# BRS: Those Damnable Paradoxes . I. Overview

1. Sep 13, 2010

### Chris Hillman

BRS: Those Damnable "Paradoxes". I. Overview

I hate talking about them! But I will force myself to try anyway.

What am I talking about? The so-called Bell spaceship "paradox" and Ehrenfest rotating disk "paradox". These paradoxes seem to come up every six to twelve months in PF, and always seem to lead to extremely long and thoroughly confused threads. As the discussion becomes increasingly heated and circular, the frequent result is that a Mentor locks the thread, and some of the disputants have even been banned from PF. These are not good outcomes.

Therefore, I think it will be useful to SA/Ms who regularly post in the relativity or astrophysics forums to have a place where we can show/discuss all the math and all the pictures without fear of interruption from possibly well intentioned newbies--- or even worse, from cranks. Without having to waste an excessive amount of time explaining standard shared mathematical background, terminology, or notation--- or even worse, trying to explain the scientific method, the role of (simple! clearly defined!) thought experiments, or the neccessity of using mathematics and pictures to avoid ambiguities and facilitate qualitative conclusions as well as quantitative reasoning.

So why do I hate talking about these "paradoxes"? Because in the past I always seemed to wind up feeling that I was simply repeating for the n-th time stuff I or others have said dozens of times before:
• in the Usenet Physics FAQ
Code (Text):

www.math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

• in Wikipedia articles
Code (Text):

en.wikipedia.org/w/index.php?title=Born_coordinates&oldid=53957524
en.wikipedia.org/w/index.php?title=Rindler_coordinates&oldid=51749949
en.wikipedia.org/w/index.php?title=Frame_fields_in_general_relativity&oldid=53048960
en.wikipedia.org/w/index.php?title=Congruence_(general_relativity)&oldid=49994474

(notice that I have linked to the last version I edited of each article; I have nothing to do with subsequent versions, which might sometimes be better, but which may be more likely to be generally worse, given the number of cranks who want to have their say in Wikipedia),
• in PF posts:
Code (Text):

www.physicsforums.com/showpost.php?p=1524303&postcount=4

(BTW, the last cited PF thread includes some good discussion between myself, pervect, and Greg Egan, who also put up some good stuff on rotating rings (easier than rotating disks!) at his website
Code (Text):

gregegan.customer.netspace.net.au/SCIENCE/Rings/Rings.html

Also, note that Wikipedia user pervect is also PF SA/M pervect, and Wikipedia user Mpatel is my former colleague in the now defunct "WikiProject Relativity", so I would expect versions of the Wikipedia articles which they edited to be much, much better than those rewritten by cranks. As for the current versions, that is anyone's guess. Note too that the old versions used old templates and figures which are in some cases no longer available, so their appearance has degraded.)

But my previous bad experiences trying to discuss these damnable "paradoxes" have occurred in "anything goes" fora like Wikipedia, or in regulated but public fora like the public areas of PF, where people with, shall we say, various levels of background and ability never hesitate to add their "two cents"--- almost invariably, with confidence inversely proportional to their understanding

Fortunately, the BRS is completely different from those venues: SA/Ms share some common background and experience and we can presume that SA/Ms don't enter discussions already convinced that "all the textbooks are wrong"--- an attitude encountered surprisingly awful in the roster of (often awful) arXiv eprints on these "paradoxes", much less public discussion forums or the Wikipedia. So this thread should hopefully be much more fun for me than any of my previous public discussions.

Since I have written extensively about these "paradoxes" many times before, let me briefly summarize in forthright language some lessons which I feel will be self-evident to anyone who has studied gtr at the graduate level and who has studied the most important contributions to the rather vast literature on "relativity paradoxes":
• The so-called Bell spaceship "paradox" is quite easily and conclusively settled by certain standard notions and techniques commonly encountered in the gtr literature:
• timelike congruences
• frame fields
• expansion and vorcitity tensor (from the kinematic decomposition of a timelike congruence)
for which see Hawking and Ellis, Large Scale Structure of Space-Time, Cambridge University Press, 1972, or Poisson, A Relativist's Toolkit, Cambridge University Press, 2003. The Ehrenfest "paradox" is considerably more delicate, and a careful discussion turns out to involve a dozen subtleties, in addition to the techniques which suffice for the Bell "paradox".
• The expansion tensor, in particular, is precisely what is needed to decisively resolve the Bell "paradox"; people who insist upon excluding it will be forced to either reinvent special cases of the expansion tensor--- a notion which is very useful for almost everything in relativistic physics--- using private notation/terminology, or else will inevitably make mistakes and will very likely wind up "talking past" other disputants who are using their own private notation/terminology.
• The physics literature on "paradoxes", particularly in the past few decades, is full of really awful papers and arXiv eprints written by persons who
• don't know the literature on the subject even slightly,
• haven't even bothered to read such essential sources as the review article of Oyvind Gron, Am. J. Phys. 43 (1975): 869
• typically don't realize that they are trying to give a "not even wrong" "answer" [sic] to a question which was correctly answered by Langevin 1927 (in the more difficult case of the Ehrenfest paradox),
• rarely know about modern and obviously relevant techniques such as expansion and vorticity tensors,
• rarely know about the multiplicity of operationally significant notions of "distance in the large"--- that's a huge problem because this multiplicity is unavoidable even in flat spacetime, at least for accelerating observers, such as observers riding on a rigidly rotating disk, or Bell or Rindler observers riding on an alleged "taut cable",
• rarely understand the distinction between induced metrics on submanifolds (e.g. spatial hyperslices) and induced metrics on quotient manifolds (e.g. in the discussion of Landau and Lifschitz of the rotating disk),
• appear incapable of drawing a good sketch to clarify the physical scenarios they have in mind,
• insist upon using inadequate or inappropriate notation, terminology, and techniques.
These deficiencies more or less guarantee their eprint a place in the dustbin of Utterly Useless Contributions to Mathematical Physics, even when the author is trying to argue for the correct resolution of the "paradox". Unfortunately, many of authors of arXiv eprints on these "paradoxes" can only be characterized as physics cranks who, in addition to the deficiencies just listed, are arguing that "all the physics textbooks are wrong". It would in fact be difficult to identify another topic in physics which has led to a comparable "density" of truly awful papers.

