Free fall coordinates in Schwarzschild?

[publius]

Forgive a question from a mere piker in GR who has got interested in something he can't find the answer to:

What is the metric and coordinates for a radial free-faller in Schwarschild? Let's specify we drop him radially from some r0 and he sets his clock to 0 and drops. I'm imagining he has his little spatial triad falling with him, and the notion of distance here is along his own local ruler.

I've seen Lamaitre coordinates, but I'm not sure if that's exactly what I'm looking for. I'm not sure if the constant "radial" coordinate there is actually fixed according to the free-fall ruler.

Here's what my gut tells me we should see here. Let one of our spatial coordiantes be aligned radially to begin with. We should see tides along here, with things "falling" away from us all along. And for small t and large enough r0, that should just reduce to the Newtonian 2GM/r^3 form. Something behind me falls back and something ahead fall fowards. I should see myself as sitting on a top of a little potential hill, no?

And that tide should increase with own time, getting stronger and stronger until something blows up and the proper time on my clock when I encounter the singularity.

Something stationary along my spatial coordinates would be accelerating to resist the tides. Something "behind" me would have to be accelerating foward to keep up, which something ahead would have to braking, accelerating backwards to stay behind.

And all that should "blow up" in some fashion as I get closer and closer to the singularity.

I'd be greatful for any guidance here. :)

-Richard

 

[Chris Hillman]

Hi, publius, and welcome to PF!

What is the metric and coordinates for a radial free-faller in Schwarschild?

The metric tensor of the Schwarzschild vacuum--- which is a particular Lorentzian manifold which happens to be a solution to the vacuum Einstein field equation (EFE)--- is part of the definition of the spacetime model (Lorentzian manifold), but it appears different when we represent (pieces) of the spacetime in different charts. If you have seen some linear algebra, this is somewhat analogous to representing one and the same linear operator by various different matrices; the matrix representation we use depends upon a choice of basis for our vector space.

That said, there are a number of coordinate charts which are particularly convenient for studying infalling observers; among these are the ingoing Lemaitre chart, ingoing Eddington chart (two forms), Nokikov chart, and ingoing Painleve chart. Since in many previous posts I have described these in detail I'll let you look through the posts listed in my sig.

I'm imagining he has his little spatial triad falling with him, and the notion of distance here is along his own local ruler.

Sounds like you are groping toward the important (even essential) notion of a frame field; see the version of the WP article on "Frame fields in general relativity" which is listed at http://en.wikipedia.org/wiki/User:H...ry:Mathematical_methods_in_general_relativity

I've seen Lamaitre coordinates, but I'm not sure if that's exactly what I'm looking for. I'm not sure if the constant "radial" coordinate there is actually fixed according to the free-fall ruler.

Constant radial coordinate? Fixed according to the frame attached to the world line of a freely falling observer?

We should see tides along here, with things "falling" away from us all along. And for small t and large enough r0, that should just reduce to the Newtonian 2GM/r^3 form. Something behind me falls back and something ahead fall fowards.

Relative to your motion, yes. More precisely, you experience radial tidal tension and tidal compression orthogonally, with the characteristic "Coulomb form" of the tidal tensor taking the same form in either Newtonian gravitation or gtr, namely in Painleve and other charts which employ the Schwarzschild radial coordinate,
$${E \left[ \vec{X} \right] }_{ab} = \frac{m}{r^3} \; \operatorname{diag}(-2,1,1)$$
or, in the Lemaitre chart
$${E \left[ \vec{X} \right]}_{ab} = \frac{2m}{9 \, (R-T)^2} \; \operatorname{diag}(-2,1,1)$$
(where $\vec{X}$ is the timelike unit tangent vector to the world line of our observer). Note that the tidal tensor is a three dimensional tensor in both theories, but in gtr it arises as one of three pieces in the Bel decomposition of the Riemann curvature tensor, relative to a timelike congruence (a "coherent" family of timelike world lines). I have discussed all this in great detail in versions of WP articles I wrote and in various PF posts.

Something stationary along my spatial coordinates would be accelerating to resist the tides. Something "behind" me would have to be accelerating foward to keep up, which something ahead would have to braking, accelerating backwards to stay behind.

Right.

And all that should "blow up" in some fashion as I get closer and closer to the singularity.

