Understanding Congruences, Rigidity: Week of Sickness & Busyness

[Fredrik]

I've been sick the past week, and also busy with some other things, so I haven't spent much time on this recently. I'm going to write down what I understand and what I don't, mainly just to get my thoughts in order.

I've been reading Wald. The relevant pages are 217 and 46. Page 46 is where he defines the "orthogonal displacement vector". It seems impossible to understand any of this without a thorough understanding of that, so I'll start there. Consider a one-parameter family of timelike curves $\{\gamma_s\}$, parametrized by proper time. I'll write the tangent vector of $t\mapsto\gamma_s(t)$ at t as $\dot\gamma_s(t)$. I'll write the curve $s\mapsto\gamma_s(t)$ as $\gamma(t)$, and its tangent vector at s as $$\dot{\gamma(t)}(s)$$. (I hope this notation isn't too confusing). We define two vector fields T and X by

$$T(\gamma_s(t))=\dot\gamma_s(t)$$

$$X(\gamma_s(t))=\dot{\gamma(t)}(s)$$

There's an extremely important step in all of this that Wald mentions without proof: It's possible to reparametrize each $\gamma_s$, without destroying the "parametrized by proper time" property, so that X is orthogonal to T at all points that the congruence passes through. If someone wants to see the details, I can type them up.

Why is X a measure of the "distance" to nearby curves in the congruence? The best answer I've been able to come up with uses a coordinate system x:

$$x^i(\gamma_{s+r}(t))=x^i(\gamma_s(t))+r\frac{d}{dr}\bigg|_0 x^i(\gamma_{s+r}(t))+\mathcal O(r^2)=\cdots=x^i(\gamma_s(t))+rX^i(\gamma_s(t))+\mathcal O(r^2)$$

$$X^i(\gamma_s(t))=\lim_{r\rightarrow 0}\frac{x^i(\gamma_{r+s}(t))+x^i(\gamma_s(t))}{r}$$

So the ith component of X in the coordinate system x is the limit of "coordinate change"/"parameter change" in the direction of X as the parameter change goes to zero.

This result is all we need from page 46. Jump to page 217. I will continue to call the timelike vector field T and the displacement vector X. I will also continue to consider a one-parameter family of curves, because the notation is already complicated enough to be annoying. We have $\mathcal L_T X=0$, because that Lie derivative is equal to the commutator [T,X], which is zero because T and X are the basis vectors associated with the coordinate system $\gamma_s(t)\mapsto(s,t)$.

A crucial step that I haven't done yet: Prove that this really is a coordinate system, given that T is a smooth vector field with integral curves $\gamma_s$, reparametrized to make X orthogonal to T.

It's not hard to show that [T,X]=0 implies $\nabla_X T=\nabla_T X$. Wald's definition of the tensor B is equivalent to

$$B(Y,Z)=g(\nabla_Z T,Y)$$

for all Y,Z. We can then define $$B^\sharp=B^i{}_j\partial_i\otimes dx^j$$, and use $\nabla_X T=\nabla_T X$ to show that $$B^\sharp(\cdot,X)=\nabla_T X$$. This is why $$B^a{}_b X^b=B^\sharp(\cdot,X)$$ "measures the failure of X to be parallel transported". Note that to get this result, we used$\nabla_X T=\nabla_T X$, which is implied by [X,T]=0.

After that, I don't understand much. I understand that $h^a{}_b=\delta^a_b+T^aT_b$ is a projection operator, but I don't understand why we're looking for a projection operator at all. When I look at the definitions of the tensors that are defined at the end, I see that they don't involve the orthogonal displacement vector X in any way. I have no idea how they can have anything to do with things like rigidity without it. Now I'm also wondering if it was a waste of time to work through the details of the orthogonal displacement vector. It doesn't seem to have anything to do with anything at the end. I really don't get any of this.

I'll give it another shot within the next few days, but if I don't start to get it soon, I'm just going to abandon it. This is taking too much time.

 

[Chris Hillman]

I've been sick the past week, and also busy with some other things, so I haven't spent much time on this recently.

You, me, and George, then. This stuxnet worm is really something, innit? Scada, scada, scada

I've been reading Wald. The relevant pages are 217 and 46. Page 46 is where he defines the "orthogonal displacement vector". It seems impossible to understand any of this without a thorough understanding of that, so I'll start there.