Speaking of pictures, DrGreg's figure in his Post #33 in
Code (Text):

illustrates Bondi's concept of radar distance, which can be our default notion of "distance in the large" should any be needed. Fortunately, in Minkowski spacetime the null geodesics are explicitly known so computing radar distance exactly should be possible in simple physical scenarios.

Next, let me move straight in to writing down two timelike congruences which are defined in certain regions of Minkowski spacetime, the Rindler and Bell congruences, and showing how to study them using their kinematic decomposition.

Last edited: Sep 13, 2010
2. Sep 13, 2010

### Chris Hillman

BRS: Those Damnable "Paradoxes". II. Rindler vs. Bell Congruences

The Cartesian chart is a global chart for Minkowski vacuum, a locally flat Lorentzian manifold with the simple line element
$$ds^2 = -dT^2 + dX^2 + dY^2 + dZ^2, \; \; -\infty < T, \, X, \, Y, \, Z < \infty$$
Which, I am sure, is not news to anyone here!

The Rindler frame is given in terms of the Cartesian chart by
$$\begin{array}{rcl} \vec{e}_1 & = & \frac{X}{\sqrt{X^2-T^2}} \; \partial_T + \frac{T}{\sqrt{X^2-T^2}} \; \partial_X \\ \vec{e}_2 & = & \frac{T}{\sqrt{X^2-T^2}} \; \partial_T + \frac{X}{\sqrt{X^2-T^2}} \; \partial_X \\ \vec{e}_3 & = & \partial_Y \\ \vec{e}_4 & = & \partial_Z \end{array}$$
where by convention the first unit vector field is timelike, and the remaining three spacelike, and all four are mutually orthogonal at each event where the frame field is defined. Notice that this frame is not defined on the entire region covered by the chart, but only on the Rindler wedge $|X| > T > 0$.

Consider the timelike congruence obtained by finding the integral curves of $\vec{e}_1$. Let us call these the world lines of (a certain family of) Rindler observers. These world lines are given as proper time parameterized curves by
$$\begin{array}{rcl} T & = & X_0 \; \sinh(s) \\ X & = & X_0 \; \cosh(s) \\ Y & = & Y_0 \\ Z & = & Z_0 \end{array}$$
Thus, these world lines obey the relation $X^2-T^2 = X_0^2$, as they should.

The Rindler congruence has the following geometric characteristics:
• the world lines are "nested hyperbolas" organized in a configuration which is rather precisely the hyperbolic trig analogue of a euclidean trig configuration: a family of concentric circles in the euclidean plane,
• acceleration vector (path curvature of world lines obtained as integral curves of $\vec{e}_1$)
$$\nabla_{\vec{e}_1} \vec{e}_1 = \frac{1}{\sqrt{X^2-T^2}} \; \vec{e}_2$$
• expansion tensor vanishes (so this congruence is rigid),
• vorticity tensor vanishes (so this congruence is hypersurface orthogonal),
• the Fermi derivatives along $\vec{e}_1$ of $\vec{e}_2, \; \vec{e}_3, \; \vec{e}_4$ vanish (so the Rindler frame is non-spinning but non-inertial; physically speaking, the Rindler observers are accelerating but gyrostabilized).
The Cartesian chart is convenient for many purposes, but there are several advantages to transforming to another chart which is "adapted" to the Rindler observers, namely the Rindler chart, which is only defined on the Rindler wedge (so it is certainly not a global chart!). The transformation we need is
$$\begin{array}{rcl} t & = & \operatorname{arctanh}(T/X) \\ x & = & \sqrt{X^2-T^2} \\ y & = & Y \\ z & = & Z \end{array}$$
The inverse transformation is
$$\begin{array}{rcl} T & = & x \, \sinh t \\ X & =& x \, \cosh t \\ Y & = & y \\ Z & = & z \end{array}$$
In the Rindler chart, the line element of Minkowski vacuum becomes
$$ds^2 = -x^2 \, dt^2 + dx^2 +dy^2 +dz^2, \; \; 0 < x < \infty, \; -\infty < t, \, y, \, z < \infty$$
where $x=0$ is a coordinate singularity, the Rindler horizon, which corresponds to the boundary of the Rindler wedge $|X|=T > 0$.

