What is the Effect of Gravity on Einstein's Train in Special Relativity?

[PeterDonis]

I'll re-check in Maxima, it's quite possible I made a transcription error.

I rechecked, and I did make an error (two, actually). When I correct them, I do get that the metric is flat--vanishing Riemann tensor.

 

[SlowThinker]

Eventually, no signal from the marble will be able to catch up to the accelerating rocket, or the block sliding around it's floor. I'm not sure if this is a sufficient answer, but it's the only one I have at the moment. This is a feature of so-called hyperbolic motion.

I'm aware of this; however I has hoping that it would be possible the other way.
So I used that: the passengers are shining at the marble, even if they can't see it. Obviously their math skills are nearly superhuman

 

[pervect]

In other words, you think it's ok to add extra terms that depend on $\tau$ to the formulas for $T$ and $X$, as long as those terms vanish at $\chi = \psi = 0$. I'll have to think about that, but I will raise a few initial concerns below.
To me, if we're going to physically interpret a chart as "rotating", we have to be able to say what is rotating relative to what. If we define a timelike congruence describing a family of observers, or points in an object, then "rotating" has a simple meaning: the congruence has nonzero vorticity, meaning that a given observer in the congruence sees neighboring observers rotating around him. But since you're only specifying one worldline--the one at your coordinates $\chi = \psi = 0$--I'm not clear on what congruence of worldlines you think describes the block (or the train), so I'm not sure what "rotating" is supposed to mean.

Let's consider a rotating disk. The "observer" would be the worldine at the center of the disk. Let's use $(T_d,X_d,Y_d)$ as the Minkowskii coordinates, and call the "rotating coordinates" $(\tau_d, \chi_d, \psi_d))$ Then we might write the congruence of worldlines by the relations

$$T_d = \tau_d \quad X_d = \chi_d \cos \omega \tau_d - \psi_d \sin \omega \tau_d \quad Y_d = \chi_d \sin \omega \tau_d + \psi_d \cos \omega \tau_d$$

Each pair of $\chi_d, \psi_d$ specifies a unqiue worldline, the collection of worldlines specified in this way form the desired congruence. But note that that only for the special "observer" worldline at the center of the disk $\chi_d = \psi_d = 0$ does $\tau_d$ equal proper time. For other members of the congruence $\tau_d$ is an affine parameter of the worldline, but it is not proper time. This does cause us various headaches when we want to compute the vorticity, etc But I'm not aware of any better/simpler way to describe a rotating congruence than above.

In the sliding block case,$\tau$ is analogous to $\tau_d$, $\psi$ is analogus to $\psi_d$ and $\chi$ is analogous to $\chi_d$. Specifying a value for $\psi$ and $\chi$ specifies a unique worldline. But $\tau$ is only proper time for the worldline at $\psi=\chi=0$. Along other worldlines, it's an affine parameter, but it's not proper time.

As an aside, it would probably be worthwhile to calculate the metric resulting from the "rotational transformation" described above. There's a webpage that does the former, but I haven't checked their results personally. http://physics.stackexchange.com/qu...to-a-relativistic-rotating-frame-of-reference. You mention the vorticity and shear, it would be worth calculating those, too.

But more than that, if this is supposed to be "the frame of the block", then it would seem like the points of the block ought to be at rest in the frame.

Points on the rotating disk are "at rest" in the rotating frame, in the sense that $\chi_d$ and $\psi_d$ are constant. In the same sense, points in the sliding block frame are constant in the sense that $\chi$ and $\psi$ are constant. None of the metric coefficients is a function of coordinate time, so the metric is stationary and has a timelike Killing vector field for both the rotating disk and the sliding block.

Certainly that's the case for Rindler coordinates: they don't describe the "viewpoint" of just one observer, they describe a common "viewpoint" of a whole family of observers, all of whom are at rest in the chart (constant spatial coordinates), and the variation in their physical measurements is entirely due to their different positions. But again, since you've only specified one worldline, I don't know if the other points of the block are supposed to be at rest in this chart or not.

I would describe the situation as using a reference worldline of an observer to build a congruence. The congruence is built from the reference worldline, but it's not the same as the reference worldline, it's a congruence.

One way to investigate these possible concerns would be to compute the kinematic decomposition for the congruence of worldlines that are at rest

I haven't attempted this - the main thing I'd watch out for is the fact that it's often assumed that the congruence is generated by a unit timelike vector field, but our time coordinate $\tau$ is a killing vector field, not a unit vector field.

 

[PeterDonis]

I'm not aware of any better/simpler way to describe a rotating congruence than above.

I agree this works for a rotating disk (and I agree with all of your statements and caveats about this example), but that's not what we're talking about here. We're talking about a train, or a sliding block, moving "horizontally" in an accelerating rocket. We shouldn't have to appeal to heuristics based on a rotating disk to specify a congruence for a different physical situation. We should be able to do it directly. See further comments below.

As an aside, it would probably be worthwhile to calculate the metric resulting from the "rotational transformation" described above.

I think that was done quite some time ago. See here:

https://en.wikipedia.org/wiki/Born_coordinates

In the sliding block case, $\tau$ is analogous to $\tau_d$, $\psi$ is analogus to $\psi_d$ and $\chi$ is analogous to $\chi_d$.

But whose $\tau$, $\psi$, and $\chi$? Yours, or mine? That's the question. It can't be both, because they're different--different coordinate transformations to/from Minkowski $T$, $X$, $Y$, and different resulting metrics. Both metrics describe Minkowski spacetime, but the congruences of observers at rest in those two charts are different congruences--different families of worldlines. (They must be, because the two 4-velocity fields derived from the two different transformations are different.) So which one is the right one to describe the block?

I'll defer further discussion of that question to a separate post, because I think computing the kinematic decomposition of the congruence of observers at rest in your chart will help to elucidate it. But see a couple of further comments on that below.

the main thing I'd watch out for is the fact that it's often assumed that the congruence is generated by a unit timelike vector field

That's not an assumption, it's the definition of a congruence. If the congruence is not a unit timelike vector field, the kinematic decomposition doesn't work.

our time coordinate $\tau$ is a killing vector field, not a unit vector field.

No, the coordinate $\tau$ generates both a KVF and a timelike unit vector field. The KVF is $\partial_{\tau}$. The unit timelike vector field is $\partial_{\tau} / | \partial_{\tau} |$, i.e., the KVF divided by its norm. (We also have to restrict the latter to the region in which $\partial_{\tau}$ is timelike, of course, which is not the entire spacetime.)

 

[PeterDonis]

I'll defer further discussion of that question to a separate post

Here is at least the start of the further discussion.

I figured out just now why I have had the feeling all through this thread that there was something nagging at me that I hadn't gotten out into the open. Let me approach it by recapping pervect's reasoning in the previous thread where he derived his "sliding block" metric, since this will establish fixed points on which we both agree.

We are considering a block sliding along the floor of a rocket that is accelerating. Here I'll continue my coordinate conventions, which are that, in a fixed inertial frame with Minkowski coordinates $T, X, Y, Z$, the floor of the rocket has proper acceleration $g$ in the $X$ direction, and the block is sliding, relative to the rocket, in the $Y$ direction. We are for now only considering the bottom of the block, i.e., the surface of the block that is in contact with the floor of the rocket. In other words, we are only looking at events on the hyperbolic "worldsheet" defined by $X^2 - T^2 = 1 / g^2$ (but with the $Y$ coordinate unconstrained); note that I am using my convention for this, not pervect's, which shifts the $X$ coordinate by $1/g$ so that the floor of the rocket is at $X = 0$ at $T = 0$, instead of $X = 1/g$.

In order to specify the motion of the floor of the block, we have to determine what the constraint is. Pervect's constraint, which I agree with, is that, in the fixed inertial frame, the momentum of the block in the $Y$ direction should be constant. This means the ordinary velocity of the block in the $Y$ direction, in the fixed inertial frame, will decrease with time, for reasons which pervect explained in the previous thread (and which I agree with). Pervect expressed this constraint using a constant $K = dY / d\tau$ (where I have switched to $Y$ instead of x and capitalized it to agree with the coordinate convention I'm using, where capital letters refer to inertial coordinates).