The tidal stresses scale as $m/r^3$ in terms of Schwarzschild radial coordinate and as $m/\tau^2$ in terms of proper time left until impact. I have used geometrized units (see the Wikipedia article in the version listed in the page cited above).

As usual when discussing a subtle theory, there are many pedantic caveats.

 

[pervect]

Aside from frame-fields (which might well be what you are looking for, but don't actually form a global coordinate system as you requested), you might want to check out:

Novikov coordinates: which assign a constant R* to a free-falling particle for one coordinate, and use the proper time tau of that free-falling particle for a time coordinate

Conceptually simple, they are mathematically messy. See MTW pg 826 or google for them, the metric is so messy I don't want to type it in. The cuve followed by an infalling particle in Novikov coordinates is very simple, though:

R* = constant, t = varies over allowable range

While they aren't what you asked for, you might (or might not) be interested in Painleve coordinates. See for instance

https://www.physicsforums.com/showthread.php?t=126307

they aren't really what you asked for however, there is no constant coordinate "attached" to a free-falling particle as there is in Novikov coordinates.

 

[publius]

Thanks for the welcome -- glad to be here, long time lurker, first time poster and all that -- and thanks for the replies.

So Chris above is one of the authors of the Wiki GR articles -- I always thought many of those were pretty good. I'm far from that level of understanding, but I can always sort of dimly get an idea of what's going on, and those articles have helped.

Yes, the "frame field" of a radial free faller was what I was after. I was imagining a little coordinate system attached to him as he fell, and wondering what the line element, ds^2, would look like in those coordinates. Ie, if I'm on a suicidal free-fall, and (trying) to hold things stationary with me, what would the clock rates of those things all around me, some spatial distance away from me in my coodinates look like for instance. And I would expect to see something "blow up" at some fixed proper time in my future, and I was just wondering what it would "look like".

But it sounds like that may be too much candy for a nickel. :) If such coordinates don't extend globally too well, or just too messy, maybe even no closed from expressions can be derived, then I'm just out of luck.

And what I meant about the Lemaitre coordiantes above is I just wasn't sure what the meaning of the "R" coordinate was. Suppose I'm at R = R1, and some other observer is at some R2. What is that R2 observer doing relative to me?

-Richard

 

[pervect]

You might also find the idea of "Fermi normal coordinates" very useful then - more genrally, any normal coordinates will do. AFAIK the frame field is associated with the tangent space, which is perfectly flat (for instance, on the 2-sphere, the tangent space would be the plane tangent to the sphere at a point.

What you do to setup Fermi normal coordinates is take a worldline, any worldline, and use the time experienced along that worldline as the time coordinate for your local coordinate system.

Space is defined as orthogonal to time. (Specifically, for example, a spatial axis is a spatial geodesic that's orthogonal to your chosen time coordinate.) This is a bit oversimplified, but should give you the basic idea. Let me know if you want a more formal presentation.

This sets up a local coordinate system. It's only valid for a region sufficiently close to the worldline, the problems come up when your spatial geodesics "sent out" from your worldline cross each other. In the region that they do NOT cross, the Fermi-normal coordinates are valid.

You can define Fermi normal coordinates for any observer, but they have a particularly simple expansion when the observer is not rotating and is free falling. This expansion is valid only to second order, however. It is:

MTW pg 332

$$ds^2 = (-1-R_{0l0m}x^l\,x^m) dt^2 - (\frac{4}{3}R_{0ljm}x^l\,x^m)dt dx^j + (\sigma_{ij}-\frac{1}{3}R_{iljm}x^l\,x^m)dx^i\,dx^j+O(|x^j|^3)dx^\alpha\,dx^\beta$$

Here the components of the Riemann are measured in the local frame-field ,(this is usually indicated by a hat, but I didn't include that hat in the latex).

Riemann normal coordinates are also of interest, they demand a free-falliing (non-accelerating) observer. (So they are also satisfy what you asked for). In general, any "normal coordinate system" would satisfy your requirements, here are some general remarks about normal coordinate systems:

Is (the Riemann normal coordinate system) the only coordinate system that is locally inertial at a point and is tied to the basis vectors there? No. But all such coordinate systems (called 'normal coordinates') will be the same to second order. Moreover, only those same to the third order will preserve the beautiful ties to the Riemann curvature tensor.