In terms of the Rindler chart, the Rindler frame is
$$\begin{array}{rcl} \vec{e}_1 & = & \frac{1}{x} \; \partial_t \\ \vec{e}_2 & = & \partial_x \\ \vec{e}_3 & = & \partial_y \\ \vec{e}_4 & = & \partial_z \end{array}$$
and
• the world lines are simply the coordinate lines $x=x_0, \; y = y_0, \; z = z_0$, and differences in t along a world line correspond to elapsed proper time measured by an ideal clock carried by the Rindler observer with that world line,
• the acceleration vector is
$$\nabla_{\vec{e}_1} \vec{e}_1 = \frac{1}{x} \; \vec{e}_2$$
which clearly brings out the very important fact that the path curvature of a Rindler observer is constant in magnitude, but different Rindler observers can have different constant values for their magnitude of acceleration (the euclidean analogue is very helpful here!)
• the expansion and vorticity tensors, which vanish when represented in the Cartesian chart, must and do vanish when represented in the Rindler chart also, with the same geometric/physical interpretations,
• the Fermi derivatives, which vanish when represented in the Cartesian chart, must and do vanish when represented in the Rindler chart also, which means that the path curvature of each Rindler observer is constant in both magnitude and direction,
• because the vorticity tensor of the Rindler congruence vanishes, there exists a unique family of spatial hyperslices which are everywhere orthogonal to the world lines of our Rindler observers; these slices are given in the Rindler chart by the coordinate planes $t = t_0$; as submanifolds they inherit from Minkowski vacuum the structure of a three-dimensional Riemannian manifold, whose Riemann curvature tensor
$$r_{2323} = r_{2424} = r_{3434} = 0$$
shows that they are homogeneous spaces, in fact, copies of euclidean three space E^3,
• the Rindler chart is a static chart and as such admits a Fermat metric on the hyperslices (this is the metric such that the geodesics in this metric are the projections to the slice of the null geodesics in the parent spacetime), in which the line element becomes
$$d{\sigma_F}^2 = \frac{dx^2+dy^2+dz^2}{x^2}$$
which gives the slices the geometry of hyperbolic three-space H^3, so that projections of null geodesics in Minkowski spacetime to the hyperslices are coordinate semicircles "orthogonal" to the Rindler horizon--- but note that only a portion of each of these null geodesics is included in the Rindler wedge (for more about the Fermat metric, see chapter 8 of Frankel, Gravitational Curvature, Freeman, 1979),
• the beacon congruence consisting of all null geodesics issuing isotropically from an event on the world line of a Rindler observer $x=a, \, y=0, \, z=0$ can be defined as the integral curves of the null geodesic vector field
$$\vec{\ell} = \frac{a}{x^2} \; \partial_t + \frac{a}{\sqrt{(x+a)^2+y^2+z^2} \; \sqrt{(x-a)^2+y^2+z^2}} \; \left( \frac{x^2-a^2-y^2-z^2}{x} \; \partial_x + 2 y \; \partial_y + 2 z \; \partial_z \right)$$
which in terms of the formalism of the optical scalars of Sachs (see the monograph of Hawking and Ellis or the textbook by Poisson) has vanishing shear and twist scalars and expansion scalar
$$\theta\left[\vec{\ell}\right] = \frac{2a}{\sqrt{(x+a)^2+y^2+z^2} \; \sqrt{(x-a)^2+y^2+z^2}}$$
which near x=a, y=z=0 is
$$\theta\left[\vec{\ell}\right] \approx \frac{1}{\sqrt{(x-a)^2+y^2+z^2}}$$
This is useful to know for various purposes; for example, if $\vec{w}$ is the tangent vector to an arbitrary world line at some event, $-E \; \vec{w} \cdot \vec{\ell}$ is the energy measured by that observer of a photon with energy E which was emitted by our Rindler observer.

Now suppose that $b > a > 0$ and consider the Rindler observers with world lines given in Rindler chart by
$$\begin{array}{l} x = a, \, y = z =0 \\ x= b, \, y =z=0 \end{array}$$
respectively. Notice that the world line of the first observer has acceleration (path curvature) 1/a, which is larger than the acceleration (path curvature) 1/b of the world line of the second observer. This might seem odd, but it is just what we should expect from considering the circular trig analog congruence, a family of concentric circles! If these two observers have stretched a cable between them, with bits of matter in the cable having world lines $x=c, \, y=z=0$ with $a < c < b$, then the expansion tensor says that nearby bits of cable maintain constant distance from each other, i.e. the cable is not being "ultralocally stretched". But this requires that each bit of cable experience a different acceleration, with trailing bits acceleration harder in order to keep up with leading bits!

It is tempting--- but not straightforward--- to try to formulate a suitable relativistic analog of Hooke's law, or even the entire theory of linear elasticity, in order to formulate a more convincing model, in which the cable consists of isotropic elastic material which is stressed but maintains a static equilibrium. Greg Egan's webpages offers some thoughts about that. There has been quite a bit of work on this over the decades, and very very briefly, one can say that the nonlinear theory of elasticity seems to be required here, and a nonlinear theory is unfortunately more difficult to work with.

Bell turned this around, and proposed to study another congruence, the Bell congruence, whose integral curves are the world lines of observers, a family of Bell observers, which again have constant path curvature, but this time all the Bell observers share the same path curvature, k. These world lines are also all orthogonal to T=0 in the Cartesian chart. (See the figure below.)

In the Cartesian chart, the proper time parameterized world lines of the Bell observers can be written
$$\begin{array}{rcl} T & = & \frac{1}{k} \; \sinh(ks) \\ X & = & X_0 + \frac{1}{k} \; \left( \cosh(ks) - 1 \right) \\ Y & = & Y_0 \\ Z & =& Z_0 \end{array}$$
Notice that for each $Y=Y_0, \; Z=Z_0$, the world line of precisely one Bell observer agrees with the world line of a Rindler observer drawn from the Rindler congruence as defined above.

The Bell frame can be given as
$$\begin{array}{rcl} \vec{f}_1 & = & \sqrt{1 + k^2 \, T^2} \; \partial_T + k \, T \; \partial_X \\ \vec{f}_2 & = & k \, T \; \partial_T + \sqrt{1 + k^2 \, T^2} \; \partial_X \\ \vec{f}_3 & = & \partial_Y \\ \vec{f}_4 & = & \partial_Z \end{array}$$
The Bell congruence has the following geometric characteristics:
• the acceleration vector is
$$\nabla_{\vec{f}_1} \vec{f}_1 =k \; \vec{f}_2$$
• the expansion tensor has only one nonzero component (wrt the Bell frame)
$$H_{22} = \frac{k^2 \, T}{\sqrt{1+k^2 \, T^2}}$$
so nearby Bell observers are moving apart along the $\vec{f}_2$ direction; furthermore, notice that
$$\lim_{T \rightarrow \infty} H_{22} = k$$
• the vorticity tensor vanishes, so the Bell congruence is hypersurface orthogonal,
• the orthogonal hyperslices are locally flat (but these slices are distinct from the hyperslices of the Rindler congruence!),
• the Fermi derivatives along $\vec{f}_1$ of $\vec{f}_2, \; \vec{f}_3, \; \vec{f}_4$ vanish (so the Bell frame is non-spinning but non-inertial; physically speaking, the Bell observers are accelerating but gyrostabilized).
Now, following Bell, consider two Bell observers sharing the coordinates $Y=Y_0, \; Z=Z_0$. Suppose they have stretched a cable between them, and suppose that each bit of cable has the same acceleration $k \; \vec{f}_2$ as the two observers (without specifying how this acceleration is imparted to bits of cable!). Then, the positive expansion tensor component $H_{22} > 0$ shows that nearby bits of cable are moving away from each other. Furthermore, as proper time increases, the rate at which nearby bits are moving away from each other approaches a positive constant. Eventually, even if it is made of rubber, the cable must break. Notice that this argument circumvents any need to worry about "distance in the large", and this is physically appropriate since real cables snap when ultralocal stresses become too large; they don't care about "distance in the large".