I expressed this constraint somewhat differently. My approach was to observe that, in the momentarily comoving inertial frame of the floor of the rocket, the ordinary velocity of the block in the $Y'$ direction (we'll use a prime for the MCIF to distinguish it from the fixed inertial frame above) will also be constant; I called this constant velocity $v$. We can easily show that my constraint is equivalent to pervect's constraint. In the MCIF, we have $\partial T' / \partial \tau = \gamma$, where $\tau$ is the proper time of the block and $\gamma = 1 / \sqrt{1 - v^2}$, and by the chain rule we have $\partial Y' / \partial \tau = ( \partial T' / \partial \tau ) \partial Y' / \partial T' = \gamma \partial Y' / \partial T' = \gamma v$. Finally, since $Y = Y'$, because the MCIF and the fixed inertial frame only differ by a boost in the $X'$ direction, we have $K = \partial Y / \partial \tau = \partial Y' / \partial \tau = \gamma \partial Y' / \partial T' = \gamma v$. So a constant $K$ implies a constant $v$ and vice versa.

Combining all this gives the 4-velocity of the block, expressed in the Minkowski coordinates of the fixed inertial frame (note that this is somewhat different from the expressions I frequently used before, since I have replaced the $\cosh$ and $\sinh$ functions, which did not have Minkowski coordinates as arguments, with their equivalent functions of Minkowski coordinates, so $\cosh g \gamma \tau = g X$ and $\sinh g \gamma \tau = g T$):

$$
U = \gamma g X \partial_T + \gamma g T \partial_X + \gamma v \partial_Y
$$

So far we are all in agreement. But now comes the key point, which was nagging at me before. Everything we've done so far only applies to the floor of the rocket, and to the bottom of the block. But pervect and I have both proposed metrics for the "block frame", and those metrics are not restricted to the floor of the rocket and the bottom of the block. They include the $\chi$ direction, which is the direction of proper acceleration, and is orthogonal to the $\tau \psi$ plane that describes the floor of the rocket and the bottom of the block. So we need to make some sort of assumption about what happens in that direction.

To put the point another way, consider the top surface of the block. It is separated, in the $X$, $X'$, or $\chi$ direction, from the bottom surface of the block and the floor of the rocket. What does the motion of this surface of the block look like, to an observer who is at rest relative to the bottom of the block?

For the rocket, we know the answer to this question: the top of the rocket is at a fixed "ruler distance" from the bottom (which is just the difference in their Rindler $x$ coordinates--note that this is not the same as the difference in their Minkowski $X$ coordinates, which decreases with time), and has less proper acceleration; the latter varies inversely with $x$. So we can describe the 4-velocity of the top of the rocket just as easily as we can that of the bottom. In Minkowski coordinates, it is

$$
U_{r} = \frac{gX}{g \sqrt{X^2 - T^2}} \partial_T + \frac{gT}{g \sqrt{X^2 - T^2}} \partial_X
$$

I have included the explicit factors of $g$ to make clear the point: we cannot just take the 4-velocity of the floor of the rocket, which would be $\sqrt{1 + g^2 T^2} \partial_T + g T \partial_X$, and "extend" it to the rest of the rocket and assume that it applies. In order to properly specify the congruence of worldlines that describes the rocket, we must look at the conditions that the congruence must satisfy, and find a 4-velocity field that satisfies those conditions. The 4-velocity $U_r$ that I wrote above does this for the rocket: as is easily verified by computation, the congruence $U_r$ has zero expansion, shear, and vorticity, and its proper acceleration varies with "altitude" (meaning, with $\sqrt{X^2 - T^2}$, since that is the constant that labels each worldline in the congruence--in Rindler coordinates it is just $x$), whereas the proper acceleration of the congruence $\sqrt{1 + g^2 T^2} \partial_T + g T \partial_X$ does not--it is always $g$. (This latter congruence, btw, is easily seen to be the "Bell congruence", which plays a key role in the Bell spaceship paradox, and of course it is known to have nonzero expansion, which is why the string in the Bell spaceship paradox eventually breaks.)

Once we find the 4-velocity field $U_r$ that describes the rocket, we then need only find a coordinate chart in which integral curves of $U_r$ have constant spatial coordinates, i.e., we want a chart $t, x, y, z$ in which $U_r = \partial_t / | \partial_t |$. This chart will be our desired "rocket frame", a non-inertial frame in which the rocket is at rest. (Of course this chart turns out to be the Rindler chart.) Similarly, once we have found a 4-velocity field $U_b$ that satisfies the conditions for a congruence describing the sliding block, we need only find a chart $\tau, \chi, \psi, z$ in which the integral curves of $U_b$ are given by $U_b = \partial_{\tau} / | \partial_{\tau} |$, and that will be our desired "block frame", a non-inertial frame in which the block is at rest.

The claim I have been making in this thread can now be stated very simply: the 4-velocity field that I presented in previous posts is the desired $U_r$, and the chart I derived in which its integral curves have constant spatial coordinates, is the desired chart for the "block frame" as described above. The one reservation I had, about the congruence having nonzero shear, I have now resolved; I rechecked my computation of the shear and found that I had made a mistake. The shear is actually zero. The vorticity is still nonzero, but that's OK; we have been in agreement that nonzero vorticity is to be expected (though I think the question of what, exactly, it means physically still deserves some discussion). In the notation I have been using in this post, that 4-velocity field is

$$
U_{b} = \frac{gX}{\sqrt{g^2 \left( X^2 - T^2 \right) - v^2}} \partial_T + \frac{gT}{\sqrt{g^2 \left( X^2 - T^2 \right) - v^2}} \partial_X + \frac{v}{\sqrt{g^2 \left( X^2 - T^2 \right) - v^2}} \partial_Y
$$

Conversely, the implied 4-velocity field pervect has been using, which corresponds to taking the 4-velocity field $U = \gamma g X \partial_T + \gamma g T \partial_X + \gamma v \partial_Y$, which is valid at the bottom of the block, and extending it over all of the block, does not work. That should be evident by comparison with the Bell congruence vs. the Rindler congruence above; my 4-velocity field $U_b$ corresponds to $U_r$ above, and pervect's 4-velocity field, which works out to $U = \gamma \sqrt{1 + g^2 T^2} \partial_T + \gamma g T \partial_X + \gamma v \partial_Y$, corresponds to the Bell congruence $U = \sqrt{1 + g^2 T^2} \partial_T + g T \partial_X$ above. So pervect's chart, in which integral curves of his 4-velocity field $U$ have constant spatial coordinates, corresponds to a "Bell chart" in which integral curves of the Bell congruence have constant spatial coordinates. I predict, therefore, that if we can compute the kinematic decomposition of pervect's congruence, we will find that it has nonzero expansion. In other words, objects at rest in his chart will not remain at a constant proper distance from each other; they will "move apart" with time.

One final note: what about the condition that $dY / d\tau$ must be constant along a given block worldline? The formula for $U_b$ above certainly does not make that evident. However, we can see that it is still true by noting that the denominator of all the terms in $U_b$, $\sqrt{g^2 \left( X^2 - T^2 \right) - v^2}$, is in fact constant along each integral curve of $U_b$. So we do in fact have $dY / d\tau$ constant on each worldline. But we do not have $d Y / d\tau = \gamma v$ along each worldline; that is only true on the bottom of the block, where the denominator of each term in $U_b$ becomes $\sqrt{1 - v^2}$, so the last term does become $\gamma v$. But at the top of the block, the denominator is larger, so we have $dY / d\tau < \gamma v$.

What does this mean? It is just a consequence of the fact that, at the top of the block, time "flows" faster than at the bottom. What does this mean for the original constraint we imposed? What is now held constant? The answer is that, in the MCIF of the floor of the rocket, the top of the block should have the same ordinary velocity $v$ in the $Y'$ direction as the bottom does. But since the top of the rocket/block has faster "time flow" than the bottom, that means that $dY / d\tau$ at the top must be less than $dY / d\tau$ at the bottom, which was equal to $\gamma v$. And that is what we see in $U_b$. And, furthermore, we do not see this in pervect's 4-velocity field, where we have $U^Y = \gamma v$ at the top of the rocket as well as at the bottom--with this 4-velocity field, the top of the block will actually be moving faster, in the MCIF of the bottom of the rocket, than the bottom is! (That was the other thing that was nagging at me all through this thread.)

Sorry for the long post to add to all the other long posts in this thread. But I wanted to get all this out while it was fresh in my mind.

 

[PeterDonis]

I rechecked my computation of the shear and found that I had made a mistake. The shear is actually zero.

