# Herglotz-Noether theorem understanding

## Main Question or Discussion Point

In this thread I seek to come to an intuitive physical understanding of the implications and conclusions of the Herglotz-Noether theorem.

In the section in Wikipedia on the Herglotz-Noether theorem it says very little beyond the summary that
The Herglotz-Noether theorem in special relativity restricts the possible linear and rotational motions of a Born-rigid object. It states that such a body may only possess a linear acceleration if it is not rotating.
That summary (in bold) is easy enough to understand, but unfortunately (after some sterling investigative work by PeterDonnis, Mentz114, WannabeNewton and pervect in this forum) appears to be dead wrong. So where does that leave us?

Proir to Herglotz and Noether, Born came up with a concept of rigidity that was applicable to relativity and demonstrated that it was possible to accelerate a 3d object in a manner that maintained this form of rigidity. Ehrenfest demonstrated that it was not possible to apply angular acceleration to a 3d object in a way that maintained Born rigidity. However, this is does not imply that an object with constant rotation cannot be Born rigid.

Herglotz and Noether independently wanted to come up with a more general formalism of rigid motion beyond the specific examples studied by Born and Ehrenfest. The original paper by Fritz Noether linked to by Wikipedia is in German so not accessible to me. The other paper http://en.wikisource.org/wiki/Translation:On_bodies_that_are_to_be_designated_as_"rigid" by Herglotz is in English and in the introduction, Herglotz states
Particularly the fact may be mentioned for the purpose of illustration, that when one of its points is fixed, the body of Born can only uniformly rotate around a fixed axis that goes through that point.
What exactly does he mean by "fixed point"? In relativity a point that is fixed in one reference frame is moving in another. Obviously he means a point that is at rest in a given reference frame, but can that be a rotating or linearly accelerating reference frame or exclusively an inertial reference frame? What does he mean by "fixed axis"? In a rotating reference frame a fixed axis is a rotating axis from the point of view of an inertial reference frame. Reading between the lines and using the information we already know, presumably he means that a rotating object can only have Born rigid motion, if its axis is fixed from the point of view of a non rotating reference frame. This does not exclude a linearly accelerating reference frame. Nevertheless, Herglotz clearly does not exclude rotation from the list of possible Born Rigid motions.

Herglotz further states:
Exactly this question will find its answer in the following lines in so far, as it will be proven that the motion of that "rigid" body is in general — i.e. neglecting special, more specified exceptions — unequivocally defined by the arbitrarily specified motion of a single of its points.
Again, this is a bit puzzling. If a disc is rotating about its x axis and linearly and inertially translating along the x axis, then a point at the centre is moving in a straight line while all other points on the disc are moving in a spiral. This is a contradiction to his statement or is one of the "specified exceptions". We are fairly sure that a disc with constant rotation can have linear inertial motion and remain Born rigid.

Elsewhere in the scant literature we see mention that the Herglotz-Noether theorem implies that a solid 3d object can ony have 3 degrees of freedom rather than the usual 6 degrees of freedom allowed in Newtonian physics. Since Ehrenfest demonstrated that it is impossible to apply angular acceleration to an object in a Born rigid manner, then these allowable 3 degrees of motion can only be linear acceleration in the 3 spatial directions.

We can now paraphrase Herglotz-Noether as "It is not possible to apply angular acceleration to a Born rigid object and amintain its Born rigidity". This is the negative (exclusive) version and does not say much more that we already learned from Ehrenfest. A more positive (inclusive) version would be "It is possible to apply linear acceleration in any direction to a Born rigid object and maintain its Born rigidity". Here a 3d object with constant rotation can be considered a "Born rigid object" and such an object can be linearly accelerated in any direction in a Born rigid manner. Since the outcome of such a "hard boost" is another Born rigid object, we can subsequently linearly accelerate the linearly accelerating and rotating object in yet another direction and still maintain its Born rigid nature and so on.

Am I too far off the mark? .. any thoughts?

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PeterDonis
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2019 Award
That summary (in bold) is easy enough to understand, but unfortunately (after some sterling investigative work by PeterDonnis, Mentz114, WannabeNewton and pervect in this forum) appears to be dead wrong.
Yes, Wikipedia's statement appears to be wrong, but that just means Wikipedia isn't always reliable. Surprise, surprise. A more careful statement of the theorem, or at least the part of it that concerns us here, is in the Giulini paper that was linked to in the other threads. It is basically this: A rigid motion with nonzero vorticity must be a Killing motion. ("Rigid motion with nonzero vorticity" is a more technically careful way of saying "rigid motion with nonzero rotation".)