• pairs of Rindler observers,
• pairs of Bell observers,
• &c
This is well worth doing, if only because it clearly shows that radar distance has some disconcerting properties:
• non-symmetric: $d(x,y) \neq d(y,x)$
• non-additive: $d(x,z) \neq d(x,y) + d(y,z)$ even when x,y,z are colinear.
(continued...)

Figures (left to right):
• a hyperbolic trig configuration, the Rindler congruence, depicted in a Cartesian chart, and its circular trig analog,
• a hyperbolic trig configuration, the Bell congruence, depicted in a Cartesian chart, and its circular trig analog,
• comparing the Cartesian and Rindler charts.

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3. Sep 13, 2010

### Chris Hillman

BRS: Those Damnable "Paradoxes". II. Rindler vs. Bell Congruences (cont'd)

Let's work in the Rindler chart. The null geodesics obey
$$\pm x \, dt = dx$$
or
$$\pm dt = \frac{dx}{x}$$
Thus we can write null geodesics with constant y,z coordinate which pass through $t=t_0, \, x=a$ in the form
$$t = t_0 \pm \log (x/a)$$
(See the figure below.)

Now let $O_\hbox{low}$ be the observer at $x=a, y=z=0$ and let $O_\hbox{hi}$ be the observer at $x=a+h, y=z=0$, where h is a small quantity. Then (see the figure below)
$$\begin{array}{rcl} \Delta t & = & 2 \; \int_{x=a}^{a+h} \frac{dx}{x} \\ & = & 2 \log (1+h/a) \end{array}$$
But the line element written in the Rindler chart shows that the proper time measured by the Rindler observer with coordinate $x=x_0$ scales like
$$\Delta s = x_0 \; \Delta t$$
Thus (see figure) the radar distance from $O_\hbox{low}$ to $O_\hbox{hi}$ (measured by $O_\hbox{low}$) is
$$\begin{array}{rcl} \frac{\Delta s}{2} & = & a \; \log \, (1 + h/a ) \\ & \approx & h - \frac{h^2}{2a} + O(h^3) \end{array}$$
But the radar distance from $O_\hbox{hi}$ to $O_\hbox{low}$ (measured by $O_\hbox{hi}$) is
$$\begin{array}{rcl} \frac{\Delta s}{2} & = & (a + h) \; \log \, (1 + h/a ) \\ & \approx & h + \frac{h^2}{2a} + O(h^3) \end{array}$$
This shows (and its obvious once you see it) that radar distance is not symmetric, but this issue only shows up at second order in h, where h is the pedometer distance in the hyperslice t=0 which would, naively, be measured by an observer who very slowly walks from $x=a+h$ to $x=a$ along $y=z=0$.

A similar computation shows that radar distance is not additive, even for "colinear" Rindler observers.

Figures (left to right):
• Radar distance is not symmetric

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4. Sep 13, 2010

### Chris Hillman

BRS: Those Damnable "Paradoxes". III. Langevin congruence

Slightly modifying our previous notation, let us write the line element of Minkowski vacuum in the cylindrical chart as
$$\begin{array}{rcl} ds^2 & = & -dt^2 + dz^2 + dr^2 + r^2 \, d\phi^2, \\ && -\infty < t, \, z < \infty, \; 0 < r < \infty, \; -\pi < \phi < \pi \end{array}$$
Via Noetherian magic or other means, one can show that affinely parameterized null geodesics satisfy
$$\begin{array}{rcl} \dot{t} & = & E \\ \dot{z} & =& A \\ \dot{r} & = & \pm \sqrt{E^2 - A^2 - L^2/r^2} \\ \dot{\phi} & = & L/r^2 \end{array}$$
for some choice of the constants (invariants of motion) E, A, L.

We can write a frame suitable for studying static observers as follows:
$$\begin{array}{rcl} \vec{g}_1 & = & \partial_t \\ \vec{g}_2 & = & \partial_z \\ \vec{g}_3 & = & \partial_r \\ \vec{g}_4 & = & \frac{1}{r} \; \partial_\phi \end{array}$$
Let us boost this static frame, at each event, in the $\vec{g}_4$ direction, by an undetermined amount depending only on r. This will give the frame appropriate for studying a congruence of observers whose world lines form helices in the cylindrical chart. For a particular choice of the r dependence, this congruence has vanishing expansion tensor and thus corresponds to a family of rigidly rotating observers. The result is the Langevin frame
$$\begin{array}{rcl} \vec{h}_1 & = & \frac{1}{\sqrt{1-a^2 \; r^2}} \; \left( \partial_t + a \; \partial_\phi \right) \\ \vec{h}_2 & = & \partial_z \\ \vec{h}_3 & = & \partial_r \\ \vec{h}_4 & = & \frac{1}{\sqrt{1-a^2 \, r^2}} \; \left( a \, r \; \partial_t + \frac{1}{r} \; \partial_\phi \right) \end{array}$$
where a > 0. Notice this frame is only defined on the ("solid cylinder") region $0 < r < a$. (But if you have sharp eyes you might have noticed that the frame makes perfect sense even at the coordinate singularity $\phi = \pm \pi$ in our cylindrical chart; to verify this, rewrite the Langevin frame in the Cartesian chart.)

