# Causal influence of mass

1. Dec 19, 2006

### SimonA

Do we know for sure that mass affects both space and time ? Could it be that mass affects time OR space but not both ?

I'm not questioning whether its appropriate to consider time as a dimension equivalent to spatial dimensions. Just looking at the influence of mass for now.

I'm wondering whether mass could (un-intuitivately) affect time alone. Then you can bring this into an extra-dimensional universe - where time is like an axis across all the dimensions. Along these lines, I can imagine both gravity and dark matter (even the nature of particles) being described at a quantum level in terms that seem more like the dynamics used to study storms and whirlpools. The standard model suffers but so many other things fall into place.

But I'm drifting off ... its the first question I need an answer to.

Thanks

Simon

2. Dec 19, 2006

### MeJennifer

Yes we are sure that it affects both.
If I am not mistaken Einstein tried to make his theory work with curvature of time alone first but he found out that that approach did not work.

What I do find odd though, and no one has ever given me an explanation as to why that is not odd, is that for instance in the RW metric, supposedly an exact solution, time does not curve at all but space obviously does.

Last edited: Dec 19, 2006
3. Dec 19, 2006

### SimonA

Hi Jennifer

Why and how are we sure ?

Again I'm curious why he thought that. Also why he came to the conclusion that it was both that were equally influenced. It seems to me that there is now enough evidence for extra dimensions. They don't appear to be temporal - but neither do they appear to be spatial in terms of our common understanding of the concept. I'd prefer it if we described our 3D space as a singular 'superdimension'. I suspect that would help the maths - but I'm having difficulty persuading myself to put in the work required to investigate this honestly.

It sounds like a good question to me. I think Einstein's brilliance in GR has blinded us to some of his assumptions.

Simon

4. Dec 19, 2006

### SimonA

And ... surely if it affects both space and time equally - it would also affect (and be affected by) the equivalent of matter that surely exists in extra dimensions, in the same way ? We know tornadoes and whirlpools can be explained on 3D dynamics. What happens when we apply the same theories to quantum particles (e.g. loop quantum gravity) and cosmological phenomena (e.g. dark matter) - and take extra dimensions as our starting point ?

Last edited: Dec 19, 2006
5. Dec 19, 2006

### cesiumfrog

It seems like we have solid experimental demonstrations that time "runs slower" near mass. On the other hand, stretching of space should be directly measurable by interferometric gravitational wave detectors. I'm guessing all the other existing evidence of GR will be difficult to explain so precisely if you propose to dramatically alter GRT.

6. Dec 19, 2006

### SimonA

This is what I'm looking for - thanks CF. Which experiment(s)? In what way can time be clearly differentiated from space in these ?

Yes and this is why I've devoted CPU time to detecting these. But AFAIK we have not yet had any solid confirmation. Could it be that EM and gravity are distinctly different entities - even though they have some archetypal mathematical similies ?

In what ways does GR break down if the direct connection with EM is moved to a more fundamental level ?

7. Dec 19, 2006

### SimonA

Of course the intuitive answer would be that mass affects non-temporal dimensions alone. Also in terms of the 'hierarchy' anomaly this is the more natural assumption - but of course that means that mass affects space-like dimensions. So what is the nature of space ? Does it have particulate nature ?

I suspect not. Can you pick any point in space, and be slightly to the left to right of the plank unit ? How does this fit with, say, Minowski space ? The Higgs Bosun field seems tenuous at best. And yet I'm tempted to accept that cosmological phenomena have similies with gas or liquid dynamics. This is another dilemma I need to resolve! Maybe the standard model will be confirmed with the LHC - but I find it doubtful at best. Like Newtons theory - it made sense at the time...

Any help with my ignorance is appreciated!

Simon

Last edited: Dec 19, 2006
8. Dec 19, 2006

### Chris Hillman

Careful, this thread immediately went off the tracks!

Hi, SimonA,

This question doesn't make sense as stated, since you haven't specified what theory you are asking about (I presume you are inquiring about general relativity), nor have you explained what you mean by "mass", "affects", "space" or "time", nor have you explained whether you are talking about a specific spacetime model (perhaps the Schwarzschild vacuum solution of the Einstein field equation?), or about all possible solutions of the EFE.