What I think we showed in the other thread is that, contrary to the Wikipedia statement (and apparently plenty of other statements in various places in the literature), a linearly accelerated object *can* also be rotating and still move rigidly--*if* the linear acceleration is absolutely constant, since only a constant linear acceleration is a Killing motion. Any change in the linear acceleration, in either magnitude or direction, violates the restriction that the motion must be a Killing motion, and therefore makes it impossible for the motion to be rigid.

(The more precise way of saying "the linear acceleration must be constant" is that the points of the object that are on its axis of rotation must describe a Rindler congruence with acceleration along that same axis. See further comments below.)

Ehrenfest demonstrated that it was not possible to apply angular acceleration to a 3d object in a way that maintained Born rigidity. However, this is does not imply that an object with constant rotation cannot be Born rigid.
Agreed. (I don't think this was ever disputed in any of the recent threads on this topic.)

Reading between the lines and using the information we already know, presumably he means that a rotating object can only have Born rigid motion, if its axis is fixed from the point of view of a non rotating reference frame. This does not exclude a linearly accelerating reference frame.
I think this is a permissible reading, yes, and it is consistent with what we found in the other thread, with the proviso that the linear acceleration must be constant and the points on the object's axis must describe a Rindler congruence, as above.

Herglotz further states:
I'm not sure what the basis is for this statement; I haven't fully read through the paper you linked to.

I think you're right that you can't completely specify the congruence we found in the other thread by specifying the motion of one of its points, since there are two degrees of freedom in specifying the congruence: there is the linear acceleration of a specified point on the rotation axis (I normalized that away in the other thread to simplify the math, but it's still there; in the correct general Rindler line element, ##ds^2 = - a^2 z^2 dt^2 + dz^2 + dr^2 + r^2 d\phi^2##, this degree of freedom is ##a##), and the angular velocity of a specified point at the same ##z## as the specified point on the axis (this was what I called ##\kappa## in the other thread). This means that the motion of at least two points must be specified; one is not enough.

these allowable 3 degrees of motion can only be linear acceleration in the 3 spatial directions.
Here's the way I would count the degrees of freedom: the possible linear accelerations along the axis take 3 degrees of freedom, and the possible angular velocities given constant linear acceleration take one more, for a total of 4. Again, I'm not sure what the basis is for the statement that there are only three; I certainly don't see that number popping out of the proof in the Giulini paper that was linked to in the other threads.

we can subsequently linearly accelerate the linearly accelerating and rotating object in yet another direction and still maintain its Born rigid nature
No, I don't think this is right; see my comments above on the linear acceleration having to be constant.

PeterDonis
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2019 Award
one of the "specified exceptions"
It looks like the exceptions are described in this paragraph (marked):

An exception only takes place, when the world-line of that point in ##R_4## has constant curvatures, namely in this case – depending on the condition that it doesn't lie in a lower space than (at the most) ##R_3## or ##R_2##, or eventually it is a straight line – there additionally exist 1 or ##\infty^{1}## or ##\infty^{3}## possible motions.
The congruence we found in the other thread meets the "constant curvature" condition; that condition is equivalent to saying that, if the points on the axis of rotation have nonzero proper acceleration, they must describe a Rindler congruence with acceleration along the same axis, as I said in my previous post.

Since there is one additional degree of freedom in this congruence (the angular velocity), I think this corresponds to the case of "##\infty^1## possible motions". However, I haven't completely figured out how he arrives at the three possibilities listed in the above quote for the "exceptions".

PAllen
2019 Award
Herglotz further states:
"Exactly this question will find its answer in the following lines in so far, as it will be proven that the motion of that "rigid" body is in general — i.e. neglecting special, more specified exceptions — unequivocally defined by the arbitrarily specified motion of a single of its points."

Again, this is a bit puzzling. If a disc is rotating about its x axis and linearly and inertially translating along the x axis, then a point at the centre is moving in a straight line while all other points on the disc are moving in a spiral. This is a contradiction to his statement or is one of the "specified exceptions". We are fairly sure that a disc with constant rotation can have linear inertial motion and remain Born rigid.
I think the meaning of this is clear in context. The key phrase is "in general — i.e. neglecting special, more specified exceptions — ". This means, arbitrary motion of one point, with changing direction and acceleration. In that general case, it is true that the motion of all other points of the object are determined by the motion of that one point, and that no vorticity is possible at all if one point is undergoing arbitrary motion.