The integral curves of $\vec{h}_1$ can be written as proper time parameterized curves (coordinate helices)
$$\begin{array}{rcl} t & = & t_0 + \frac{s}{\sqrt{1-a^2 \, r_0^2}} \\ z & = & z_0 \\ r & = & r_0 \\ \phi & = & \phi_0 + \frac{as}{\sqrt{1-a^2 \, r_0^2}} \end{array}$$
Notice that $d\phi/dt = a$ for all these integral curves, they are helices of uniform pitch. These helices form the Langevin congruence, and we say that they are the world lines of a family of Langevin observers. The Langevin congruence has the following geometric characteristics:
• the acceleration vector is
$$\nabla_{\vec{h}_1} \vec{h}_1 = \frac{-a^2 r}{1-a^2 \, r^2} \; \vec{h}_3$$
which is "purely radial" and pointing inward, so that each Langevin observer must accelerate radially inward, with constant magnitude of acceleration, in order to maintain his circular orbit around r=0,
• different Langevin observers can have different (constant) accelerations; the magnitude of acceleration depends on r and it diverges as $r \rightarrow a^-$,
• the expansion tensor vanishes (by construction), so that the Langevin congruence is rigid,
• the vorticity tensor is nonzero; equivalently, the vorticity vector is
$$\vec{\Omega} = \frac{a}{1-a^2 \, r^2} \; \vec{h}_2$$
which is "purely axial", so that the world lines of nearby Langevin observers are twisting about each other in a sense aligned with the axis r=0,
• because the vorticity tensor is nonzero, the Langevin congruence is not hypersurface orthogonal; there exists no family of hyperslices in Minkowski vacuum which are everywhere orthogonal to the world lines of the Langevin observers,
• the Fermi derivatives along $\vec{h}_1$ of $\vec{h}_2, \; \vec{h}_3, \; \vec{h}_4$ shows that the Langevin frame is spinning about $\vec{h}_2$ (parallel to the axis r=0); by introducing a suitable secular spin rate depending only on r, we can "despin" this frame, giving a frame which is non-spinning but non-inertial,
• although the Langevin congruence admits no family of orthogonal hyperslices, each world line in the congruence is carried by some rotation about r=0 to another such, and they are all equivalent in the sense of having identical path curvature and torsions (there are four-dimensional and Lorentzian generalizations of curve theory on E^3), and we can therefore form the quotient manifold, a Riemannian three-manifold whose points correspond to the world lines of our observers (you can think of collapsing each one-dimensional curve to a point).
Quotient manifolds are tricky: if you try making one from any old congruence of integral curves of some vector field, if you can even construct a quotient manifold at all, you will almost certainly come up with a manifold which has utterly horrid topology (see Appendix A in Massey, Algebraic Topology: an Introduction, Springer, 1967). It's just our good fortune that this works out for the Langevin congruence.

The result is (with the inessential z coordinate suppressed) the Einstein disk
$$ds^2 = \frac{dr^2}{(1+a^2 \, r^2)^3} + r^2 \, d\phi^2$$
which has Gaussian curvature
$$R_{1212} = -3 a^2 \; ( 1 + a^2 \, r^2)^2$$
which approximates H^2 near r=0, as Einstein guessed would happen as early as 1908. But
• this must be understood as a quotient manifold, specifically the quotient of Minkowski spacetime by the Langevin congruence (right!),
• not as "the physical metric of a rigidly rotating disk" (wrong!)
(Compare the discussion in Landau and Lifschitz, Classical Theory of Fields, Pergamon, section 84.)

Incidently, the despun Langevin frame turns out to be
$$\begin{array}{rcl} \vec{b}_1 & = & \frac{1}{\sqrt{1-a^2 r^2}} \; \left( \partial_t + a \; \partial_\phi \right) \\ \vec{b}_2 & = & \partial_z \\ \vec{b}_3 & = & - \sin \left( \frac{at}{\sqrt{1-a^2 r^2}} \right) \; \frac{1}{\sqrt{1-a^2 r^2}} \; \left( ar \; \partial_z + \frac{1}{r} \; \partial_\phi \right) + \cos \left( \frac{at}{\sqrt{1-a^2 r^2}} \right) \; \partial_r \\ \vec{b}_4 & = & \cos \left( \frac{at}{\sqrt{1-a^2 r^2}} \right) \; \frac{1}{\sqrt{1-a^2 r^2}} \; \left( ar \; \partial_z + \frac{1}{r} \; \partial_\phi \right) + \sin \left( \frac{at}{\sqrt{1-a^2 r^2}} \right) \; \partial_r \end{array}$$
Note the spin rate of this frame wrt the original Langevin frame is $a/\sqrt{1-a^2 r^2}}$, and this frame is itself spinning wrt "the distant fixed stars" at the rate a. The difference
$$\frac{a}{\sqrt{1-a^2 r^2}} - a \approx \frac{a^3 r^2}{2}$$
is the Thomas precession. (A more straightforward derivation: consider parallel transport around a small loop in the velocity space of Minkowski space.)

Earlier, we "straightened out" the world lines of the Rindler observers (which are coordinate hyperbolas in the Cartesian chart) in order to construct the Rindler chart. Here, we can "straighten out" the world lines of the Langevin observers (which are coordinate helices in the Cartesian chart) in order to construct a new chart, the Born chart. This is obtained from the cylindrical chart very simply, by putting
$$\bar{\phi} = \phi - a t$$
The line element becomes
$$ds^2 = -(1-a^2 r^2) \, dt^2 + 2 a \, r^2 \, dt \, d\bar{\phi} + dz^2 + dr^2 + r^2 \; d\bar{\phi}^2$$
(note that these are not "orthogonal" coordinates). The Langevin frame becomes
$$\begin{array}{rcl} \vec{h}_1 & = & \frac{1}{\sqrt{1-a^2 \; r^2}} \; \partial_t \\ \vec{h}_2 & = & \partial_z \\ \vec{h}_3 & = & \partial_r \\ \vec{h}_4 & = & \frac{a \, r}{\sqrt{1-a^2 \, r^2}} \; \partial_t + \sqrt{1-a^2 \, r^2} \; \frac{1}{r} \; \partial_{\bar{\phi}} \end{array}$$
Null geodesics satisfy
$$\begin{array}{rcl} \dot{t} & = & E \\ \dot{z} & = & A \\ \dot{r} & = & \pm \sqrt{E^2-A^2-L^2/r^2} \\ \dot{\bar{\phi}} & = & L/r^2 - a E \end{array}$$

Last edited: Sep 13, 2010
5. Sep 13, 2010

### Fredrik

Staff Emeritus
Re: BRS: Those Damnable "Paradoxes". I. Overview

I would say that it's much easier than that. First consider the following version of Bell's spaceship paradox: (I won't describe the string; you know that part). Let S be an inertial frame. Unless I say otherwise, all the coordinates I mention are components of S. Consider two rockets that at t=0 are identical in every way, except for being located at different x coordinates. The coordinate distance between them at t=0 is L. Each rocket is equipped with a clock and a computer that controls the engine. The rockets are at rest for all t<0. At t=0, the engines are switched on, and at some later time they are switched off.