Good idea. FWIW: reference sections in some useful books:

Hawking & Ellis, Large Scale Structure of Space-Time, hereafter HE:
4.1: Fermi propagation, decomposition of timelike unit vector fields, expansion and vorticity tensors, evolution/propagation equations

Wald (note unusual notation for Riemann tensor):
3.3: deviation vector, Jacobi formula
9.2: decomposition of timelike congruences (timelike unit vector fields)

MTW:
Exc. 22.6 decomposition of timelike unit vector fields

Stephani et al., Exact Solutions of the Einstein Field Equations, hereafter EXACT:
2.9 connection, torsion, Ricci rotation coefficients
2.10 Ricci identity
6.1.1 decomposition of timelike unit vector fields
6.2 evolution of decomposition of timelike unit vector fields, rigid motions
8.1: structure coefficients for commutators

Poisson, A Relativist's Toolkit, hereafter Poisson:
2.3 decomposition, Frobenius theorem, evolution of decomposition

de Felice and Clarke:
2.2 structure coefficients
4.4 Fermi propagation
4.5 Ricci rotation coefficients, expansion and vorticity tensors
8.2, 8.3 evolution of expansion and vorticity

(I hope this notation isn't too confusing).

I'm already confused. Isn't s your proper time parameter? In Wald's notation, t is an affine parameter but not in general proper time parameter.

In the "BRS: Timelike Congruences" I am using X for tangent vector (to world lines in congruence, later the timelike unit vector field e_1 in some frame field) and $\xi$ for deviation vector (aka connecting vector).

Maybe you should take geodesic deviation for granted until you "get the idea" of expansion and vorticity? Just an idea--- maybe exactly the opposite will work better for you, since I agree that deviation for neighboring curves in a congruence is what we are talking about, obviously. Also, Wald's discussion is much more limited than that in other books. We really need to know how to study nongeodesic congruences in order to study Rindler and Langevin congruences in Minkowski vacuum, with an eye towards studying "rigidity".

With regard to parallel transport, you can parallel transport a single vector along any curve, but you can parallel transport an entire frame field only along a geodesic curve. You can however Fermi transport a frame along any curve, which is why Fermi derivatives are relevant to the nongeodesic case.

It's not hard to show that [T,X]=0 implies $\nabla_X T=\nabla_T X$.

Right, holds by definition for a torsion-free connection, and the Levi-Civita connection is the unique metric compatible torsion-free connection.

Wald's definition of the tensor B is equivalent to
$$B(Y,Z)=g(\nabla_Z T,Y)$$

Right, putting $\vec{T} = \vec{X}$, $\vec{Z} = \partial_{x^b}$, $\vec{X} = \vec{\xi}$, I was writing, in a coordinate basis
$$Y^a \, J_{ab} = Y^a \, X_{a ;b}$$
so his B is closely related to what HE write as B_{ab} and what I was writing as J_{ab}.

I don't understand why we're looking for a projection operator at all. When I look at the definitions of the tensors that are defined at the end,

Does it help to point out that the expansion and vorticity tensors are orthogonal to the tangent vector, even if the congruence is not a geodesic congruence (but is parameterized by arc length)? In the notation of "BRS: Timelike Congruences":
$$\theta_{ab} \, X^b = 0, \; \; \omega_{ab} \, X^b = 0$$

I see that they don't involve the orthogonal displacement vector X in any way.

Does it help to point out that Wald's discussion in the paragraph containing (9.2.4) is intended to motivate the tensor B, in the case of timelike geodesic congruences? After that you can forget about the connecting vector if you like.

I have no idea how they can have anything to do with things like rigidity without it. Now I'm also wondering if it was a waste of time to work through the details of the orthogonal displacement vector. It doesn't seem to have anything to do with anything at the end. I really don't get any of this.

I think you are making progress, but it sounds like you might run out of time unless you refocus on understanding what Wald says in section 9.2 in the paras after (9.2.4).

Looks like you may have already noticed this, but I don't find the shear tensor particularly useful; the expansion tensor
$$\theta_{ab} = \sigma_{ab} + \frac{\theta}{3} \, h_{ab}$$
gives more immediate insight into the physical experience of a family of observers whose world lines belong to some timelike congruence. E.g. in cosmology this gives the generalization of "Hubble parameter" to models which are not isotropic or homogeneous. So I recommend focusing on seperately deriving

• the Raychaudhuri equation for the evolution of the expansion tensor,
• the evolution of expansion and vorticity tensors

Does it help to point out that we can characterize the world lines of a body undergoing rigid motion in terms of Lie dragging the projection tensor? In the notation of "BRS: Timelike Congruences" (so X is timelike unit tangent vector field from the congruence, not deviation vector):
$${\cal L}_{\vec{X}} h_{ab} = 0$$
Equivalent characterization: the expansion tensor vanishes.