So the expression for the metric I give above for general Fermi normal coordinates, should be the same expression for Riemann normal coordinates since your worldline isn't accelerating. Furthermore, any sort of normal coordinates should have the above expression for the metric to second order (and the expression is only valid to second order).

 

[smallphi]

Taylor and Wheeler's book 'Exploring black holes' has some free fall coordinates for Schwarzschild metric.

 

[Chris Hillman]

Since this just came up in another thread: unfortunately, different textbooks seem to use different terminology, not always consistent with the mathematical literature, but IMO good terminology runs roughly like this:

• A Riemann normal chart (section 11.6 in MTW) is constructed using a kind of generalization of polar coordinates on $E^2$ (where radial lines, which are geodesic arcs, radiate from "the origin") in which we consider all geodesics issuing from one event; in the context of Riemannian geometry these are often called Gaussian normal charts; we can expand the metric tensor in a power series around "the origin" and then the components of the Riemann tensor give "second order corrections" to the metric,
• A Fermi normal chart (end of section 13.6 in MTW) is constructed using a kind of generalization of cylindrical coordinates on $E^3$ (where radial lines, which are geodesic arcs, radiate from "the symmetry axis", which is a curve), in which we consider geodesic rays issuing from one timelike curve; we can expand the metric in a kind of power series around the timelikc curve and then the components of the tidal tensor (and the other pieces appearing in the Bel decomposition of the Riemann tensor wrt our timelike curve) give "second order corrections" to the metric.

Thus, these charts offer one of many alternative intrinsic characterizations of the curvature tensor. In particular, in the frame field approach, the Bel decomposition of the Riemann curvature tensor (wrt a timelike congruence, or sometimes just one timelike curve or world line, in which case it is only defined on events on this one world line) is analogous to the familiar decomposition of the EM field tensor (wrt a timelike congruence, or sometimes just one timelike curve or world line) into electric and magnetic vector fields.

 

[pervect]

The end goal here is to assign 4 real numbers to every point in a 4-d manifold in a manner that's very close to a rectangular (and not polar) Cartesian coordinate system.

The question in my mind, is what do we (and the textbooks, so the interested student can look it up) call this construction?

The main part of the mechanism is that we create a map from the tangent space to the physical space. We can then use a Cartesian coordinate system on the tangent space to define a Cartesianl-like coordinate system on the physical space.

The way the mapping process works is that every vector in the tangent space has a magnitude and a direction. The direction part of the vector picks out some particular geodesic in physical space that goes through the specified origin. The magnitude part of the vector tells us how far along the geodesic (arc length) to go. The end result is a point in physical space that corresponds to the point we picked in tangent space.

We can specify the coordinates in the tangent space in any manner we like, but since the goal is to end up with a Cartesian-like coordinate system, we set up a Cartesian coordinate system in the perfectly flat tangent space, and use the above mapping technique to identify some particular point in our non-flat physical space that's associated with the point we picked out in the tangent space. We then utilize the coordinates of the point in tangent space to also label the point in physical space.

 

[publius]

Again, thank you all for taking the time to respond here. It is much appreciated, and given me a lot to chew over.

-Richard

 

[pervect]

Hope we're not getting too advanced for you, let us know if there's something that looks interesting that you need clarified.

It might be useful, for instance, to think about how you'd go about assigning purely spatial coordinates to a point on the surface of the curved Earth using some of the techniques we've talked about.

Note that when you have a way of reducing the Earth's surface to a flat map, you can use the coordinate system of a plane (any style) to assign coordinates to the surface of the Earth by finding the same point on the flat map. Similarly, if you have a coordinate system that describes coordinates on the Earth, you can plot them to a plane to create a map. In general there are a large number of ways of creating such flat maps, however. The method I talked about at some length above by tracing geodesics is just one of them.

 

[Chris Hillman]

Riemann and Fermi Normal Charts

This seems to keep coming up, so I'll add another citation (you will not find a more authoritative citation anywhere, I think, certainly not on-line): see sections 3.1 and 3.2 from Eric Poisson, "The Motion of Point Particles in Curved Spacetime", Living Rev. Relativity 7 (2004), URL (cited on 11 Dec 2007) http://www.livingreviews.org/lrr-2004-6; see http://relativity.livingreviews.org/Articles/lrr-2004-6/index.html . LRR is comparable in quality with RMP (Reviews in Modern Physics, a prestigious journal which publishes review papers by acknowledged experts [manuscripts are commisioned by the editors, not submitted] and then rigorously peer reviewed.)