Just for completeness, here are the corrected results for the vorticity (the shear, as I said, is zero):

$$
\omega_{TX} = - \omega_{XT} = \frac{g v^2}{\left[ g^2 \left( X^2 - T^2 \right) - v^2 \right]^{3/2}}
$$

$$
\omega_{TY} = - \omega_{XT} = - \frac{g^2 v T}{\left[ g^2 \left( X^2 - T^2 \right) - v^2 \right]^{3/2}}
$$

$$
\omega_{XY} = - \omega_{YX} = \frac{g^2 v X}{\left[ g^2 \left( X^2 - T^2 \right) - v^2 \right]^{3/2}}
$$

 

[PeterDonis]

we cannot just take the 4-velocity of the floor of the rocket, which would be $\sqrt{1 + g^2 T^2} \partial_T + g T \partial_X$, and "extend" it to the rest of the rocket

It may be worth expanding on why the "extended" 4-velocity of the floor of the rocket is the formula given in the quote above--i.e., why we cannot just say $U = g X \partial_T + g T \partial_X$ everywhere in the rocket, not just on the floor. The answer is that this $U$ is not a unit vector except on the floor of the rocket. So we have to normalize it somehow. But there are at least two ways to do that.

One way is to take $g X \partial_T + g T \partial_X$ and simply divide it by its norm, which is $g \sqrt{X^2 - T^2}$. That way gives the 4-velocity field $U_r$, i.e., the Rindler congruence. But $U_r$ does not have the same ordinary velocity in the $X$ direction, in the fixed inertial frame, for all worldlines; the ordinary velocity in the $X$ direction is $T / X$, which varies from worldline to worldline. So in the fixed inertial frame, different worldlines in this congruence will be moving at different speeds, at a given time $T$.

The other way is to keep the $X$ component of $U$ constant, and adjust the $T$ component so it is a unit vector. That way gives the 4-velocity field $U = \sqrt{1 + g^2 T^2} \partial_T + g T \partial_X$. This congruence has an ordinary velocity in the $X$ direction of $T / \sqrt{1 + g^2 T^2}$, which is a function of $T$ only, so at a given time $T$ in the fixed inertial frame, every worldline in the congruence is moving at the same speed.

The question is which of the above ways is the "right" way to "extend" the 4-velocity of the floor of the rocket to other altitudes in the rocket. And the counterintuitive thing that this question brings to the fore is that in SR, unlike in Newtonian mechanics, we have to choose which of two desirable properties we want the congruence describing the rocket to have: either the proper distance between the worldlines can be constant, or each worldline can have the same speed in a fixed inertial frame. There is no congruence that has both properties; only the Rindler congruence has the first property, and only the Bell congruence has the second.

The same issue arises in the case of the sliding block, and I believe I have shown that in this case, the 4-velocity field $U_b$, and the metric I derived from it, is the one corresponding to the Rindler congruence in the example above--i.e., the one in which the proper distance between the worldlines in the block is constant. Conversely, I believe that pervect's metric is the one in which the worldlines at rest in the metric correspond to the Bell congruence in the example above--i.e., the one in which each worldline has the same speed at a given time in a fixed inertial frame. So which one is the "right" one depends on which property you think is the right one for the "block frame" to have.

 

[pervect]

I'll take the opportunity to recap a short summary of the procedure by which the transforms were created, one that I edited once before (I did the edits the next day, so they might have been overlooked), and I will edit even more now in an effort to be as clear as I can.

In vector notation we are just saying that the position pointed to by the (spatial) block coordinates $(\chi, \psi)$ at time $\tau$ is equal to $\vec{P_0 }+ \chi \vec{e_\chi} + \psi \vec{e_\psi}$. Here $\vec{P_0}$ is the position of the point-like origin of the block at time $\tau$, $\chi$ and $\psi$ are coordinate values, and $\vec{e_\chi}$ and $\vec{e_\psi}$ are basis vectors. $\vec{P_0}$ was obtained by integrating the 4-velocity of the pointlike origin of the block, U.

$\vec{e_\chi}$ and $\vec{e_\psi}$ are unit spatial vectors, and their components were found by Peter. The coordinate basis in which the components of these vectors were specified is the cartesian (T,X,Y) coordinate basis. I have replaced the numeric indices with symbolic ones, so that $\vec{e_\chi}$ is equivalent to $\vec{e_1}$ and $\vec{e_\psi}$ is equivalent to $\vec{e_2}$ in the original. $\vec{e_\chi}$ has 3 components, which we can write as $(e_\chi)^T, (e_\chi)^X, (e_\chi)^Y)$. $\vec{e_\psi}$ is similar. Hopefully using the symbolic rather than the numeric indices for these vectors will be helpful rather than confusing.

$\vec{e_\chi}$ and $\vec{e_\psi}$ lie entirely in a surface of constant $\tau$, so that in this surface, they are purely spatial vectors. However, due to the relativity of simultaneity, $(e_\chi)^T$ and $(e_\psi)^T$ are not zero, because T is not the same thing as $\tau$.

Multiplying these purely spatial basis vector by the spatial coordinate values gives us a spatial offset. We add this spatial offset to the spatial position of the block to find the cartesian coordinates pointed to by block coordinates $(\chi, \psi)$ at block time $\tau$.

This is the standard way in which we use a set of basis vectors (usually a triad, but for simplicity I've ignored z, so it's a pair of basis vectors) to find spatial coordinates in any affine space - it's not terribly exotic.

 

[PeterDonis]

the position pointed to by the (spatial) block coordinates $(\chi, \psi)$ at time $\tau$ is equal to $\vec{P_0 }+ \chi \vec{e_\chi} + \psi \vec{e_\psi}$.

This works as long as $\vec{e_{\chi}}$ and $\vec{e_{\psi}}$ are orthogonal to the worldline of $\vec{P_0}$. This is only true for the "reference" worldline that you are labeling as $\chi = \psi = 0$. It's not true for other values of $\chi$, and I'm not sure whether it's true even for $\chi =0$ for all values of $\psi$. I'm not sure if your procedure still works if this orthogonality condition is violated.

$\vec{e_\chi}$ and $\vec{e_\psi}$ are unit spatial vectors, and their components were found by Peter.

Yes, but $\vec{e_{\psi}}$ is not orthogonal to $\vec{e_{\tau}}$ everywhere--although $\vec{e_{\chi}}$ is orthogonal to the $\tau \psi$ plane everywhere, so that part is ok.

Also, while the method I used to find $\vec{e_\chi}$ is unambiguous--it's just picking a unit vector in the direction of the proper acceleration--the method I used to find $\vec{e_\psi}$ is not. There are multiple ways to "extend" my definition of $\vec{e_\psi}$ away from the coordinate values $\chi = \psi = 0$ (in your convention). The "canonical" way would be to use $\vec{e_\psi}$ at $\psi = 0$ to determine a spacelike geodesic, and then parallel transport $\vec{e_\psi}$ along that geodesic to other values of $\psi$. However, I don't know if using the formula I gave for $\vec{e_\psi}$ everywhere, not just at $\psi = 0$, satisfies that requirement.

Multiplying these purely spatial basis vector by the spatial coordinate values gives us a spatial offset. We add this spatial offset to the spatial position of the block to find the cartesian coordinates pointed to by block coordinates $(\chi, \psi)$ at block time $\tau$.

Does this still work if the basis vectors are functions of $\tau$? Perhaps it might help to express the basis vectors you are using purely in terms of Minkowski coordinates; I don't know if that has been done anywhere in this thread.

Also, it seems to me that there is something missing from the procedure as you describe it. What 4-velocity vector is supposed to correspond to a pair $(\chi, \psi)$, for pairs other than $\chi = \psi = 0$ (where we agree on what that vector is)? (This kind of gets into the worldline issue that I'll discuss further below.)

This is the standard way in which we use a set of basis vectors (usually a triad, but for simplicity I've ignored z, so it's a pair of basis vectors) to find spatial coordinates in any affine space - it's not terribly exotic.

I understand the procedure, but this leaves open a key question. The block is not just a single point. The whole point of a "block frame", as I said in a previous post, is to find a chart in which each point of the block is at rest, not just the center of the block. In other words, for some range of $\chi$ and $\psi$, each pair $(\chi, \psi)$ labels a unique, fixed point of the block. (This assumes that the set of worldlines so labeled forms a rigid congruence, but as I've shown in previous posts, there does exist a rigid congruence that can be held to describe the block.) So the question is, if we take the points labeled by a given pair $(\chi, \psi)$, for all values of $\tau$, to form a worldline, and consider the set of such worldlines for some range of $\chi$ and $\psi$, do they form a rigid congruence? If they do, then the coordinates $\tau, \chi, \psi$ can indeed be held to describe a "block frame". But if not, then they can't, even if the coordinates are derived using your procedure.