I don't want to give away the punchline I had in mind for the "BRS: Timelike Geodesics", but so far I am (not entirely on purpose) writing a debauch of indices. Does it help to admit now that it will turn out that using the frame field we are interested in, instead of some coordinate basis, will make the computation of expansion/vorticity much easier?

 

[Fredrik]

I'm already confused.

Sorry about that. Finding a good notation was a major problem for me when I worked through the details to prove that the curves can be reparameterized to make X orthogonal to T. I find this notation less confusing than any alternatives I tried. I will continue to call the timelike vector field T here, and use X for the orthogonal displacement vector.

Isn't s your proper time parameter?

My proper time parameter is t. The tangent vectors of the curves $t\mapsto\gamma_s(t)$ make up my timelike vector field T, and the tangent vectors of the curves $s\mapsto\gamma_s(t)$ make up the orthogonal displacement vector field X.

In Wald's notation, t is an affine parameter but not in general proper time parameter.

On page 46, yes, but on page 217, he says that his $\xi$ (my T) is normalized to $\xi^a\xi_a=-1$. This implies that it's parametrized by proper time. I got the result that we can reparametrize the $\gamma_s$ to make X orthogonal to T without destroying the normalization of T. (I'll post the details if you or someone else requests them). I'm pretty confident about this result, so I don't really need to discuss it, but I will, if someone wants me to.

Wald's discussion is much more limited than that in other books. We really need to know how to study nongeodesic congruences...

He doesn't actually use the assumption that T is a geodesic in the calculations before the definitions of expansion/shear/vorticity (the only exception is the result $B_{ab}T^b=0$, which isn't used later), so I don't think it has caused any of my confusion so far. But I'm aware that his definitions of those tensors are different because he considers a geodesic congruence. Bcrowell's summary here is useful.

I think you are making progress, but it sounds like you might run out of time unless you refocus on understanding what Wald says in section 9.2 in the paras after (9.2.4).

What he says immediately after (9.2.4) is that nearby geodesics are stretched and rotated by $B^a{}_b$. That I don't understand at all. I do see that $B^a{}_bY^b=\nabla_TY$ (abstract index notation on the left, coordinate free notation on the right), for any Y that commutes with T, and I think I understand why the orthogonal displacement vector X commutes with T, but that's all I see. I don't see why $B^a{}_b$ "stretches" or "rotates" anything.

I tried to focus specifically on the claim at the beginning of page 218 that $\theta$ "indeed measures the average expansion of the infinitesimally nearby geodesics". This is supposed to follow from (9.2.4), but we have

$$\theta=B^{ab}h_{ab}=B^{ab}g_{ab}+B^{ab}T_aT_b=B^a{}_a$$

by (9.2.5) and (9.2.2) so I don't see how (9.2.4) even enters into it.

Edit: I found that

$$\theta=B^a{}_a=(\nabla_T\partial_i)^i=T^i\Gamma^j_{ij}$$

but I'm not sure what that means.

 

[Chris Hillman]

[Wald] says that his $\xi$ (my T) is normalized to $\xi^a \, \xi_a=-1$. This implies that it's parametrized by proper time.

Right, on p. 217 he is using $\vec{\xi}$ to denote an arbitrary timelike unit vector field, and the integral curves of this vector field will be proper time parameterized curves. Unfortunately, I was using $\vec{\xi}$ for the connecting vector field (deviation vector field), which is spacelike and not normalized to have unit length... So you, I, and Wald are all using wildly conflicting notations.

What he says immediately after (9.2.4) is that nearby geodesics are stretched and rotated by $B^a{}_b$.

A bit more precisely, one could say: the connecting vectors pointing to "the present location" of neighboring observers are stretched and rotated, as the proper time of the fiducial observer increases, in a manner described by the expansion and vorticity tensors.

I don't see why ${B^a}_b$ "stretches" or "rotates" anything.

I know you didn't like Poisson's book on your first pass, but he really tries hard to motivate precisely this by first studying in detail lower dimensional expansion and vorticity tensors.

