How to Express the Schwarzschild Metric in Fermi Normal Coordinates?

[Gleeson]

TL;DR Summary

Schwarzschild metric in Fermi normal coordinates about a radially infalling, timelike geodesic.

I have been learning a bit about Fermi normal coordinates in Eric Poisson's "A Relativist's Toolkit". Problem 1.10 in this book is to express the Schwarzschild metric in Fermi normal coordinates about a radially infalling, timelike geodesic.

I know that in the Fermi normal coordinates (denoted {T, X^a} where a = 1, 2, 3), the metric to quadratic order will be:

$$g_{TT} = -1 + R_{TaTb}(T)X^aX^b $$
$$g_{Ta} = -2/3R_{Tbac}(T)X^bX^c $$
$$ \delta_{ab} - 1/3 R_{abcd}(T)X^c X^d $$

T is the proper time of along the geodesic.

So to solve this I believe I need the above components of the Riemann tensor, in the Fermi normal coordinates, evaluated on the geodesic of interest.

I'm not sure where to start, and would appreciate some suggestions please.

I was thinking that I should first consider Schwarzschild metric in Schwarzschild coordinates. In these coordinates then I should find the radially infalling geodesic(s). Then, still in these coordinates, find the orthonormal basis attached to this geodesic (the time like vector corresponding to tangent vector to the geodesic). Then I should calculate:

$$ R_{Tabc}(T) = R_{\mu \nu \alpha \beta}e^{\mu}_T e^{\nu}_a e^{\alpha}_b e^{\beta}_c $$

where the greek indices denote the Scharzschild coordinates.

Does this seems correct? Is there a better way to do this?

 

[PeterDonis]

I was thinking that I should first consider Schwarzschild metric in Schwarzschild coordinates.

For a radially infalling timelike geodesic, either Painleve coordinates (if the geodesic is falling from rest at infinity) or Novikov coordinates (if the geodesic is falling from rest at a finite altitude) would probably be a better choice as a starting point. Schwarzschild coordinates are better adapted to observers who are "hovering" at a fixed altitude.

 

[Gleeson]

Thanks. But I was not aware of either of those coordinate systems. And I doubt they were they intended ones to start from in this case, because I think it would be absurd for this book to have assumed we would know this.

Also, whether or not the geodesic is falling from rest at infinity was not specified. It just said radially ingoing and timelike.

 

[PeterDonis]

I was not aware of either of those coordinate systems.

They're covered later on in the same book...right along with Schwarzschild coordinates. The only coordinates that are covered in Chapter 1 of the book are Fermi normal coordinates.

I doubt they were they intended ones to start from in this case, because I think it would be absurd for this book to have assumed we would know this.

If this argument is valid (and I'm not saying it isn't), it would apply equally well to Schwarzschild coordinates, since those have not yet been covered in the book either. So if you are correct here, then the book does not intend for you to use any other coordinate charts to solve this problem.

whether or not the geodesic is falling from rest at infinity was not specified. It just said radially ingoing and timelike.

That would imply that the problem can be solved without having to specify this, i.e., that the form of the metric does not depend on it. You might end up with an undetermined constant in the metric that would reflect the initial conditions of the geodesic.

 

[Gleeson]

"If this argument is valid (and I'm not saying it isn't), it would apply equally well to Schwarzschild coordinates, since those have not yet been covered in the book either."

The book assumes some knowledge of the Schwarzschild metric. I contend it is therefore totally reasonable to assume knowledge of Schwarzschild coordinates. More exotic ones, like you mentioned, I think are unreasonable to assume knowledge of.

"So if you are correct here, then the book does not intend for you to use any other coordinate charts to solve this problem."

Sorry, I don't follow what you mean here.

 

[PeterDonis]

The book assumes some knowledge of the Schwarzschild metric.

The Schwarzschild metric (spacetime geometry) is not the same thing as Schwarzschild coordinates.

I contend it is therefore totally reasonable to assume knowledge of Schwarzschild coordinates. More exotic ones, like you mentioned, I think are unreasonable to assume knowledge of.

Your description of these other coordinates as "more exotic" is not a good one. However, your assumptions about the book's expectations are of course up to you and I'm not going to argue with them.

That said, you came here asking for help, so you might want to at least consider the fact that the responses you are getting include different assumptions from the ones you are making.

"So if you are correct here, then the book does not intend for you to use any other coordinate charts to solve this problem."

Sorry, I don't follow what you mean here.

What is unclear about it? Chapter 1 is all about constructing Fermi normal coordinates from scratch, without using any other coordinates. That's all I'm referring to.

 

[Gleeson]

"The Schwarzschild metric (spacetime geometry) is not the same thing as Schwarzschild coordinates."

I know..

"Your description of these other coordinates as "more exotic" is not a good one. However, your assumptions about the book's expectations are of course up to you and I'm not going to argue with them.
That said, you came here asking for help, so you might want to at least consider the fact that the responses you are getting include different assumptions from the ones you are making."

I appreciate your responses. And I don't want to keep wasting time discussing this point. I just am in disbelief at your point of view. (And that doesn't mean I am not willing to consider I am wrong.) Sure, precisely how "exotic" the coordinates you mentioned are, is subjective, but one could easily take a couple of GR courses without hearing about them. But what student has ever seen the Schwarzschild metric without seeing Schwarzschild coordinates? That is all I am saying.

"What is unclear about it? Chapter 1 is all about constructing Fermi normal coordinates from scratch, without using any other coordinates. That's all I'm referring to."

Doesn't the construction in the book start with some arbitrary coordinates, and then show how to transform from those arbitrary ones into the Fermi ones (This is why I conjectured the above approach to the problem.)? If not, then I have significantly misunderstood the book's discussion. Which is possible.