Furthermore, if I hand you a chart $\tau, \chi, \psi$ in which the set of worldlines at constant spatial positions in the chart, labeled by pairs $(\chi, \psi)$ for some range of $\chi$ and $\psi$, forms a rigid congruence, and whose 4-velocity matches the 4-velocity we have agreed describes the center of the block at $\chi = \psi = 0$, then it seems to me that this chart meets all the requirements for a "block frame", whether it was derived using your procedure or not. Given such a chart, one ought to be able to reconstruct it using your procedure, but that doesn't mean your procedure is the only way to get to such a chart. And I claim that the chart I have presented in this thread is exactly such a chart (with one change, that I'm labeling the center of the block by $\chi = 1/g)$ instead of $\chi = 0$).

 

[PeterDonis]

$\vec{P_0}$ was obtained by integrating the 4-velocity of the pointlike origin of the block, U.

I also expressed concern about this before: the integration you did, even if it is correct, is not unique. The coordinate functions that I obtained are also valid integrals of the 4-velocity I gave. (Note that our functions agree for the worldline at the center of the block; the non-uniqueness only comes in when we extend things to other worldlines.) So it seems to me that your procedure does not uniquely determine a coordinate chart. Which brings us back to the question I posed before: which chart is the "right" one to represent the "block frame"?

 

[pervect]

I also expressed concern about this before: the integration you did, even if it is correct, is not unique. The coordinate functions that I obtained are also valid integrals of the 4-velocity I gave. (Note that our functions agree for the worldline at the center of the block; the non-uniqueness only comes in when we extend things to other worldlines.) So it seems to me that your procedure does not uniquely determine a coordinate chart. Which brings us back to the question I posed before: which chart is the "right" one to represent the "block frame"?

Because the derivative of a constant function is zero, integration is never unique, two functions that differ by the same amount have the same derivatives, hence they are both integral of the same function.

When we specify an observer by specifying the observer's worldline, though, it's a matter of convention to say that the the spatial origin of the "observers frame" is at (0,0). When we follow this convention, we fix the value of the integration constant.

 

[pervect]

I'm aware of this; however I has hoping that it would be possible the other way.
So I used that: the passengers are shining at the marble, even if they can't see it. Obviously their math skills are nearly superhuman

I think I've got a better answer as to what happens to a laser pointed "ahead" of the rotating block, in the direction of it's 4-acceleration. The direction of the 4-acceleration (which we've dubbed $\vec{e_X}$) does not rotate with respect to the fixed stars in the Minkowskii frame, so the laser beam won't rotate with respect to the fixed stars either, and will always point in the same direction in the Minkowskii frame. To be really specific, the laser beam emitted from the sliding block will aberrate slightly due to the sliding motion of the block, but it won't rotate.

The way I would describe this in more ambiguous English is to say that the sliding block doesn't rotate relative to the fixed stars, even though it DOES rotate relative to a torque-free gyroscope that's carried along with the sliding block, as described in the paper I quoted much earlier in the thread.

Another way of saying this in ambiguous English - in the presence of frame dragging effects such as Thomas precession, gyroscopes may (and will) rotate with respect to the fixed stars, so the notion of "rotating with respect to the fixed stars" and the notion of "rotating relative to a gyroscope" are different.

My English description of the siutatuion is hopefully more understanadable, though I doubt it's unambiguous :(.

 

[PeterDonis]

Because the derivative of a constant function is zero, integration is never unique, two functions that differ by the same amount have the same derivatives, hence they are both integral of the same function.

When we specify an observer by specifying the observer's worldline, though, it's a matter of convention to say that the the spatial origin of the "observers frame" is at (0,0). When we follow this convention, we fix the value of the integration constant.

That isn't the issue. The issue is that you are integrating a function of multiple variables $(\tau, \chi, \psi)$ with respect to only one of them at a time. That means a "constant" might not be a function of nothing at all; it might be a function of the other variables. The same goes for functions that appear in the arguments of other functions. I pointed this out before, but let's look at it again.

The function we want to integrate has the form $\partial T / \partial \tau = \gamma \cosh \left( \gamma g \tau \right)$, $\partial T / \partial \chi = \sinh \left( \gamma g \tau \right)$, and $\partial T / \partial \psi = \gamma v \cosh \left( \gamma g \tau \right)$ when $\chi = \psi = 0$. Your ansatz for the integral is:

$$
T = \left( \frac{1}{g} + \chi \right) \sinh \left( \gamma g \tau \right) + \gamma v \psi \cosh \left( \gamma g \tau \right)
$$

My ansatz for the integral is:

$$
T = \left( \frac{1}{g} + \chi \right) \sinh \left( \gamma g \tau + \gamma g v \psi \right)
$$

Both of our proposed integrals give the same three partial derivatives at $\chi = \psi = 0$. So your ansatz for the integral is not unique. (Note that there are no integration constants in either of our formulas; as you say, fixing $\chi = \psi = 0$ fixes their values. But that doesn't change the fact that both formulas give the same partial derivatives.) A similar process shows that there are two possible integrals for $X$. (We agree on the transformation for $Y$ so that's not an issue.)

 

[PeterDonis]

The way I would describe this in more ambiguous English is to say that the sliding block doesn't rotate relative to the fixed stars, even though it DOES rotate relative to a torque-free gyroscope that's carried along with the sliding block, as described in the paper I quoted much earlier in the thread.

Another way of saying this in ambiguous English - in the presence of frame dragging effects such as Thomas precession, gyroscopes may (and will) rotate with respect to the fixed stars, so the notion of "rotating with respect to the fixed stars" and the notion of "rotating relative to a gyroscope" are different.

This looks like a good description to me. The only quibble I would have is that the term "frame dragging" is not normally applied to Thomas precession (or even de Sitter precession in Schwarzschild spacetime); it's usually reserved for effects like the Lense-Thirring precession that occur only in stationary but non-static spacetimes (i.e., in somewhat ambiguous English, in cases where spacetime itself, not just an object or a family of observers, is "rotating").

The more technical way of saying it is that the congruence describing the sliding block has nonzero vorticity but is non-rotating relative to a fixed global inertial frame.

 

[SlowThinker]

While in this rotating digression, should not every rotation have a center, that can be identified from within the system?
Where is the center of this rotation? Or is it some funny kind of rotation that is the same everywhere?

 

[PeterDonis]

While in this rotating digression, should not every rotation have a center, that can be identified from within the system?
Where is the center of this rotation? Or is it some funny kind of rotation that is the same everywhere?

This is a key reason why I insisted in an earlier post on distinguishing between a rotating disk and the "rotation" of the sliding block.

In the case of the rotating disk, yes, there is a unique center of rotation; that is, there is a point on the disk which is not rotating--it has zero angular velocity, and it feels zero proper acceleration, unlike every other point on the disk. Going along with that, the center of the disk stays at the same point in space, and all the other points on the disk follow paths that are spatially closed: each one comes back to the same point in space once per rotation (all as viewed from a fixed inertial frame).

In the case of the sliding block, however, there is no such center; there is no point of the block that is not "rotating" in the sense pervect described. And, of course, no point on the block stays at the same point in space, and no point on the block has a path that is spatially closed.

What, then, does the term "rotation" refer to that is the same in both cases? That's where the more technical language I used comes in. I said the congruence describing the object (disk or block) has nonzero vorticity. What does that mean?

Consider a particular point on the disk or block, and suppose that it has little rods connecting it to neighboring points. And suppose the point also carries a gyroscope along with it. "The congruence describing the object has nonzero vorticity" means that the set of little rods rotates relative to the gyroscope as the object moves. This is true of both the disk and the block.

 

[pervect]

While in this rotating digression, should not every rotation have a center, that can be identified from within the system?
Where is the center of this rotation? Or is it some funny kind of rotation that is the same everywhere?

I would say the later "some funnny kind of rotation" is the closest answser. In my approach, at least, the center of rotation is wherever you put your reference observer, which is to some extent an arbitrary choice. The value of the rotation, in radians/unit proper time, should be independent of your choice of origin of the $\psi$ coordinate, though it DOES depend on your choice of origin to the $\chi$ coordinate . To understand the dependence on the $\chi$ coordinate (which I would informally call heigh). Note that in either the elevator frame or the sliding block frame, two clocks at the same $\psi$ coordinate but different $\chi$ coordinates do not tick at the same rate.

 

[pervect]

An additional comment. If we consider a rotating Newtonian bar, the center of the bar does not accelerate, it's in an inertial frame. The ends of the bar do accelerate, they are not in an inertial frame, so there is a clear choice for the "center of rotation". But in the sliding block case, every point on the block is accelerating, and a torque-free gyroscope placed at any point on the block will rotate relative to the block.

 

[PeterDonis]

I would describe the situation as using a reference worldline of an observer to build a congruence.

There is more than one way of doing this; that's what my point about the non-uniqueness of the integrals illustrates.