The way you phrased the question suggests that you might not yet understand how gtr is built upon the geometrical notion of a Lorentzian manifold, together with a physical interpretation of such a manifold (as a "model of spacetime"). Einstein also explained how gtr generalizes his earlier theory of relativistic kinematics, namely str; briefly, gtr incorporates gravitational dynamics, modeled using the curvature of the spacetime. Repeat: the spacetime.

MeJennifer and the other respondents, please take note and reconsider your responses to date!

As I very recently noted in another thread, under some conditions one can slice up a particular spacetime into "spatial hyperslices" which are nonintersecting and which smoothly fill up the spacetime. Some such hyperslicings may have the property that the world lines of some "interesting" family of inertial observers (say) in our spacetime model are everywhere orthogonal to the slices. Then we obtain a dynamical picture in which "space" evolves over "time", but this picture refers to this specific family of observers. It is also true that in some such cases--- in particular, in the Schwarzschild vacuum, or in some FRW dust models--- one can find such a hyperslicing in which all the hyperslices are (intrinsically) flat three-dimensional euclidean spaces. In these cases, in this particular sense, one could say that the spacetime is built from "stacked with a twist" flat three-planes, somewhat like a ruled surface.

For example, the Schwarzschild vacuum solution is given in the Painleve coordinate chart by the line element:
$ds^2 = -(1-2m/r) \, dt^2 + 2 \, \sqrt{2m/r} \, dt \, dr + dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),$
$-\infty < t < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi$
(note that the range on the Schwarzschild radial coordinate $$r$$ is not a typo!--- this chart is valid right down to the curvature singularity). Here, you can see by inspection that the "constant Painleve time slices" $$t=t_0$$ are all intrinsically flat (down to the curvature singularity). This time coordinate has a simple physical interpretation in terms of proper time kept by clocks carried by infalling Lemaitre observers, i.e. non-spinning and inertial observers who fall in freely and radially "from rest at infinity".

But such statements would be seriously misleading without specifying precisely what one means, and I know of no general forumulation which could reasonably be interpreted as "in gtr, allowed spacetime models are always curved only along time". Quite the contrary, in fact: there are "solutions" in which, in a sense, only "space" is curved.

For example, using a global coordinate chart such as the Kruskal-Szekeres chart, or even better, using a "conformal compactification" (illustrated by a Carter-Penrose diagram) shows that the evolution of "space" over "time" given by the Painleve slicing is in a sense seriously misleading, despite enjoying the appealing property of having a strikingly simple relation to our Newtonian expectations. In the classic textbook Gravitation by Misner, Thorne, and Wheeler, aka "MTW", you can find some nice diagrams showing some less misleading pictures of the evolution of "space" over "time" in the Schwarzschild vacuum solution, based upon some alternative slicings.

Last edited: Dec 20, 2006
9. Dec 19, 2006

### cesiumfrog

GPS satellites, atomic clocks on planes, atomic emission/absorption from separate heights in a building..

10. Dec 19, 2006

### Chris Hillman

Time runs slower? No.

Hi again, SimonA,

I know what you mean, and I appreciate the quotes, and I also admit that many authors who know better use similar language. But talking like this is a really bad idea, unless you have reason to believe that all readers will correctly understand your meaning (never the case in a public popsci forum with posts available years later!). Of course time does not run slower here there or anywhere in any Lorentzian manifold--- that wouldn't even make sense! (What would it run slower with respect to?)

Rather, when we compare the time kept by a pair of ideal clocks, we will probably effect the comparison using something like radio time signals, whose world lines will be null geodesics (in a vacuum region), and then divergence of these null geodesics due to the spacetime curvature will generally mean that if we use some procedure to synchronize the clocks, after a while using the same procedure will indicate that they are "now" out of synch again.

Last edited: Dec 19, 2006
11. Dec 19, 2006

### cesiumfrog

Heh. Mathematically you're completely correct of course.

But physically, I can put an atomic clock in the basement, another on the top floor, and a couple at ground level. I can check they're synchronised today, and take another reading next week.

