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In this paper, we investigate a certain property of curvature which differs in a remarkable way between Lorentz geometry and Euclidean geometry. In a certain sense, it turns out that rotating topological objects may have less curvature (as measured by integrating the square of the scalar curvature) than non-rotating ones. This is a consequence of the indefinite metric used in relativity theory. The results in this paper are mainly based of computer computations, and so far there is no satisfactory underlying mathematical theory. Some open problems are presented.

The purpose of this paper is to draw attention to a certain property of curvature which differs in a remarkable way between Lorentz geometry and Euclidean geometry. However, the author’s ambition is rather to pose interesting questions than to give ready answers. Coarsely speaking, the main idea is that in Lorentz geometry, rotating objects may be less curved (as measured by the integral of the square of the scalar curvature) than non-ro- tating ones. The original motivation for this mathematical problem comes from physics, but the author does not make any claims about implications in the other direction, except that any kind of deeper understanding of Lorentz geometry may eventually turn out to be useful for uniting general relativity and quantum mechanics. In any case, the problems that arise also have an independent mathematical interest.

It has been known for more than 85 years that, through the work of P. A. M. Dirac [

In [

In the following we will only present a simple example of this phenomenon, but even in this situation the computations are by far too complicated to be carried out by hand. In fact, the results in this paper depend heavily on the use of computers and so far there is no complete, satisfactory mathematical theory behind them. On the other hand, for the understanding of the general case, computers will certainly not be enough.

Considering some topologically non-trivial manifolds attached to an otherwise flat three-dimensional space to form a new manifold M, the reader may think of some kinds of wormhole (see Section 2). In this paper, we will refer to such manifolds as topological objects. We now want to study what happens if we let a topological object rotate around some axis. However, since we are mainly interested in Lorentz geometry, it is not at all clear from the beginning what the word rotation should refer to, so in Section 3 we will introduce the idea of a quasi-rota- tion, and in Section 4 we discuss how to measure curvature. In Section 5 we then apply the ideas of the previous two sections to the wormhole of Section 2.

In the case of usual Euclidean geometry, it seems very reasonable (although mathematically non-obvious) to expect that the total amount of scalar curvature should be minimized when the object is not rotating, and that it should grow with increasing speed of rotation. In fact, the very process of rotating should tend to bend the geometry inside the object, which ought to generate curvature. This is also what will come out in the example which we will consider here. What is curious, however, is that this does no longer seem to hold if we replace Euclidian geometry with Lorentz geometry (see Sections 6 and 7).

This leads us to formulate some open questions in Section 8. For example, is it a common fact of Lorentz geometry that rotating topological objects may have smaller total curvature in this sense than the corresponding non-rotating ones? In particular, one may wonder if topological objects which minimize the integral of the square of the scalar curvature are always rotating in some sense. To make this question somewhat more precise, we will in this paper consider minimizing under the extra condition that the total space-time volume is fixed.

Let us also mention that, in the example to be discussed, the fact that the total curvature may decrease when we start to rotate does not appear to be a local geometric property: at some points of the manifold the scalar curvature may increase and at other points it may decrease, as will be seen in (6.3) below. Thus, this is a global property of the scalar curvature, rather than a local one.

We start by making the idea of a “wormhole” in Euclidian three-space more precise. We first remove the two open balls

where we have adopted the convention that

Having thus defined the manifold structure and corresponding coordinates, we now proceed to define a metric on M. In fact, outside the cylindrical region, we will simply use the usual Euclidian metric

On the other hand, inside the cylindrical part C, we want to define the metric in a way which makes it reasonably regular at the boundaries of the cylinder. The reader may check that the following simple choice makes the metric

Finally, we note that the 3-dimensional volume element for

In this section, we study what happens when we rotate a topological object, defined by a manifold M as in Section 2. A very natural idea would be to say that the best method should be to find exact solutions to the field equations as has been done e.g. for rotating black holes in the form of the Kerr metric (see [

For these reasons, we will make use of a much more simpleminded approach. We will in this paper only investigate very slow rotations, and in this situation we can therefore start from the usual idea of a rotation in Euclidian space and then modify it slightly for our needs. This is clearly not the ideal way to proceed. Nevertheless, it will turn out that it is still good enough for illustrating the main point of this paper.

We first introduce some notation. Let B be an open sphere, centered at the origin in

For example, in the case of the simple wormhole of Section 2, B is a sphere which contains

We can now trivially extend M to four dimensions by adding a time-coordinate t to get

To see how to define a rotation, we start with the trivial case where M is simply equal to

where

To rotate a more general topological object, we need to construct a new map and a new image manifold which outside of

To define the four-manifold describing a rotation with angular velocity

where the union over

In a little bit less formal language, this can be viewed as just a stack of copies of the manifold M (one for each

Rather than pursuing the mathematical formalism further, let us just state some more or less obvious properties of this construction which can in fact be used to characterize

1) For each

2) For each t,

These two conditions essentially characterize the rotation, and they can in fact also be used to define the differentiable structure on