If we restrict ourselves to small values of $\chi$ and $\psi$, then our two methods agree (because to first order in their arguments, $\cosh$ is equal to $1$ and $\sinh$ is equal to its argument; and we can use the sum identities for the hyperbolic trig functions to separate out $\cosh \gamma g \tau$ and $\sinh \gamma g \tau$ in my formulas so that we can expand the functions of $\psi$ to first order to get expressions equivalent to yours). So in a small enough world tube about the reference worldline, our two transformations result in the same congruence.

But beyond that small world tube, our methods diverge, and I think the reason for this is that there is no single congruence that has all of the properties that intuitively might seem desirable. Your congruence has the property that surfaces of constant $\tau$ are flat, not just in the small world tube around the reference worldline (where both of our methods result in locally flat surfaces of constant $\tau$--basically this is because our methods are both locally equivalent to Fermi-Walker transport, plus a rotation of basis vectors due to nonzero vorticity), but everywhere. But I suspect that this comes at the price of the congruence being non-rigid; which means that worldlines separated by constant values of $\chi$ or $\psi$ do not stay at the same proper distance from each other.

My congruence is, as I have shown, rigid, which I think is a necessary property to have if we are going to identify worldlines in the congruence with points in the block. But this comes at the price of surfaces of constant $\tau$ not being flat globally. So it looks like we have to pick one or the other; we can't have both.

 

[PeterDonis]

I suspect that this comes at the price of the congruence being non-rigid

In order to get more of a handle on this, I will go ahead and post an analysis of pervect's metric similar to what I've already posted for mine, so we have a good basis for comparison and can see exactly what properties are and are not satisfied by each metric and the associated congruence of worldlines at rest in it. This will take multiple posts as I work through the steps.

The coordinate transformation between Minkowski $T, X, Y$ and pervect's $\tau, \chi, \psi$ is:

$$
T = \left( \frac{1}{g} + \chi \right) \sinh \gamma g \tau + \gamma v \psi \cosh \gamma g \tau
$$

$$
X = \left( \frac{1}{g} + \chi \right) \cosh \gamma g \tau + \gamma v \psi \sinh \gamma g \tau
$$

$$
Y = \gamma v \tau + \gamma \psi
$$

And the metric is:$$
ds^2 = - \gamma^2 \left[ \left( 1 + g \chi \right)^2 - \gamma^2 v^2 g^2 \psi^2 - v^2 \right] d\tau^2 + 2 \gamma^2 v g d\tau \left( \psi d\chi - \chi d\psi \right) + d\chi^2 + d\psi^2 + dz^2
$$

The coordinate basis vectors derived from the above are:

$$
\partial_{\tau} = \left[ \gamma \left( 1 + g \chi \right) \cosh \gamma g \tau + \gamma^2 v g \psi \sinh \gamma g \tau \right] \partial_T + \left[ \gamma \left( 1 + g \chi \right) \sinh \gamma g \tau + \gamma^2 v g \psi \cosh \gamma g \tau \right] \partial_X + \gamma v \partial_Y
$$

$$
\partial_{\chi} = \sinh \gamma g \tau \partial_T + \cosh \gamma g \tau \partial_X
$$

$$
\partial_{\psi} = \gamma v \cosh \gamma g \tau \partial_T + \gamma v \sinh \gamma g \tau \partial_X + \gamma \partial_Y
$$

Let's first compute the proper acceleration of the 4-velocity field $U$ of worldlines at rest in this chart. That requires us to divide $\partial_{\tau}$ by its norm, which is $| \partial_{\tau} | = \gamma \sqrt{ \left( 1 + g \chi \right)^2 - \gamma^2 v^2 g^2 \psi^2 - v^2 }$. So the 4-velocity is

$$
U = \frac{1}{\sqrt{ \left( 1 + g \chi \right)^2 - \gamma^2 v^2 g^2 \psi^2 - v^2 }} \left( \left[ \left( 1 + g \chi \right) \cosh \gamma g \tau + \gamma v g \psi \sinh \gamma g \tau \right] \partial_T + \left[ \left( 1 + g \chi \right) \sinh \gamma g \tau + \gamma v g \psi \cosh \gamma g \tau \right] \partial_X + v \partial_Y \right)
$$

The proper acceleration is then

$$
A = U \cdot \frac{dU}{d\tau} = \frac{g \gamma}{\left( 1 + g \chi \right)^2 - \gamma^2 v^2 g^2 \psi^2 - v^2} \left( \left[ \left( 1 + g \chi \right) \sinh \gamma g \tau + \gamma v g \psi \cosh \gamma g \tau \right] \partial_T + \left[ \left( 1 + g \chi \right) \cosh \gamma g \tau + \gamma v g \psi \sinh \gamma g \tau \right] \partial_X \right)
$$

(As a sanity check, we can see that for $\chi = \psi = 0$, $U$ and $A$ reduce to the forms we have previously agreed on for the reference worldline.)

If we then express $A$ in terms of the coordinate basis vectors, we find that it is

$$
A = \frac{g \gamma^2}{\left(| \partial_{\tau} |\right)^2} \left[ \left( 1 + g \chi \right) \partial_{\chi} + g \psi \left( \partial_{\psi} - \gamma \partial_Y \right) \right]
$$

So, for $\psi \neq 0$, $A$ no longer points purely in the $\chi$ direction. (Note also that this is a "mixed" formula, since $\partial_Y$ appears; I won't be able to clean it up until I have the inverse of the above coordinate transformation.)

For completeness, we can also check the unit vectors in the coordinate directions; we find that $\partial_{\chi}$ and $\partial_{\psi}$ are unit vectors everywhere, so we have:

$$
\hat{e}_{\tau} = U = \frac{1}{\gamma \sqrt{ \left( 1 + g \chi \right)^2 - \gamma^2 v^2 g^2 \psi^2 - v^2 }} \partial_{\tau}
$$

$$
\hat{e}_{\chi} = \partial_{\chi}
$$

$$
\hat{e}_{\psi} = \partial_{\psi}
$$

The next thing will be to obtain the inverse of the above coordinate transformation, so I can express $U$ and $A$ above purely in terms of Minkowski coordinates and then compute the kinematic decomposition. (Pervect, if you already have the inverse transformation, it would help to post it, since I'm currently having trouble deriving it.)

 

[PeterDonis]

The next thing will be to obtain the inverse of the above coordinate transformation, so I can express $U$ and $A$ above purely in terms of Minkowski coordinates and then compute the kinematic decomposition.

Well, I haven't yet been able to invert pervect's coordinate transformation, but I have done something that makes it moot as far as computing the kinematic decomposition is concerned: I have found that the 4-velocity field $U$ in his chart is the same as the one in my chart! In other words, his $U$ and my $U$ describe the same congruence of worldlines; the only difference is how we parameterize them. I'll give a sketch of the proof below.

Of course this means that many of the potential issues I had raised in previous posts are moot, since they depend only on the congruence, not on how we parameterize it. (It also gives me a much better handle on comparing the two charts, but I'll save that for a future post.)

Here is how we can see that the two charts both have the same congruence of worldlines at rest in them. For this proof I'll switch to my convention for $X$ and $\chi$, where the floor of the rocket is at $\chi = 1/g$ instead of $\chi = 0$. This will make the formulas simpler since we'll just have $g \chi$ everywhere we used to have $1 + g \chi$. It will also make it easier to see how the proof goes.

So with the new convention, we have the following:

$$
X^2 - T^2 = \chi^2 - \gamma^2 v^2 \psi^2
$$

That expression should look familiar; it occurs in the norm of pervect's $U$. Using it, we can re-express that norm as $| U | = \sqrt{g^2 \left( X^2 - T^2 \right) - v^2}$.

That expression should also look familiar: it is the same as the norm of my $U$, expressed in Minkowski coordinates (see post #81). Guided by that hint, we then look at the rest of the expression for pervect's $U$, and compare it with his coordinate transformation, and find that

$$
U = \frac{1}{\sqrt{g^2 \left( X^2 - T^2 \right) - v^2}} \left( g X \partial_T + g T \partial_X + v \partial_Y \right)
$$

This is, of course, the same formula as I derived in post #81 for my 4-velocity field. So at every point in in Minkowski coordinates, which means at every point in spacetime, pervect's $U$ and my $U$ are the same vector. That means both 4-velocity fields must generate the same congruence of worldlines.

 

[PeterDonis]

his UU and my UU describe the same congruence of worldlines; the only difference is how we parameterize them.