Last edited: Dec 19, 2006
12. Dec 19, 2006

### Chris Hillman

Yes, but my point is that when you "check that they are synchronized today", you must transfer time signals somehow, and this information can move at best along a null geodesic. The so-called "gravitational time dilation effect" (more properly, "gravitational red shift") arises only in case the null geodesics representing successive time signal checks diverge. This happens if you repeat the famous Pound-Rebka experiment). If these null geodesics happen to converge, you would have gravitational blue shift.

13. Dec 20, 2006

### MeJennifer

Am I correct in understanding that this not an exact metric but a weak field metric?

14. Dec 20, 2006

### Chris Hillman

Hi, MeJennifer,

No, that is exact! See for example http://www.arxiv.org/abs/gr-qc/0001069 or the well if microscopically illustrated discussion in Black Hole Physics by Frolov and Novikov.

To see this, you need only know a coordinate transformation to some chart with which you are already familiar. Almost certainly the Schwarzschild chart is the obvious choice. So let me write these two line elements, using $$\tau$$ for Painleve time coordinate and $$t$$ for Schwarzschild time coordinate:

Schwarzschild chart:
$ds^2 = -(1-2m/r) \, dt^2 + 1/(1-2m/r) \, dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),$
$-\infty < t < \infty, \; 2m < r < \infty, 0 < \theta < \pi, \; -\pi < \phi < \pi$

Painleve chart:
$ds^2 = -(1-2m/r) \, d\tau^2 + 2 \, \sqrt{2m/r} \, d\tau \, dr + \, dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),$
$-\infty < \tau < \infty, \; 0 < r < \infty, 0 < \theta < \pi, \; -\pi < \phi < \pi$

The coordinate transformation from the Schwarzschild chart is given by
$\tau = t + \int \frac{\sqrt{2m/r}}{\sqrt{1-2m/r}} \, dr = t + \sqrt{8mr} - 4 m \, \operatorname{arctanh}(\sqrt{r/2m})$

The frame field defining the Lemaitre observers (with nonspinning local Lorentz frames attached to each observer) is given in the Painleve chart by:
$\vec{e}_0 = \partial_\tau - \sqrt{2m/r} \, \partial_r = \vec{X}$
$\vec{e}_1 = \partial_r$
$\vec{e}_2 = \frac{1}{r} \, \partial_\theta$
$\vec{e}_3 = \frac{1}{r \, \sin(\theta)} \, \partial_\phi$
(The coefficients of the coordinate derivatives are the components of the four vector fields.) In terms of this frame, the acceleration vector of our observers is $$\nabla_{\vec{X}} \vec{X} = 0$$, confirming that these observers are inertial (free-falling), and from the form of $$\vec{e}_0$$ you can see that they are radially infalling, and also that as r goes to infinity, $$\vec{e}_0$$ becomes more and more "upright", i.e. the Lemaitre observers "start falling from rest at infinity".

Since these are simply four vector fields, and since the notion of a pair of orthonormal vector fields is a geometric (or "coordinate-free") notion, such frame fields are geometric structures which do not depend upon using any coordinate chart, although to write them down explicitly we must adopt some coordinate chart. But once we have a frame in hand we are free to change to any other chart should that be convenient. For example, you can easily represent the above frame field in the original Schwarzschild coordinates.

The so-called expansion tensor (see MTW, or Hawking and Ellis, or the newer book by Eric Poisson, A Relativist's Toolkit) is
$H[\vec{X}]_{\hat{a} \hat{b}} = \sqrt{m/2/r^3} \, \operatorname{diag} \, (1,-2,-2)$
The first component is positive, which means that each Lemaitre observer is pulling away from his neighoring Lemaitre observers radially. (You can verify this visually by plotting the integral curves of $$\vec{e}_0$$ in the $$\tau, r$$ plane.) The second two components are negative, which means that each Lemaitre observer is approaching his neighbors orthogonally to his direction of infall toward the hole.

The expansion tensor is a three dimensional tensor because it has been projected into a hyperplane element orthogonal to $$\vec{e}_0$$, which in this situation happens to correspond to being projected into the hyperslice $$\tau=\tau_0$$. In the expression for the expansion tensor, the hats on the indices are often used to emphasize that the components are taken with respect to a frame field (orthonormal basis, anholonomic basis) rather than a coordinate basis. The vector field in brackets emphasizes that the expansion tensor is evaluated with respect to a particular timelike congruence, in this case the one given by the timelike unit vector field $$\vec{e}_0$$ from our frame field.