The requirement that the rockets are identical implies that the world lines of the rockets are identical too, except for their positions. (The clocks are synchronized and the computers are running the same programs; otherwise the rockets wouldn't be identical). More precisely, if $A:\mathbb R\rightarrow M$ and $B:\mathbb R\rightarrow M$ are the world lines, and $S:M\rightarrow\mathbb R^2$ is the coordinate system, we have $S\circ B(\tau)=S\circ A(\tau)+(0,L)$. The spacetime diagram associated with S looks roughly like the image I have uploaded. The coordinate distance between the world lines is =L at all times (even though I wasn't able to get that exactly right in Paint). The lines I drew above the IRRELEVANT box just show one example of what the world lines might look like after the engines are shut off. They must be timelike geodesics; they must start at events that are timelike separated from the events where they engines are switched on (at t=0); they must be a coordinate distance L apart (at each t above the box). But apart from that, there are no restrictions on what they can look like.

Now consider a simultaneity line of the comoving inertial frame at an event E on A's world line above the box. Let F be the event where it intersects B's world line. The length contraction formula tells us that the distance in the comoving inertial frame between E and F is gamma*L. So a string that's at rest in this inertial frame (in which both rockets are at rest), can't be attached to both rockets unless its length is at least gamma*L>L. The string we attached had rest length L and couldn't be stretched to a longer rest length, so it must have snapped.

Now consider the scenario with constant proper acceleration that goes on forever. How would that change the spacetime diagram? We only need a small part of the answer: It would bend B's world line to the right. That would make the simultaneity line longer than in the first scenario, so the string must break in this scenario too.

This is a complete resolution of both the scenario with constant proper acceleration that goes on forever, and a scenario with an arbitrary acceleration pattern, and it doesn't require the reader to understand frame fields, the expansion tensor, or any other techniques based on differential geometry. He/she only has to understand simultaneity and length contraction.

The text should say "simultaneity line of comoving inertial frame"

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6. Sep 13, 2010

### bcrowell

Staff Emeritus
Re: BRS: Those Damnable "Paradoxes". I. Overview

The trouble with using sophisticated mathematics to resolve these paradoxes, in one of the public forums on PF, is that at least half of the participants (and possibly 95% of them) won't be able to follow the math, so they'll just go on creating confusion. Also, some people who have the math background have been known to misapply or misinterpret the fancy mathematics.

One of the reasons I love the Weiss and Baez usenet faqs is that they often show how the same problem can be attacked with lowbrow mathematical methods, and then with more highfalutin' ones.

It's not necessarily a bad thing to have a 300-post thread on the Ehrenfest paradox every couple of months or so. I participated in one of those, and it helped me to wrestle with the ideas and come to a better understanding of them. Sure, it was redundant with previous threads on the same topic -- but it wasn't redundant *for me*, because I was participating in it.

That is a very useful paper, but it is not freely available online. There are PF users who don't have access to journals. (E.g., they may live in a rural area far from any university library where they could walk into the stacks.)

Last edited: Sep 13, 2010
7. Sep 14, 2010

### Staff: Mentor

Re: BRS: Those Damnable "Paradoxes". II. Rindler vs. Bell Congruences

Can you demonstrate the third point? The one about the expansion tensor.

8. Sep 15, 2010

### Chris Hillman

Hi all,

Only DaleSpam's post was along the lines I was expecting, so I may need some time to pick myself off the floor.

That's a hallmark of Baez's posts and something I much admire, so I'd encourage SA/Ms to try to emulate that aspect in posts in the public areas.

But this requires mastery on the part of the author of all relevant material, which is something I am hoping to facillate here!

I must demur however on one point: I don't know if you meant to characterize expansion and vorticity as "sophisticated", but you referred to

Just to make my position clear: if $\vec{X}$ is a timelike vector field defining a timelike congruence, the expansion tensor is the projection via the projection operator $h_{ab} = g_{ab} + X_a X_b$ to the orthogonal hyperplane element of the symmetrized covariant derivative $X_{(m:n)}$:
$${H\left[ \vec{X} \right]}_{ab} = {h^m}_a \; {h^n}_b \; X_{(m;n)}$$
That's really no more "sophisicated" than concepts like the Riemann curvature tensor or other tensorial objects which are often discussed in PF public areas!

I would agree that expansion and vorticity appear to be unfamiliar to most regulars, and I hope to change that here!

My hope is to raise the level of understanding of SA/Ms who regularly post in the relativity forum, in the belief that benefits will ultimately accrue in the public areas.

Did you have anyone specifically in mind?

Not sure what your point is, since I was talking about authors of arXiv eprints who have a responsibility (as I see it) to perform a literature search and obtain/study the most important papers (including all relevant review papers!) well before they start writing their own paper.

I agree that when citing review papers it is preferable to cite ones which are available on-line, but in this case there is apparently none available on-line, so it makes sense to cite the most important published review paper, if only to emphasize that a large literature had already accumulated by 1975.

A much more recent updated review by Gron appeared as a chapter in a book (c. 2003?) which was briefly on-line but appears to have been taken down.

I don't want to get into some philosophical argument, but as you may know, many mathematicians have a strong liking for allegedly "elementary proofs" which avoid sophisticated methods. The trouble is, if you compare say an "elementary proof" of the Prime Number Theorem with the "sophistiated proof" of Hadamard, the latter is much shorter and easier to understand. That is, in such cases, it is usually easier to learn whatever is needed to understand the "sophisticated proof" plus that proof than to read an "elementary proof", and almost always much more efficient, in the sense that the "sophisticated" background will be useful for many other purposes.