Just to summarize where I am currently on this, the obvious change from my coordinates to pervect's is the $\chi$ coordinate. If we use $\bar{\chi}$ for my coordinate, then we have $\bar{\chi}^2 = \chi^2 - \gamma^2 v^2 \psi^2$. So if we pick a worldline in the congruence that pervect labels with $(\chi, \psi)$, I will label it instead with $\bar{\chi} = \sqrt{\chi^2 - \gamma^2 v^2 \psi^2}$.

What I haven't yet figured out is whether that is the only reparameterization. The fact that we both have the same transformation formulas for $Y$ strongly suggest that it is--i.e., that pervect's $\tau, \psi$ and my $\tau, \psi$ are the same. But I haven't yet been able to get everything else to match up under that assumption.

If we assume for a moment that the above is the only reparameterization, what does it mean? It means that our surfaces of constant $\tau$ are the same, and we label each worldline in the congruence with the same $\psi$, so in each surface of constant $\tau$, our $\psi$ "grid lines" are the same. But our $\chi$ "grid lines" are not; pervect's $\chi$ grid lines and my $\bar{\chi}$ grid lines are tilted with respect to each other (they match only on the reference worldline, the one he labels with $\chi = 0$, $\psi = 0$ and I label with $\bar{\chi} = 1 / g$, $\psi = 0$--but the $1/g$ is just a translation and doesn't affect the tilting of the grid lines, and I have left it out of the discussion above). Furthermore, the "tilt" is not linear; in pervect's chart, his $\chi$ grid lines are straight (because his spatial metric is Euclidean), but my $\bar{\chi}$ grid lines are curved (so the increment of $d\psi$ needed to move along my $\bar{\chi}$ grid lines between the same two worldlines is different, hence my $g_{\psi \psi}$ is not $1$).

So why do my $\bar{\chi}$ grid lines curve, as seen in pervect's chart? Because they have to remain perpendicular to the direction of proper acceleration, so that the proper acceleration vector is always a multiple of $\partial_{\bar{\chi}}$. In pervect's metric, the proper acceleration vector is not always perpendicular to his $\chi$ grid lines. (On the reference worldline, it is, and pervect's $\chi$ grid lines and my $\bar{\chi}$ grid lines are parallel--or, to put it another way, on that worldline, my curved grid lines are just tangent to his straight ones). Also, my $\bar{\chi}$ grid lines are the ones that are at a constant "altitude" in the rocket--for example, the grid line I label $\chi = 1/g$ is the one at the altitude of the floor of the rocket. In pervect's chart, the floor of the rocket, and any other line at a constant altitude in the rocket, will look curved and will not be at a constant $\chi$ coordinate.

I think the above helps to illustrate the differences between our two charts, and what features each one has. But, as I said, I haven't confirmed that the $\chi$ change is the only one, so this is all tentative at this point.

 

[pervect]

In order to get more of a handle on this, I will go ahead and post an analysis of pervect's metric similar to what I've already posted for mine, so we have a good basis for comparison and can see exactly what properties are and are not satisfied by each metric and the associated congruence of worldlines at rest in it. This will take multiple posts as I work through the steps.

The coordinate transformation between Minkowski $T, X, Y$ and pervect's $\tau, \chi, \psi$ is:

$$
T = \left( \frac{1}{g} + \chi \right) \sinh \gamma g \tau + \gamma v \psi \cosh \gamma g \tau
$$

$$
X = \left( \frac{1}{g} + \chi \right) \cosh \gamma g \tau + \gamma v \psi \sinh \gamma g \tau
$$

$$
Y = \gamma v \tau + \gamma \psi
$$

I don't _think_ it makes any difference to the subsequent calculations, but I wanted to point out that using the above, $\tau=\chi=\psi=0$ transformed to $X=/1g, T=0$ whereas in my original version there was an additional offset in the transform so that $\tau=\chi=\psi=0$ transformed to $X=T=0$ instead.

The next thing will be to obtain the inverse of the above coordinate transformation, so I can express $U$ and $A$ above purely in terms of Minkowski coordinates and then compute the kinematic decomposition. (Pervect, if you already have the inverse transformation, it would help to post it, since I'm currently having trouble deriving it.)

I don't have an inverse.

 

[pervect]

I think the above helps to illustrate the differences between our two charts, and what features each one has. But, as I said, I haven't confirmed that the $\chi$ change is the only one, so this is all tentative at this point.

I rather belatedly got around to noticing that your $g_{00}$ was only a function of $\chi$. For an observer "at rest", i.e. one with constant $\chi$ and $\psi$ coordinates, the only non-zero component of the 4-velocity U will the $U^{\tau}$ component. Because $g_{00}$ is only a function of $\chi$, when we normalize U by making $g_{00} u^{\tau} u^{\tau}$ have a magnitude of 1 (a value of -1 in the sign conventions I favor), this implies that $U^\tau$ is only a function of $\chi$, and thus $dU/d\tau$, the 4-acceleration, only has components in the $\partial_\chi$ direction.

I tend to think of surfaces of constant $g_{00}$ in explicitly stationary or static metrics (i.e. coordinates in which none of the metric coefficients is a function of time) as equipotential surfaces. I'm not sure I've seen this exact terminology explicitly used in a textbook, though it was inspired by the example of the rotating Earth's geoid being both both an equipotential surface and a surface on which $g_{00}$ is constant. Anyway, as the above argument suggests, choosing coordinates such that equipotential surface are a function only of a single coordinate automatically makes the 4-acceleration of an observer "at rest" in those coordinates perpendicular to the equipotential surface. So expressing the result in coordinate independent terms, we can say that the 4-acceleration of an observer "at rest" in a static or stationary space-time is always perpendicular to an equipotential surface.

So if we consider the floor of the rocket on which the block is sliding, it is an equipotential surface, and this feature is true in all coordinate systems. Your coordinate choices keeps the (appearance?) of the floor of the rocketr "flat" in the sense that the rocket floor has a constant $\chi$ coordinate. If we imagine the rocket having multiple floors at different heights, your transformation would keep all the floors of the rocket "flat" in the sense that each floor would have a characteristic and constant value for $\chi$. This simplicity in the description of the floor comes at the expense of introducing a curved space.

My coordinates make the floor of the rocket "curve", because equipotential surfaces do not have constant $\chi$ coordinate, the surface is a function of both $\chi$ and $\psi$. However, the advantage of my coordinate choice is that it was constructed in such a manner as to make the spatial part of the flat.

 

[PeterDonis]

I don't _think_ it makes any difference to the subsequent calculations, but I wanted to point out that using the above, $\tau=\chi=\psi=0$ transformed to $X=/1g, T=0$ whereas in my original version there was an additional offset in the transform so that $\tau=\chi=\psi=0$ transformed to $X=T=0$ instead.

You're right, there should be an extra constant term in the $X$ transform. It doesn't affect any of the other calculations. (I did all the computations with my convention anyway, where there is just $\chi$ instead of $(1/g) + \chi$ in the transforms and no extra offsets, and the floor of the rocket at $\psi = 0$ is at $\chi = 1/g$ at $\tau = 0$.)

 

[PeterDonis]

Because $g_{00}$ is only a function of $\chi$, when we normalize $U$ by making $g_{00} u^{\tau} u^{\tau}$ have a magnitude of 1 (a value of -1 in the sign conventions I favor), this implies that $U^\tau$ is only a function of $\chi$, and thus $U/d\tau$, the 4-acceleration, only has components in the $\partial_\chi$ direction.

Yes.

I tend to think of surfaces of constant $g_{00}$ in explicitly stationary or static metrics (i.e. coordinates in which none of the metric coefficients is a function of time) as equipotential surfaces.

I called them "surfaces of constant altitude", but "equipotential surfaces" is just as good.

I'm not sure I've seen this exact terminology explicitly used in a textbook

IIRC MTW use it in some places, but I don't think they put a lot of emphasis on it.

choosing coordinates such that equipotential surface are a function only of a single coordinate automatically makes the 4-acceleration of an observer "at rest" in those coordinates perpendicular to the equipotential surface.

No, that is always true, because the 4-acceleration of the observer "at rest" must be in the direction of the gradient of the potential, which is orthogonal to the equipotential surface. That is an invariant statement, independent of coordinates. If you compute the inner product of the 4-acceleration vector in your chart with a vector that is tangent to the equipotential surface, expressed in your chart, you will find that that inner product is zero.

The special feature of my chart, where the equipotential surfaces are surfaces of constant value of a single coordinate $\chi$, is that the vectors tangent to the equipotential surface have no $\partial_{\chi}$ component. So they are linear combinations, in my chart, of $\partial_{\psi}$ and $\partial_z$ only. In your chart, vectors tangent to the equipotential surfaces can have components in all three of the spatial directions (but, as above, they will still be orthogonal to the 4-acceleration everywhere).