I know what Wald means on p. 218 but right now I don't seem to see a way of explaining which differs from what he said. However, I do think that Poisson's discussion of lower dimensional expansion tensor should help. Remember that expansion scalar is just trace of expansion tensor, so by general math stuff, you should expect a connection with logarithmic derivative of "volume" (or "area" in lower dimensions) and also with averaging "distance" over a sphere of test particles.

In "BRS: Static Axisymmetric "Gravitationless" Mcmsf Solutions" I discuss an example of exact solutions containing congruences with zero acceleration, expansion tensor, and vorticity tensor. I mentioned matching conditions in the context of matching across -++ signature slices. You can also set up matching conditions across spatial hyperslices and then the extrinsic curvature tensor of the hyperslice is the negative of the expansion tensor of the vector field of normals to the family of hyperslices to which the one we are interested in belongs. Dunno if that helps at all, but thought I'd mention it!

Another general mathy thing which might help: if you recall how to exponentiate matrices, try exponentiating traceless symmetric and antisymmetric 3x3 matrices. You get respectively "volume-preserving shears" and rotations. So our shear tensor and vorticity tensor are describing infinitesimal shears and rotations. The expansion tensor $\theta_{ab} = \sigma_{ab} + \theta/3 \, h_{ab}$ then exponentiates to give linear transformations which change volume and also have shear like effects.

When I am feeling better, I hope to discuss in "BRS:Timelike Congruences" some examples of expansion tensors and vorticity tensors of simple but nontrivial timelike congruences in Minkowski vacuum, which I hope will help make these tensors seem more intuitive.

 

[George Jones]

Right, on p. 217 he is using $\vec{\xi}$
I know you didn't like Poisson's book on your first pass

I quite like section 31.2 from the third edition of Relativity: An Introduction to Special and General Relativity by Stephani. Also, sections 15.1 - 15.3 from An Introduction to General Relativity and Cosmology by Plebanski and Krasinski look good, but I haven't taken a detailed look at this.

 

[Chris Hillman]

The book by Stephani has a German language edition, but I am not sure whether there were later editions. The English translation went through at least three editions, with the last one adding some additional material. I have the second edition of the translation.

 

[George Jones]

The book by Stephani has a German language edition, but I am not sure whether there were later editions. The English translation went through at least three editions, with the last one adding some additional material. I have the second edition of the translation.

There were three German editions, 1977, 1980, and 1989. I have the third (2004) English edition. From its back cover,

Throughly revised and updated, and now also including special relativity ...

 

[George Jones]

I tried to focus specifically on the claim at the beginning of page 218 that $\theta$ "indeed measures the average expansion of the infinitesimally nearby geodesics". This is supposed to follow from (9.2.4), but ...

$$\begin{array}{rcl} u^a \, u_a & = & -1 \\ \dot{u}^a & = & {u^a}_{;b} \, u^b = 0 \\ \xi_a \, u^b & = & 0 \\ {u^a}_{;b} \, \xi^b & = & {\xi^a}_{;b} \, u^b \end{array}$$
(the second illustrates the "dot notation"

From equation (9.2.4) and the dot notation, we have

$$\begin{equation*} \begin{split} \dot{\eta}^a &= \frac{D}{d\tau} \eta^b \\ &= B^a {}_b \eta^b \\ &= \frac{1}{3}\theta h^a {}_b \eta^b \\ &= \frac{1}{3}\theta \eta^a . \end{split} \end{equation*}$$

In order to see the effect of $\theta$ alone, I have set $\sigma_{ab} = \omega_{ab} = 0$. $\eta$ is orthogonal to 4-velocity $\xi$, so is position in position in space of nearby streaming particles with respect to a fidicual particle. The above equation says that spatial velocity of these particles with respect to the fidicual particle is proportional to spatial distance.

Handwaving Explanation
Treat "dot" as a normal derivative, Let $\eta$ by the spatial radius of a nearby small sphere of streaming particles. The spatial volume of the sphere is

$$\begin{equation*} \begin{split} V &= \frac{4}{3} \pi \eta^3 \\ \dot{V} &= 4 \pi \dot{\eta} \eta^2 , \end{split} \end{equation*}$$

and

$$\frac{\dot{V}}{V} = \frac{3 \dot{\eta}}{\eta} = \theta.$$

Thus, $\theta$ gives a sort of averaged expansion for the sphere of particles.