 

[PeterDonis]

@Gleeson please us the quote function to quote what you are responding to. There are Quote and Reply buttons at the lower right of each post window. I find the easiest thing usually is to highlight what you want to respond to, then click "Reply", and the text you highlighted will be automatically put inside a quote block in the response window.

 

[Gleeson]

@Gleeson please us the quote function to quote what you are responding to. There are Quote and Reply buttons at the lower right of each post window. I find the easiest thing usually is to highlight what you want to respond to, then click "Reply", and the text you highlighted will be automatically put inside a quote block in the response window.

Thanks. I couldn't figure out how it worked.

 

[PeterDonis]

Doesn't the construction in the book start with some arbitrary coordinates

Yes, it does. I'm just not sure if the solution to the problem you are asking about requires making a particular choice about which arbitrary coordinates they are. You might be able to work out what the components of the Riemann tensor are in an orthonormal basis whose timelike unit vector is tangent to the geodesic (since that's what you actually need) by geometric and physical arguments using the special properties of Schwarzschild spacetime, without having to compute them in an arbitrarily chosen coordinate chart (Schwarzschild or any other).

 

[Gleeson]

Yes, it does. I'm just not sure if the solution to the problem you are asking about requires making a particular choice about which arbitrary coordinates they are. You might be able to work out what the components of the Riemann tensor are in an orthonormal basis whose timelike unit vector is tangent to the geodesic (since that's what you actually need) by geometric and physical arguments using the special properties of Schwarzschild spacetime, without having to compute them in an arbitrarily chosen coordinate chart (Schwarzschild or any other).

If you have any suggestions what those geometric and physical arguments would be, please let me know. This would be a better way to do it.

 

[Gleeson]

And if anyone else has any ideas about this, please let me know.

 

[pervect]

If you want the exact formulation, and not to use the approximation you already mentioned, it'll be very involved. First you'd need to map the Fermi-normal coordinates to whatever coordinate system you want to use. Once you have the coordinate map from one set of coordinates to the other, you can use the tensor transformation rules to transform the metric tensor - or, depending on your approach, you can use algebraic means to perform this transformation, though you may have to find the inverse coordinate transform, depending.

To find the map from Fermi-normal coordinates to, say, Schwarzschild coordinates (this may not be the best choice, but I want to give the coordinate system a name), you'll need to follow the coordinate-free construction process used to define the Fermi-Normal coordinates. MTW's textbook "Gravitation" goes through this in some detail if your text doesn't.

The first thing to do is to assign a set of basis vectors to your reference geodesic, the radially infalling one you mentioned, at some point P. Then you need to Fermi-Walker transport these basis vectors along your geodesic. In this case, though, Fermi-Walker transport is just parallel transport, because your reference worldline is a geodesic. And if you also know that Fermi-Walker transport is what a gyroscope does, you probably have some intuition about how the basis vectors are transported already. You'll have to watch out for Thomas precession, but given the lack of proper acceleration of a geodesic, I don't think there will be any.

The Fermi-normal coordinates specify a specific geodesic, different from and orthogonal to your reference infalling geodesic, and how far along that geodesic you travel. So given the Fermi-Normal coordinates, you can in principle construct this new geodesic curve, representing it with a parameterized worldline, then from the distance you travel along the worldline, you can identify a specific point with specific Schwazschild coordiantes.

The end result of this process is to generate the map you need, the one that maps Fermi coordinates to the Schwarzschild coordinates. Depending on how you go about transforming the metric, you may need or want the inverse transformation - i.e., you now have the transformation from Fermi-normal coordinates to Schwarzschild coordiantes, but if you want to use for instance algebraic substitution (rather than tensor manipulation) to do the coordinate transformation, you'd want the inverse transformation. First you need to address the issue of what region this inverse exists in. Then there's the issue of whether or not the inverse has a closed form expression (you might need to use series methods).

Note that you'll run into various issues if this inverse doesn't exist at all - in general, this means that you won't have a one-one correspondence between points and coordinates. You run into this issue when the worldlines you generated in the previous step (the orthogonal to your reference geodesic) cross, this identifies the same point on the manifold with multiple Fermi-Normal coordinates.

 

[pervect]

If my previous post seems like entirely too much work, I agree. As most authors stress, Fermi-Normal coordinates are most useful very near the reference worldline, and it's too much work for too little return to go above the second-order approximation which has already been given in terms of the Riemann tensor. In the region where the coordinates are useful, the higher order terms are not needed.

If the Schwarzschild problem seems too hard, I'd suggest a simpler one. Imagine that the Earth was not rotating or gravitating, and consider only its surface. Construct Fermi-normal coordinates for the resulting 2-space 1-time geometry along a worldline with fixed latitude and longitude. The time aspects of the geometry are (deliberately) trivial for the zero rotation case.

If we suppress the trivial time dimension, we basically have a map of the Earth's surface. We can note that it will be close to scale near the reference origin (the fixed latitude and longitude point), but will become infinitely distorted as it approaches the antipodal point.

If we make the origin reference point the north pole, the area around the pole will be faithfully to scale. When we reach the equator, we see significant scale issues. Imaging the Earth as spherical (rather than an elipsoid) with a circumference of 24,000 miles, the circumference of the equator drawn to scale on the map will be incorrect at 38,000 miles (2*pi*6000 miles) a bit above 50 percent error. The distortions grow rapidly below the equator, approaching an infinite distortion as one nears the south pole (which is the antipodal point). Note that the south pole has the issue I mentioned previously of having multiple coordinates assigned to a single point.