3) The map

Clearly, we can not in general demand that the map

If the four-manifolds

Clearly, such a quasi-rotation does not in itself involve Lorentz geometry in any essential way, and in fact, the idea makes equally good sense for a positive definite metric as for a pseudo-metric. Also, it is not assumed to represent any kind of physical geometry that in itself would have physical significance. A more realistic geometry for a rotating object will probably involve the time-coordinate t in a much more complicated way. In addition, there is nothing unique about the way in which we define a quasi-rotation, even if the axis of rotation and the angular speed

Since the spacial coordinates and the axis of rotation is at our choice, we will for definiteness always use a fixed axis. Hence, we will drop

In this section, we discuss how to measure curvature. In the simplest case of an object N at rest, with a metric of the simple form

where R denotes (4-dimensional) scalar curvature, and where

In the next simplest case of a quasi-rotation as in Section 3, we can, using condition II in the definition of a quasi-rotation, still measure the total curvature by integrating

In more complicated situations, (4.1) may no longer make any sense, since there may be no natural unique way of defining surfaces of constant time if there is no natural choice for the time-coordinate.

Nevertheless, it is possible to measure curvature for more general objects. If we consider the situation where space-time outside the object is flat and the object itself is, except for internal movements, at rest, then clearly we can unambiguously speak of time in the exterior in the sense of the time in the rest-frame of the object.

Suppose that we can extend the surfaces

where

It is worth noting that the extended surfaces

The purpose of this section is to sketch how to construct a quasi-rotation of the wormhole in Section 2. This essentially boils down to defining a metric on

Firstly, we will only work with very slow rotations and, as a consequence we will, just as in Section 3, partly treat the motion in a non-relativistic way, even in the case of an indefinite metric. This may appear odd at first sight, but since the theory of general relativity gives no unique way of constructing the geometry of a rotating object, the Lorentz transform may be no more natural than the Galileo transform in this case. As a matter of fact, to illustrate the point that we want to make, either one of them would do, which is why we have chosen the least technical one.

Secondly, we will only consider the case

As in Section 3, we extend the manifold M of Section 2 to four dimensions by adding a time-variable t, to obtain a 4-manifold

on

on

To construct the quasi-rotation of M, we assume the rotation to take place around the y-axis with uniform speed in the counter-clockwise direction. We therefore assume that the two boundaries located at

We note that the movement of the wormhole destroys the connection between the metric (2.3) of the cylindrical region and the Minkowski metric of the flat region. Hence, we must next investigate how the metric should be modified for

To this end, we note that from (the right hand part of) (5.3) we obtain

Next we note from (2.1) that

(which is analogous to (2.2)) when substituted into (5.4) gives

where we have again dropped the + sign and written t instead of

Carrying out the similar computations at the other end of the wormhole where

From (2.3), (5.6) and (5.7) it is now evident that the easiest way to interpolate between

for all

To fulfill the third condition, we will have to modify the metric slightly. The volume element of the metric in the interior is given by

where from (5.8) we have that

Hence we see that the function G in (5.9) is given by

where furthermore the function

equals 1 to first order at the endpoints

then clearly we get a metric with the same volume element as in the non-rotating case,

independently of v, and furthermore the regularity properties at

In this Section we will now attempt to compute the curvature integral I in (4.1) for the metrics

To compute this integral we make use of the built-in tensor calculus of Maple, since a direct computation by hand is virtually impossible (the Maple output for the scalar curvature covers 15 pages). Thus, using first Taylor expansion in v to order 5 and then numerical integration, Maple gives

Hence, we can see that

We now turn to the question whether the behavior in (6.2) is global or local. To this end, we simply split the

domain of integration in (6.1) up into 2000 small rectangles of the form

with

Thus, the fact that

For comparison we can also carry out the analogous computations in the case of a positive definite metric

where now

Again using Maple, we similarly obtain

Thus we see that in this case,

It is the belief of the author that the difference between formulas (6.2) and (7.3) reflects an important difference between Lorentz geometry and Euclidian geometry. However, the computations in this paper are very far from giving any definite solutions to all the problems that arise. In particular, the simple wormhole in Section 2 is too much simple to say something definite about the general case. Let us therefore here summarize some of the open questions:

Question 1. In the case of Lorentz geometry, is it the case that for any manifold M with metric g and curvature integral

Remark 1. Formulated in this way, it is quite likely that the question does not have an affirmative answer. For instance, it is quite possible that the behavior of

Nevertheless it could be that the question could have an affirmative answer for some reasonably large classes of manifold.

But there is also another way of formulating the problem. From the point of view of this paper, the most interesting topological objects are those which minimize the curvature in the sense of (4.1). Thus one could try to ask the following

Question 2. In the case of Lorentz geometry, is it true that a non-rotating manifold with curvature integral

If this is true, a reasonable way of proving it would be to construct some kinds of quasi-rotations of all such manifolds. Similarly one may ask, in the case of Euclidean geometry, if it is true that minimizing manifolds are always non-rotating.

The point is that the question itself is now independent of the somewhat technical definition of a quasi-rotation in Section 3. In fact, the word non-rotating manifold can be replaced by invariance under time-translation, and also we can allow for more general metrics and replace the condition of flatness outside B by a weaker asymptotic condition.

It is however a problem with the objects in this paper that it is not at all clear under what circumstances I have a minimum, even if we restrict ourselves to the case where the space-time volume is fixed in some appropriate sense. Hence, there are many existence and regularity questions which have to be considered if we want to investigate rotating objects more deeply.