimental difficulties in stepping outside the universe to "take a look".

The curvature of space-time can be reduced to the intrinsic (aka Gaussian) curvature of all possible space-time slices. See for instance https://en.wikipedia.org/wiki/Gaussian_curvature. So we can regard the Riemann tensor as giving us the Gaussian curvature of any two dimensional subspace. And if we know the Gaussian curvature of all two dimensional subspaces, we can find the Riemann tensor.

To make a precise statement about "space being curved" one needs a definition of space. A good candidate for this definition would be the Bel decomposition of the Riemann tensor, https://en.wikipedia.org/wiki/Bel_decomposition. One could regard the "topgravitic" part of this tensor as representing the idea of "space curvature".

GIven this precise definition of what "spatial curvature" is, one can additionally say that in a vacuum region of space-time, the topogravitic part of the Riemann is equal to the electrogravitic part. So the presence of any tidal forces (which are represented by the electrogravitic part of the Riemann tensor) in a vacuum space-time implies that "space is curved", in the sense that the topogravitic part of the tensor must be non-zero as well.

The tidal for components would be, for instance $R_{txtx}, R_{tyty}, R_{tztz}$. If we orient or coordinate axis correctly, we can make all the non-diagonal terms of the electrogravitic tensor zero, leaving only these three diagonal terms. The corresponding numerically equal topogravitic components of the tensor would be given by the Hodges dual, where (tx)->(yz), (ty)->(xy), (tz)->(yz), so that the numerically equal components would be $R_{txtx}->R_{yzyz}$, and similarly for the other two.

Thus if (tx) plane of space-time has a non-zero Gaussian curvature, we can say that $R_{txtx}$ is nonzerom which implies that $R_{yzyz}$ is nonzero (assuming a vacuum space-time), and thus we can conclude that the purely spatial (yz) plane has a non-zero Gaussian curvature.

 

[Dream Relics]

I love the fact that I am finding the same ideas that I have had being put forth on here by other people. This is an interesting topic. Trying to understand the curvature of space seems like a tricky thing. The classic illustration that people always put forward of the stretched sheet with a bowling ball in the middle, as a sort of analogy, seems to me to be somewhat lacking. Or at least, as a two dimensional model for a 3 dimensional reality, the picture it paints only goes so far.

is it not true that we only talk about the curvature of space because we deduce this idea from some observable effects, like gravity on light, and so forth? But I wonder if curvature is really the best word to use. They describe the bending of light as lensing. I can easily picture space as being made of glass and being denser closer to massive objects. Though, the idea that space can curve (whatever it is we really mean by curving) begs the question that space then must also be something and not just the absence of somethings. And so I wonder with regard to that does anyone have any working theories as to what space is? or is is still like gravity itself, where we can only describe its effects but haven't much of a solid idea of what it actually is?

Another notion that I wonder about is the effect that gravity has on time. It is stated that the closer you are to mass the slower time goes relative to farther away from that mass. Hence near a black hole time goes really slow, or on the surface of the Earth time goes a tinsy bit slower than up where the satellites are. So this makes me wonder, at places in space that have zero amounts of mass in them, or that are at least very far away from lots of mass, like out between the galaxies for instance, might we suppose that time there is much faster relative to close to a galaxy? And in fact, might we also suppose the degrees of differences in the speed of time from places with lots of mass and no mass could be considered in the same sense that we think of light waves as having a spectrum, or range, in which there is a lower and upper limit, or frequency? does it make sense to suppose that if all the mass and energy was condensed into one spot ( the singularity ) that the gravity in that situation would be so strong that time itself would cease? Do we suppose that massive enough black holes can stop time completely?

If we envision the curvature of space around a massive object like the sun as ripples emanating, like a pebble thrown into a pond. Then imagine the same also for the Earth or saturn or whatever. What happens when the curvature bending in one direction meets the curvature bending in another direction? Do they interact?

If mass causes space to curve and the curvature of space is what causes gravity, and if mass is all made from vibrating strings, then might the effect of the curve of space or gravity or both be some kind of resonance of all that vibration together effecting the material of whatever space is? Like the bending of space is the intensity and wavelength of that vibration emanating outward in all direction, like sound waves do when you ring a bell?

 

[A.T.]

curvature of space is what causes gravity

Spacetime, not space.

 

[Drakkith]

Though, the idea that space can curve (whatever it is we really mean by curving) begs the question that space then must also be something and not just the absence of somethings. And so I wonder with regard to that does anyone have any working theories as to what space is? or is is still like gravity itself, where we can only describe its effects but haven't much of a solid idea of what it actually is?

I think the key here is to understand that GR is a theory of geometry. Geometry is, in a short, "a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space." There's no reason, that I know of, to assume that just because the geometry of space isn't "flat" that space has to be something other than framework.

So this makes me wonder, at places in space that have zero amounts of mass in them, or that are at least very far away from lots of mass, like out between the galaxies for instance, might we suppose that time there is much faster relative to close to a galaxy?

Indeed. I believe I did the math once, and the difference between intergalactic space and the surface of the Earth was negligible even after 13 billion years. Something on the order of a few seconds to a few days difference at best if I remember correctly.

And in fact, might we also suppose the degrees of differences in the speed of time from places with lots of mass and no mass could be considered in the same sense that we think of light waves as having a spectrum, or range, in which there is a lower and upper limit, or frequency? does it make sense to suppose that if all the mass and energy was condensed into one spot ( the singularity ) that the gravity in that situation would be so strong that time itself would cease? Do we suppose that massive enough black holes can stop time completely?

The difference in the rate that time passes is observer dependent and ranges from one second per second on one end, to nearly zero on the other end. I say "nearly" zero because there is no known way for time to cease passing completely. The rate can be any value close to zero, but it cannot be zero without causing some... problems.

Then imagine the same also for the Earth or saturn or whatever. What happens when the curvature bending in one direction meets the curvature bending in another direction? Do they interact?

Curvature doesn't work like that. Gravitational waves might, but not a "static" situation. In the case of gravitational waves, I'm not sure what happens.