Also, the Riemann tensor can be decomposed into three pieces (actually only two are independent, since this is a vacuum solution), the electrogravitic and magnetogravitic tensors:
$E[\vec{X}]_{\hat{m}\hat{n}} = \frac{m}{r^3} \; \operatorname{diag} (-2, \, 1, \, 1)$
$B[\vec{X}]_{\hat{m}\hat{n}} = 0$
Here, the electrogravitic tensor is exactly the same as the tidal tensor for the analogous static spherically symmetric gravitational field in Newtonian gravitation! The first component is negative, which means that our observers experience a radial tidal tension. The second two components are positive, which means that our observers experience tidal compression orthogonally. As this shows, the famous "sphaghettification" is nothing unknown to Newtonian physics, just more intense inside a black hole than what we experience at the surface of the Earth. The magnetogravitic vanishes, which means that our observers measure no "gravitomagnetic effects". However, other observers, such as Hagihara observers, who are in stable circular orbits in the exterior region, and are represented by a different frame field, would observe such effects.

BTW, if we compute these tensors with respect to the timelike unit vector of the frame modeling the physical experience of static observers (this frame is of course only defined outside $$r=2m$$), we obtain the same "Coulomb form" for the electrogravitic tensor, and the magnetogravitic tensor again vanishes, so the static observers do not measure magnetogravitic effects.

We can also compute the three-dimensional Riemann tensor giving the "intrinsic curvature" of the hyperslices (see the books cited above), which turns out to be
$r_{\hat{m}\hat{n}\hat{p}\hat{q}} = 0$
as we should expect from the spatial line element
$d\sigma^2 = dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),$
$0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi$
which is of course simply the usual spatial line element for a flat euclidean three-space (in a polar spherical chart).

By the way, the normals to one of our slices $$\tau=\tau_0$$ are given by our timelike unit vector field (tangents to the world lines of the Lemaitre observers where they intersect $$\tau=\tau_0$$), and the expansion tensor wrt $$\vec{X} = \vec{e}_0$$ then gives (the negative of) the extrinsic curvature tensor; see again the books cited above.

(Personal note referring to my post on how I learned gtr: the Painleve chart is the chart which I rediscovered when I was first reading MTW, using only high school "circular" trig -> "hyperbolic" trig. Basically, I drew curves orthgonal to the world lines of the infalling Lemaitre observers in the Schwarschild chart, and figured out what integral I needed to evaluate to pull down these curves to form straight lines, i.e. to pull the hyperslices into the shape of coordinate planes. Finding the line element in the new chart doesn't actually require evaluating the integral I wrote out above. Then of course I noticed something unexpected: in the new chart, the spatial line element is just the usual polar spherical line element!)

To repeat: all of these expressions are exact; no approximations are involved.

What is the difference between a holonomic (e.g. a coordinate basis) and an anholonomic basis (frame field)? Simply that these are made up of vector fields, so we can compute the commutators (Lie brackets) of pairs of these vector fields. In a coordinate basis such as
$\partial_\tau, \; \partial_r, \; \partial_\theta, \; \partial_\phi$
(the coordinate basis of the Painleve chart), these commutators all vanish. But the commutators of our frame field do not all vanish.

Last edited: Dec 20, 2006
15. Dec 20, 2006

### Chronos

Perhaps time and space are no more meaningful concepts in a universe devoid of matter, than one dominated by matter - at least by our current understanding of mathematics. I suspect we do not yet have the proper mathematical tools to resolve these issues. Gravity is an elusive creature. It is the physics equivalent to the ancient chinese riddle - the sound of one hand clapping. Most scientists are repulsed by the notion of a 'universe from nothing', but it is a devilishly clever trap. Unless you let go of the notion of causality, it is impossible to avoid this question.

16. Dec 20, 2006

### Chris Hillman

Huh?

Hi, Chronos,

I for one have no idea what you are talking about.