So if we consider the floor of the rocket on which the block is sliding, it is an equipotential surface, and this feature is true in all coordinate systems. Your coordinate choices keeps the (appearance?) of the floor of the rocket "flat" in the sense that the rocket floor has a constant $\chi coordinate$. If we imagine the rocket having multiple floors at different heights, your transformation would keep all the floors of the rocket "flat" in the sense that each floor would have a characteristic and constant value for $\chi$. This simplicity in the description of the floor comes at the expense of introducing a curved space.

The term "the floor of the rocket" is ambiguous. In Minkowski coordinates, if we leave out the $Z$ coordinate, the floor of the rocket is described by the "worldsheet" $X^2 - T^2 = 1 / g^2$, with $Y$ unconstrained. This is an infinite set of hyperbolas running in the $Y$ direction, and each one of those hyperbolas is the worldline of a "rocket observer", sitting at rest on the floor of the rocket at some fixed value of $Y$.

To one of those rocket observers (say the one with $Y = 0$ for definiteness, since that's the "reference" observer for Rindler coordinates), the floor of the rocket at some instant of his time is a line in the $Y$ direction "cut" out of the worldsheet at some fixed value of $T$ (and $X$, since $T$ determines $X$ on the worldsheet). This is obviously, once we put the $Z$ dimension back, a flat plane.

Now, since the bottom of the sliding block is always in contact with the floor of the rocket, the congruence of worldlines that describes the bottom of the block (the one that is at rest in both of our charts) must lie in that same worldsheet I described above. But the worldlines sit in that worldsheet at an angle, so to speak. For example, the "reference" observer of the block, the one who is at $\psi = 0$, has a worldline that intersects that of the reference rocket observer above at $T = Y = 0, X = 1/g$ (using my coordinate convention, since that's the one I used for the equation of the worldsheet above). At more and more positive values of $T$, this worldline increases in $X$, but it also increases in $Y$--and for more and more negative values of $T$, the worldline also increases in $X$ (since now it is moving in the $-X$ direction, but decelerating), but it decreases in $Y$. So it curves around the worldsheet at an angle.

Therefore, the intersection of the worldsheet with surfaces of constant $\tau$ in either of our charts (and, as far as I can tell, these are the same surfaces for both charts) will also curve around the worldsheet at an angle; these surfaces will not be the same surfaces I described above as the "floor of the rocket at an instant of time" for the rocket observers. That means the intersection of the worldsheet with surfaces of constant $\tau$ generates surfaces that are in fact curved, not flat, as a matter of geometry. So the curvature is not a matter of appearance.

My chart makes these surfaces, the equipotential surfaces from the point of view of the sliding block, "appear flat" only in the sense that they are surfaces of constant $\chi$; but my metric makes clear that they are in fact curved, geometrically, because $g_{\psi \psi}$ is not $1$. Your chart makes the curvature clear in a different way, by making the equipotential surfaces explicitly curved, i.e., curved surfaces in a Euclidean spatial metric. Either way works, and both agree that the surfaces, geometrically, are curved.

My coordinates make the floor of the rocket "curve", because equipotential surfaces do not have constant $\chi$ coordinate, the surface is a function of both $\chi$ and $\psi$. However, the advantage of my coordinate choice is that it was constructed in such a manner as to make the spatial part of the flat.

Yes, but there's another tradeoff besides making the equipotential surfaces functions of two coordinates. The "apparent" direction in your chart of the proper acceleration vector changes with $\psi$. But, as you noted in a previous post, the actual direction of that vector does not change relative to the "fixed stars"--more precisely, it doesn't change relative to Minkowski coordinates. In those coordinates it always points, spatially, in the $X$ direction. My chart reflects that by keeping the proper acceleration always in the $\chi$ direction.

In short, in a situation like this, as I've said before, it is impossible to have a single chart that directly represents all of the properties we are interested in. We have to pick and choose which properties we want the chart to directly represent, and which ones we want to have it represent only indirectly.

 

[PeterDonis]

What I haven't yet figured out is whether that is the only reparameterization.

On thinking this over, I don't think it can be; at the very least, $\psi$ needs to change also from my metric to pervect's.

Consider pervect's $\chi \psi$ plane at some constant value of $\tau$ (and $z$), and think about how my $\bar{\chi}$ and $\bar{\psi}$ "grid lines" would look. We know that my $\bar{\chi}$ grid lines look curved. But my $\bar{\psi}$ grid lines are everywhere orthogonal to my $\bar{\chi}$ grid lines (because there is no cross term in my metric between those two coordinates). That means my $\bar{\psi}$ grid lines must also curve, as seen in pervect's chart. But pervect's $\psi$ grid lines are straight lines in his chart, so his and mine can't be the same. The best we can hope for is that on the floor of the rocket, his $\psi$ and my $\bar{\psi}$ will be the same.

I'm afraid this quite possibly means that the $\tau$ coordinate needs to transform as well, because, as I observed before, both of us have the same transformation equation for $Y$ in terms of $\tau$ and $\psi$. So the comparison between our two charts in the $\chi \psi$ plane, as above and in previous posts, may only be of limited validity, since it relied on our two charts having the same surfaces of constant $\tau$.

 

[pervect]

I concur that PeterD's time coordinate, which I'll refer to as $\tau^\prime$ in this post, and my time coordinate, which I'll refer to as $\tau$ in this post, are not the same. To check this, I performed the following test.

I took my expressions for $T(\tau,\chi,\psi)$, $X(\tau,\chi,\psi)$, $Y(\tau,\chi,\psi)$, as modified to be compatible with Peter's choice of origin as in post https://www.physicsforums.com/threads/gravity-on-einsteins-train.835994/page-6#post-5260046.

Then I applied Peter's inverse transformation from post https://www.physicsforums.com/threads/gravity-on-einsteins-train.835994/page-4#post-5256530, to get $\tau^\prime(T,X,Y)$.

The result did not appear to simplify to $\tau^\prime = \tau$, but the expression was complicated enough that it was difficult to be absolute sure they were not the same. So what I did was consider the much easier case to test if the surface $\tau=0$ mapped to the surface $\tau^\prime=0$. Substituting $\tau=0$ into my expressions for T,X,Y gives:

$$
T = \gamma \chi \quad X = (1/g + \chi) \quad Y= \gamma \psi
$$

I then substituted these values for T,X,Y into Peter's expression for $\tau^\prime$

$$
\tau^\prime = \gamma (\frac {arctanh(T/X)}{g} - v Y)
$$

I won't give the result here, but it was clearly not equal to zero as it should be if our time coordinates were the same. The results were differ by so much that I wonder if there could be some error - but if so, I haven't spotted it. I do have a simple verbal description of my time coordinate - it's the time coordinate of an inertial frame co-moving with the reference observer. I don't have any such simple verbal description for Peter's time cordainte $\tau^\prime$.

 

[PeterDonis]

I do have a simple verbal description of my time coordinate - it's the time coordinate of an inertial frame co-moving with the reference observer. I don't have any such simple verbal description for Peter's time cordainte $\tau^\prime$.

Both of our time coordinates meet that description, because they both match on the worldline of the reference observer. (If you plug in $\psi = 0$ and $\chi = 1/g$ to my formulas, and plug $\psi = 0$ and $\chi = 0$ into yours, they will match.) They only differ off of that worldline. There is no unique way to define "the time coordinate of an inertial frame co-moving with the reference observer" globally, because there is no such frame globally since the reference worldline is accelerating. We are simply doing the extension of the obvious local time coordinate for the reference observer to a non-inertial frame covering the rest of spacetime (or at least the region of spacetime covered by our scenario) in different ways. Your way makes the surfaces of constant coordinate time Euclidean; my way makes them always orthogonal to the proper acceleration vector.

 

[pervect]

Both of our time coordinates meet that description, because they both match on the worldline of the reference observer. (If you plug in $\psi = 0$ and $\chi = 1/g$ to my formulas, and plug $\psi = 0$ and $\chi = 0$ into yours, they will match.) They only differ off of that worldline. There is no unique way to define "the time coordinate of an inertial frame co-moving with the reference observer" globally, because there is no such frame globally since the reference worldline is accelerating.

Sure there is. While the reference observer is, as you note, accelerating, at any given instant of proper time there is a non-accelerating observer who is co-located with the accelerating observer, and who has zero velocity relative to the accelerating observer. That observer is the co-moving inertial observer, and they have a corresponding inertial frame.