In the case of something like the Bell "paradox", to repeat, the expansion tensor is precisely the right mathematical tool for the job, and it is equally useful in many other gtr-related discussions (e.g. almost anything in cosmology). This is why it is so important, in my view, for SA/Ms who regularly post in the relativity, cosmology, and astrophysics sections to be familiar with expansion and vorticity.

Hope that clarifies why I am urging SA/Ms to learn about the fun, not particularly sophistiated, and remarkably useful kinematic decomposition. I hope that anyone who has never given it a try will do so!

Yes, absolutely! I'll try to get to it tommorrow. I plan try to try to clearly motivate/explain the definitions and to write out the computations in arbitrary detail, which is pretty ambitious, so I should start when I am fresh.

9. Sep 15, 2010

### Fredrik

Staff Emeritus

I'm surprised that you're surprised. I felt that my post was necessary because of all those PMs you sent me in February where you insisted that it isn't possible to prove that the string breaks without the fancy techniques.

This is true. However, most PF users won't understand any argument that requires some knowledge of differential geometry. I also think you underestimate how difficult it is for me and the other SAs to learn these techniques. Last time we talked about this, I had to spend a considerable amount of time refreshing my memory about covariant derivatives and other stuff that I didn't know as well as I should. It took me several hours just to verify a comment in Wald about a reparametrization of the curves in the congruence. I'm not saying that that time wasn't well spent, but I am saying that these things aren't nearly as simple as you are suggesting.

I share that view about many things. For example, I have often said that I can teach someone special relativity and the basics of linear algebra in less time than I can teach the same person just special relativity. However, in the case of Bell's spaceship paradox, I think there are lots of people who can understand my argument perfectly in 15 minutes, but would need 15 days to really understand yours. I don't doubt that there are good reasons to study this stuff, but wanting to understand why the string breaks isn't one of them.

I'm interested in learning it, mainly because I want to see and understand a coordinate independent definition of rigid motion. I'm going to take another look at the definitions today, and see if I can't understand them this time. (I feel like I did 90% of the work last time, so I expect it to be easier now). I will focus on understanding the definition of the expansion tensor, and on why it can be interpreted as a measure of non-rigidity.

I probably won't use it to dig deeper into either Bell or Ehrenfest. Another thing I probably won't do is to try to read Poisson. I have tried a couple of times, but I think it's extremely badly written. I often can't make any sense at all of what he's saying. That book just makes me angry (at the author), so I'll try Wald first, and check out Hawking & Ellis if I need more.

10. Sep 15, 2010

### bcrowell

Staff Emeritus

11. Sep 15, 2010

### bcrowell

Staff Emeritus

It could be me, on a bad day. My motivation for going back and really digging into GR recently was that I had never been satisfied with the level of conceptual understanding I'd achieved in the one-semester graduate course I took. As a grad student, there were a lot of times when I needed to get my field theory and GR problem sets done, so I just cranked out the calculations, without feeling good about really understanding in detail what they *meant*. Anyone who has taught freshman mechanics has seen something similar happen over and over, at a more elementary level. E.g., students use calculus to analyze velocity and acceleration, but conceptually they still believe that a force is necessary in order to produce a velocity. They know ten times the math that Galileo knew, but they have failed to absorb the meaning of the math. Any time you want to get really depressed, try sketching a wiggly graph of x(t) for someone who has finished a year of calculus and asking them to sketch v(t). I assign these exercises every semester, and in a class of 25 students I typically get none at all who can do it without extensive remediation.

I'm glad that you've motivated me to dig into these ideas. For instance, I found it interesting to try to figure out if they helped in the case of the angularly accelerating one-dimensional Born-rigid ruler. I think the answer was that they were the wrong tool for that job, but coming to that conclusion was a good example of the process one has to go through to learn new techniques and really understand what they mean.

12. Sep 16, 2010

### Chris Hillman

Re: BRS: Those Damnable "Paradoxes". I. Overview

Sorry, Fredrik, I seem to have a chronic disk space problem so I delete frequently and I had completely forgotten whatever happened in February, and I take the point about not all SA/Ms having time to master or relearn covariant derivatives and all that. I hope that in this thread we can make this easier for you to relearn and use (particularly if you install and use Ctensor under Maxima, probably with some custom scripts, or even better GRTensorII under Maple), but since Ben Crowell reminded us of one of the most impressive features of John Baez's expositions (starting with very elementary if not always fully precise ideas and gradually motivating/introducing a much more sophisticated viewpoint), perhaps in this thread we can collectively work out some elementary approach which we can all agree is "good enough" for some purposes in answering PF queries.

Just to stress the point, when I ranted about authors who haven't done a literature search and who haven't read such basic sources as the two reviews by Gron in my Post #1 above, I was talking about authors of (bad) arXiv eprints and papers published in conference proceedings and junk journals.

Oh, excellent! I guess all "old-timers" will agree that never mastered anything until they had to teach it in a formal course, or had some other reason to revisit it a year or more after their first exposure. So this is indeed a great opportunity for us all, since I have always found that no matter how much one knows (or thinks one knows), there is always more to learn and to try to share. Your twisting filament is for example new to me, I think, so I'll have to think about it...

I won't have time until later, but in case anyone wants to jump ahead, here's the first thing I plan to try: study the explicit world lines of a specific example in Minkowski vacuum. E.g. construct by hand some time-varying pitch "helical" world lines which form the world sheet of a conveniently "twisting filament", accelerations not otherwise specified, fit these into a convenient congruence, and compute the kinematic decomposition and see if my guess is correct in the example. Then think about what the results mean physically.

Last edited: Sep 16, 2010
13. Sep 18, 2010

### Chris Hillman

Trying again to entice SA/Ms to learn about expansion tensor & all that

Hi, Fredrik,

I am sorry that you had a (really really) bad experience reading Poisson's book, but if you are still up for it, I'd like to try to encourage you to try again to master the expansion and vorticity tensors, maybe using the much shorter discussion in Hawking & Ellis, coupled with working some computations. Another possibly useful resource are the Wikipedia articles I cited in the other BRS thread, in the versions I last edited.

Remember, you can always use coordinate bases for computations and re-express your answer in terms a frame field at the very end (using linear algebra for change of basis in each tangent space). This should help in checking against any results I state anywhere!