Then there are the special cases, which I think are much better covered by the modern formulation by Giulini: if there is vorticity, the motion must be killing motion. The problem found in these threads is that a number of sources have produced overly restricting descriptions of what constitutes killing motion.

On degrees of freedom, my take of the key distinction is how many functions can you specify freely to describe the motion. In Newtonian mechanics, for a rigid body, you could specify 6 freely: 3 components of velocity of the COM, 2 describing evolution of axis of rotation, 1 describing evolution of angular speed. Note, these are 6 functions of time, not constants.

Herglotz-Noether says, in SR, in a formulation I've seen:

- if there is no rotation, you get to pick only 3 free evolution function.
- if there is rotation, you get have no free functions.

What I think has happened is that overly restrictive conclusions have been drawn from the second part, especially, because stated in this form, it is genuinely unclear exactly what is allowed. Thus, the killing motion formulation is a lot clearer. Thus, uniform acceleration is not a free function, but when is it allowed when not? The pure degrees of freedom formulation doesn't say. Our current understanding is that constant acceleration is ok along the spin axis of a rotating body, but not in any other direction. We find no exceptions Ehrenfest's conclusion of angular speed must be fixed (except possibly for a zero thickness disk if Epp et al are right; but 2-surfaces were never the subject of Herglotz-Noether).

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PeterDonis
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2019 Award
functions of time, not constants.
Hm, yes, I should have taken that into account when I was specifying degrees of freedom earlier. See below.

- if there is no rotation, you get to pick only 3 free evolution function.
Yes, these would be the 3 components of acceleration of one point of the body, which, if there is no rotation, is sufficient to specify the entire rigid motion.

- if there is rotation, you get have no free functions.
But you do get to freely pick initial conditions, which is what I really should have said in my previous post. In the case of rotational motion, you can freely choose two initial conditions, which appear as free parameters (but which must be constant once chosen):

(1) The proper acceleration of one specific point on the rotation axis (which must be in the same direction as the rotation axis);

(2) The angular velocity of rotation as measured by an observer comoving with that same specific point.

What I think has happened is that overly restrictive conclusions have been drawn from the second part
Yes, instead of my #1 a lot of statements in the literature basically assume, without proof (and as we now believe, erroneously), that only zero proper acceleration is allowed on the axis of rotation, when in fact nonzero but constant proper acceleration, since it is also a Killing motion (and can therefore be combined with constant angular velocity to yield a motion that is also a Killing motion--but note that this only works if the linear acceleration is along the rotation axis, which is why that restriction must be present), is also allowed. More generally, the way to look for rotating congruences that can be rigid is to look for motions of the rotation axis that are Killing motions--as is, for example, "hovering" at a constant radius in Schwarzschild spacetime.

PAllen
2019 Award
--but note that this only works if the linear acceleration is along the rotation axis, which is why that restriction must be present), is also allowed. More generally, the way to look for rotating congruences that can be rigid is to look for motions of the rotation axis that are Killing motions--as is, for example, "hovering" at a constant radius in Schwarzschild spacetime.
That's really interesting and suggests the surprising conclusion that a disk in radial free fall in SC geometry is not an allowed rigid motion! But a gyroscope, in circular orbit is ok; other orbits are not ok.

PeterDonis
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2019 Award
That's really interesting and suggests the surprising conclusion that a disk in radial free fall in SC geometry is not an allowed rigid motion!
Yes, it does. However, this case is more complicated to analyze because (unlike the case of constant proper acceleration) we can't carry over the corresponding analysis from flat spacetime for anything more than a small local patch of the motion.

But a gyroscope, in circular orbit is ok
I'm not even sure this is true in general, because this motion combines two rotations: the orbital motion of the gyro's center of mass is already a combination of two KVFs, ##\partial_t## and ##\partial_{\phi}## (in the standard Schwarzschild coordinates), and adding in the gyro's own rotation on top of that will, in general, add a rotation about a different axis, so the combined rotation will not (I think) be a linear combination of KVFs with constant coefficients. The only case in which I think this would work is if the gyro's own spin axis were oriented in exactly the same direction as the axis of its orbit (i.e., exactly perpendicular to the plane of its orbit), so the combined rotation is just a rotation about the same axis with a different (but still constant) angular velocity.