 

[publius]

Too advanced? Nah, I understand it all perfectly. Like I do brain surgery -- give me a call if need me to poke around your noggin. I'm cheap. Seriously, my education is just a B.S in physics, and of late, I got interested in GR and have been sort of knocking around with the concepts, just for fun and my own edification.

The notion of mapping a 2D surface by using tangent planes is a picture I can see, or even the idea of using the tangent line to a curve (in a plane) has helped paint a lot of pictures for me. And I realize that's limited (how many ways can something curve -- for example take a curve that curves in 3D, you may need more higher embedding dimensions to get a N-D curved surface, and the whole point is to describe things without resorting to them, no?)

For example, I know a "horizon" is a coordinate dependent thing, and using the 2D spherical surface example, I always imagined a horizon was the point where the surface curved such that the tangent plane was now othorgonal to the one you were using. Ie, if z^2 = 1 - x^2 - y^2 (this would at least be parallel to the tangent plane at the pole), then you've got problems at r^2 >= 1. The surface curves back underneath you, and you'd have to choose the negative square root.

Is that roughly a way to think of "what happens" at an horizon? And, I think the spatial geodesics, the great circles launched would all converge there.

-Richard

 

[pervect]

I sometimes use an embedding that's somewhat similar to yours, but I reverse it. I make the "north pole" of the Earth the event horizon, and the equator represents points far away from the event horizon. Latttude corresponds to radial distance, and longitude corresponds to time. This represents only a subset of the black hole space-time, and isn't really very good for visualizing the event horizon. But it has the correct metric properties, i.e. it can be used to demonstrate gravitational time dilation.

I was trying to find where I worked this out with the supporting mathematics in a post here on PF, but I couldn't find it :-(.

One of my favorite visual aids for black holes (including event horizons) is more complex, but it's based on a paper (by Marolf). There's an old thread on this at

https://www.physicsforums.com/showthread.php?t=149932&page=3

the related part starts around post 36. This has the links to Marolf's paper, some web pages by Marolf, and other good stuff that I won't repeat here.

The embedding diagram embeds the r-t plane of a Schwarzschild black hole (along with the Kruskal extension) such that you could draw a standard space-time diagram on this curved surface. Formally, this means that that it embeds the space-time of the r-t plane of the black hole not in a Euclidean space, but in a Minkowskian space with 2 space + 1 time dimensions.

I've attached a copy of the embedding diagram here. The green region is our normal space-time, and the red region is the black hole. The line between the green and red regions is the event horizon. Time runs up the page. You can see that the event horizon is a trapped light beam (which is one of the standard defintions of an event horizon, a trapped null surface).

As I mentioned, you can imagine curved space-time for this black hole as drawing your standard space-time diagrams on this curved surface, rather than on the usual flat sheet of paper. (The coordinate on the vertical axis is the 't' coordinate.)

What is all the rest of the stuff on the diagram? Well, the Schwarzschild black hole with the Kruskal extensions is really a wormhole connecting two differnt asymptotically flat space-times, so the pink region is the "white hole", and the blue region is the "other" asymptotically flat space-time.

The entire event horizon is then the region (pair of crossed lines) where the different colors meet. You can see there's nothing really special about it locally.

 

[Chris Hillman]

Ditto pervect's recommendation of the Marolf paper, but only after you have mastered the basics of surface theory and some elementary gtr, I think!

The embedding picture you are thinking of, I think, is the Flamm paraboloid, which is an embedding of a hyperslice orthogonal to the world lines of static observers. Such a slice never makes into the interior but its equator corresponds to a two-sphere worth of events on the horizon. There is an excellent discussion in Misner, Thorne, and Wheeler, Gravitation, one of the best books ever published on any topic and by common consent one of the classics of scientific literature and a milestone in the development of physics, arguably comparable to books by Newton or Galileo.

Note that as MTW stress, to understand the geometry of the spacetime itself from a sequence of embeddings of "succesive" spatial hyperslices, Carter-Penrose diagrams are extremely helpful.

BTW, other penny dropped, you must be "Publius" from BAUT, where I have occasionally lurked for some time.

 

[publius]

BTW, other penny dropped, you must be "Publius" from BAUT, where I have occasionally lurked for some time.

Yes, that's me, warts and all. :)

-Richard