Yes, the whole point is that these quantitites are just what one needs to understand what a congruence is doing in coordinate-free, geometric terms. Very much analogous to how, in elementary curve theory in E^3, analyzing an arc length parameterized curve in terms of path curvature and torsion as functions of the parameter gives the geometric characteristics of how the curve is bending and twisting, which may not be obvious from a coordinate description. And it generalizes to other 3-manifolds where the problem of interpreting coordinates may become more acute. In both cases, it is helpful to start with flat manifolds because then you can draw and interpret pictures more easily, and move on later to curved manifolds.

I am pondering a kind of Schaum's Outline chapter in a BRS devoted just to timelike congruences, written to some extent with you and DaleSpam in mind, with some judiciously chosen worked examples in Minkowski spacetime and some other simple spacetimes, with the goal of trying to convey how to read off valuable physical insight from these quantities. This will require me to muster some energy and other resources, so bear with me.

This comment helped me to recognize a point I failed to bring out in my Post #1: I hope to use the obvious interest in the Bell and Ehrenfest "paradoxes" to try to motivate SA/Ms who haven't learned about acceleration, expansion, vorticity of a timelike congruence to do so. Because these are so important everywhere in gtr.

Last edited: Sep 18, 2010
14. Sep 18, 2010

### bcrowell

Staff Emeritus
Re: Trying again to entice SA/Ms to learn about expansion tensor & all that

Attempting to wade through Born's treatment of rigid motion, with equations in old-fashioned non-tensorial coordinate notation, was enough to convince me of how much nicer it is to work with a nice, concise modern definition. It's good to see that Hawking and Ellis have a treatment (p. 82) explicitly discusses the expansion tensor, not just the expansion scalar as Wald does. I think Fredrik's point was orthogonal to yours, however. It's great if some of us get hip to the expansion tensor, but that won't help with resolving "paradoxes" in the public forums.

15. Sep 18, 2010

### Chris Hillman

Can mastery of important techniques "trickle down" to physics grad students at PF?

Ben, you gladden my heart with this:

Fredrik, you probably already appreciate what I am trying to say, but let me restate it yet again: once you learn expansion and vorticity you will appreciate how much effort it saves you in all kinds of gtr-related discussions.

Also
I can't say how or when, but I can't help believing that the more SA/Ms are comfortable with expansion and vorticity, the less likely that some among the next generation of current students of gravitation physics will omit to learn this. I hypothesize that PF has a certain "reach" even into serious physics programs, but my evidence is anecdotal (points at marcus and some others who joined PF, I think, while enrolled in physics/astro Ph.D. programs.)

Last edited: Sep 18, 2010
16. Sep 18, 2010

### Fredrik

Staff Emeritus
Re: BRS: Those Damnable "Paradoxes". I. Overview

Thanks Chris. I've had at least 7 beers too many to look at it tonight (hm, it's 5 AM, so maybe I should say this morning), but I'll probably have a look at it Sunday evening. I have to spend 5 hours on a train anyway, so I might as well try to learn something.

17. Sep 18, 2010

### Chris Hillman

Re: BRS: Those Damnable "Paradoxes". I. Overview

And how many did Chronos have?

18. Sep 19, 2010

### Fredrik

Staff Emeritus
Re: BRS: Those Damnable "Paradoxes". I. Overview

Haha...I was wondering the same thing when I saw his post about studying general relativity at age 2.

I don't doubt that. I'm just not sure I will be involved in many discussions of that sort, since I'm more interested in other things. I am however interested enough to study the basics right now.

I'm using Wald as my main source. I have worked my way through the stuff on page 217 now, in both abstract index notation and coordinate independent notation. I also studied my notes about the stuff on page 46 (where the term "orthogonal displacement vector" is explained), so I think I understand all the calculations on page 217 now. But I still don't understand why $\theta$, $\sigma$ and $\omega$ can be interpreted as expansion, shear and vorticity, or why the first two being =0 is an appropriate definition of rigid motion.

By the way, in the other thread, you and bcrowell discussed the significance of the fact that Wald is talking about a congruence of geodesics instead of a general congruence. As far as I can tell, the only result on page 217 that only holds for geodesics is $B_{ab}\xi^b=0$, and it isn't used for anything later. Edit: I have now read George Jones's post in the public thread, where he says that Wald's definition of $\omega$ is simplified by assumption that the curves are geodesics.

Last edited: Sep 19, 2010
19. Sep 19, 2010

### George Jones

Staff Emeritus
Re: BRS: Those Damnable "Paradoxes". I. Overview

I don't have either with me right now, so I could be wrong, but it seems to me that Hawking and Ellis does a better job of explaining this than does Wald. Some people (but maybe not Fredrik ) might like the explanation (for geodesic congruences) in Chapter 2 of

http://www.physics.uoguelph.ca/poisson/research/agr.pdf,

which is a late draft of Poisson's book.

20. Sep 19, 2010

### Chris Hillman

Re: BRS: Those Damnable "Paradoxes". I. Overview

I was interrupted by local circumstances, but hopefully have found a few minutes of peace to try to start work on the new BRS.

After reviewing some notes and rereading the appropriate sections of some books, I agree with George: Hawking and Ellis are probably clearer than Wald, and--- dare I say it?--- Poisson's approach is probably best of all. So I am planning to try to focus on reorganizing the material presented by Poisson in a way which clarifies (I hope) the distinction between notations which came up--- basically, for timelike geodesic congruences, some stuff is "already projected" to the normal hyperplaces, but for the general case (needed for rigid motions!), you do need to use the projection tensor.

I think I'll first try to get the definitions and basic computations down, using index gymnastics since that is more familiar, and trying to make some sketchy motivations using Lie dragging and such. Then if I can find time and energy to be careful not to make any silly mistakes, I might try to provide a guide to the notation in the books discussed, because they all do things a little differently. Then the really important part: discussing simple but nontrivial examples, with pictures, which I think is the best way to really understand what acceleration, expansion, shear and vorticity "really mean" in gtr.

Eventually I have in mind a similar approach for null geodesic congruences and the optical scalars.