 

[PeterDonis]

While the reference observer is, as you note, accelerating, at any given instant of proper time there is a non-accelerating observer who is co-located with the accelerating observer, and who has zero velocity relative to the accelerating observer. That observer is the co-moving inertial observer, and they have a corresponding inertial frame.

Yes, but it's a different inertial frame at each event on the reference observer's worldline. So there is no such thing as "the" time coordinate of the reference observer's comoving inertial frame. There are an infinite number of such coordinates, one for each event on the reference observer's worldline.

What you may be trying to say here is that $\tau$ is intended to be like the time coordinate in Rindler coordinates, where each surface of constant Rindler time $t$ is orthogonal to each worldline in the Rindler congruence. Of course this isn't possible for the entire congruence $U$ that describes the sliding block, because that congruence has nonzero vorticity; but it is possible for each surface of constant $\tau$ to be orthogonal to the reference observer's worldline. But both of our $\tau$ coordinates have that property, because the cross terms in both of our metrics vanish on the reference worldline.

There is another property that surfaces of constant time in Rindler coordinates have, which is that they are geodesic surfaces, i.e., if we pick any point on such a surface and look at the three spatial basis vectors there, the geodesics determined by those basis vectors (and linear combinations of them) generate the entire surface. However, there is no way to "read off" that property just from the metric; you would need to actually compute the geodesic equation and see. I haven't done that for either of our metrics, so I don't know if either of our charts have the property I just described.

 

[pervect]

What you may be trying to say here is that $\tau$ is intended to be like the time coordinate in Rindler coordinates, where each surface of constant Rindler time $t$ is orthogonal to each worldline in the Rindler congruence.

The congruence is not the Rindler congruence. It's a congruence of _inertial observers_. Obsevers who are not accelerating . They're all relatively at rest in flat space_time. It's not that complicated, really. It's just Einstein clock synchronization. Wiki has a bit on it, https://en.wikipedia.org/wiki/Einstein_synchronization, wiki even mentions the "no-redshift" condition which is one of two necessary conditions for clock synchronization to occur.

Inertial observers in inertial reference frames are also mentioned in http://arxiv.org/abs/0708.2490v1, which I mentioned in post #49 of this threead. I'll quote one section which talks about it briefly.

Because a torque-free gyroscope precesses relative to
the train, it follows that the train has a proper rotation,
meaning that the train rotates as observed from its own
momentary inertial rest frame.

Note that this paper talks about an inertial observer in inertial frame S, which is at rest relative to the track, and another inertial observer in frame S', which frame is moving along the track with some velocity v. They do not see the need to complicate the analysis any more than that.

Note that they make points similar to those I've been attempting to make several times - that the proper frame of the train is rotating, and that in that proper frame, the track is curved.

 

[PeterDonis]

The congruence is not the Rindler congruence.

Which congruence are you talking about? If you mean the congruence of observers at rest in Rindler coordinates, that congruence is accelerated, not inertial. So is the congruence $U$ of observers at rest in either of our metrics. Those are the only congruences we've discussed in this thread, as far as I know.

this paper talks about an inertial observer in inertial frame S, which is at rest relative to the track, and another inertial observer in frame S', which frame is moving along the track with some velocity v

These two inertial observers do not form a congruence. Their worldlines cross. A congruence is a family of worldlines that do not cross anywhere, so every event in the region of spacetime covered by the congruence lies on one and only one worldline in the congruence.

they make points similar to those I've been attempting to make several times - that the proper frame of the train is rotating, and that in that proper frame, the track is curved.

But this "proper frame" is not an inertial frame, and the congruence of worldlines of observers at rest in this frame is not an inertial congruence.

More precisely: if by the "proper frame" you mean an inertial frame in which the train is momentarily at rest, then that frame is not rotating; it can't be. Inertial frames are by definition non-rotating. In this frame, at the instant in which the train is at rest in it, different parts of the track are moving at different velocities in the "vertical" direction--the part towards the rear of the train is moving down, and the part towards the front of the train is moving up. (Fig. 11 in the paper and its accompanying discussion show this.) So in this frame, the track can be thought of as "rotating" around an axis perpendicular to the plane spanned by the "vertical" direction (the direction of proper acceleration) and the direction of the train's and track's relative motion. But the frame itself is not rotating; it's inertial.

Also, in the momentarily comoving inertial frame of the train, the track appears curved. As the paper explains, this can be understood as being due to relativity of simultaneity; in this frame, the events at which the different parts of the track are at the same "vertical" coordinate (the $X$ coordinate in my nomenclature) are not all simultaneous. In other words, whether the track appears curved depends on how you "cut" a surface of constant time out of the entire worldsheet describing the track. The MCIF of the train cuts "at an angle" through that worldsheet, and therefore cuts out a curved surface.

If you want to define a non-inertial "proper frame" in which the train is at rest continuously (which is what I thought both of us were trying to do), then whether or not the track is curved in that frame works the same way as the above: it depends on how the surfaces of constant $\tau$ in that non-inertial frame cut the worldsheet describing the track. As far as I can tell, both of our metrics define surfaces of constant $\tau$ that cut at an angle, so the track appears curved in both of our metrics. Are you saying that the surfaces of constant $\tau$ in one of our metrics (presumably yours) are all the same as surfaces of constant time in some inertial frame? If so, which one?

 

[pervect]

Certain aspects of your (PeterDonis's) metric continue to bother me. The piece I'm missing that would help me the most in resolving my questions is an expression for the inertial coordinates (T,X,Y) as functions of the block coordinates ($\tau,\chi,\psi$), which I don't seem to find. I could be missing this piece due to the length of the thread, if this has actually been posted somewhere, a pointer to it would be appreciated.

That said, let me review the textbook support for the approach I outlined, which may be both clearer and more convincing than my own exosition. It's also much more lengthly, but the length appears to be needed to address all the little issues that keep popping up in our attempts to discuss it here on PF. The approach is basically straight out of MTW's textbook, "Gravitation". The case of the accelerating and rotating observer in flat space-time is first discussed in exercise 6.8 on pg 174. Because it's only an exercise, it's not as easy to follow as the similar but more leisurely discussion of the accelerating and rotating observer in curved space-time starting on page 327,. The title of this discussion is "The Proper reference frame of an accelerated observer", in section $12.6. As a side note, the title of this section shows that these coordinates are important enough to merit a name of their own.

The discussion in section $12.6 gives us the line element of the metric for the proper reference frame to the first order. I won't go into details, but the argument is based on considering certain of the Christoffel symbols along the worldline of the accelerating observer. This approach resulting line element given in 13.32

$$
ds^2 = -(1 + 2a^\hat{j}x^\hat{j}) (dx^\hat{0})^2 -2 (\epsilon_{\hat{j}\hat{k}\hat{l}}x^\hat{k}\omega^\hat{l}) (dx^\hat{0} dx^\hat{j})+\delta_{\hat{j}\hat{k}} (dx^\hat{j}dx^\hat{k})
$$
plus terms of order $O(|x^\hat{j}|^2)$Here $j,k,l$ range from 1 to 3, making $a^j$ and $\omega^l$ both three-vectors, which are respectively the proper acceleration and proper rotation of the frame. $\epsilon_{jkl}$ is the 3-d Leva-Civita symbol, and $\delta$ is the Kronecker delta.

The line element given by MTW agrees with my line element to the first order, where a is the proper acceleration $\gamma^2 g$ of a point on the block, and $\omega$ is the proper rotation, which has a magnitude of $\gamma^2 g v$ (I haven't worked out the sign of the rotation) and is oriented about the $\hat{z}$ axis. This proper rotation is due to Thomas preccession as discussed in http://arxiv.org/abs/0708.2490v1, which also gives us the magnitude quoted above.

This agreement with the textbook results to the first order, plus the flatness of the resulting Riemann tensor, convinces me both that the line element is correct, and that the proper reference frame of the sliding block fits neatly into the paradigm of an "accelerating and rotating observer", i.,e an accelerating observer who chooses to use orthonormal basis vectors, but not to Fermi-walker transport them.

 

[PeterDonis]

The piece I'm missing that would help me the most in resolving my questions is an expression for the inertial coordinates (T,X,Y) as functions of the block coordinates $(\tau,\chi,\psi)$, which I don't seem to find.

It's in post #68, which is indeed quite a while ago now:

https://www.physicsforums.com/threads/gravity-on-einsteins-train.835994/page-4#post-5254985

The inverse transformation is in post #80:

https://www.physicsforums.com/threads/gravity-on-einsteins-train.835994/page-4#post-5256530

[Edit: deleted the rest of what I originally posted here until I can do some more computations.]