Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor

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As the Nambu—Goto action simply equals the world-sheet area up to an overall factor of the string tension , it is natural to speculate on the possibility that string theory admits space-time manifolds on which only the notion of area is defined, in the absence of a metric to define the notion of length. This topic is also interesting from a purely mathematical point of view. Perhaps the notion of area is more fundamental than that of length in physics from the perspectives of string theory.

The aim of this paper is to write down an equation for the area metric analogous to the Einstein equation. In fact, an analogue of the Einstein equation was given in Ref.

Tensor Calculus Lecture 8: Embedded Surfaces and the Curvature Tensor

Instead, we hope to deal with geometric structures that cannot be properly characterized by any metric or effective metric , and we would like to provide an alternative approach in which the area metric will be the only quantity that defines geometry in our formulation. This means that we need to find a way to associate a particular area connection to any given area metric.

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Such an area connection has not so far been discussed. The plan of the paper is as follows. In Sect. The focus of the paper is the scenario of string theory, on which we have already briefly commented above. We note in Sect. The notion of area connection is introduced in Sect.

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In contrast to the uniqueness of the Levi—Civita connection in Riemannian geometry, the area metricity condition and the area torsion-free condition are not sufficient to uniquely fix the area connection, because there are a lot more covariant degrees of freedom in the area connection.

We show in Sect. Area curvature is defined in Sect. We find static, spherical solutions to the generalized Einstein equation in vacuum in Sect. Finally, we summarize and comment in the last section. There are at least two scenarios in which the geometry of space-time is characterized not by the notion of length, but by the notion of area. The first scenario occurs in the context of quantum gravity when the quantum state of the space-time is a superposition of several coherent states with different metrics.

The second scenario happens in string theory as a more general background for the Nambu—Goto action of a free string. A simple way to see this is the following.

Gilkey Peter B

The second scenario in which area replaces length as the fundamental geometric notion of space-time occurs in string theory. In string theory, forces are mediated by strings, and the only necessary notion of geometry for the string world-sheet action is the area, not the length. While a metric always defines an area metric, a priori a well defined area does not necessarily imply a unique well defined metric. Hence we expect that, in string theory, it is sensible to consider space-time background geometries characterized by the notion of area, in the absence of the notion of length.

In the background in which the string world-sheet action is fully determined by an area metric, the complete information of space-time geometry is encoded in the area metric. The second scenario will be the focus of this work, although most of the discussions below apply to both scenarios. In this paper we focus on the second scenario string theory , and we will comment in Sect.

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It is possible to generalize the notion of area further, just like the notion of length can be generalized in Finsler geometry. However, it will be restricted by other constraints. We will see below that this definition of area is suitable for the application to string theory. The difference between metric and area metric is minimal. We discuss this issue separately for even and odd dimensions.

Furthermore, the effective metric is in general not covariantly constant because the volume form is not, regardless of how one defines the area connection, as we will see below in Appendix A. In the perturbative string theory, the dynamics of the background fields can be derived from the string world-sheet theory by requiring conformal symmetry, but one has to first rewrite the Nambu—Goto action in another way before the quantization of the theory can be carried out.

We will leave the quantization of this action for future work. Instead, we make some comments here.

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First we recall that the Nambu—Goto action of a string in a generic Riemannian space-time is classically equivalent to the Polyakov action, which can be quantized as a perturbation theory around the Minkowski background. Einstein's equation is the constraint on the background at the leading order to ensure that the beta functions vanish.

In principle, this procedure of deriving Einstein's equation from the string world-sheet action can be generalized to derive a field equation for the area metric. We hope that, even without an actual calculation of the beta functions, it is possible to make a good guess about the generalized Einstein equation if we know how to define geometric quantities that generalize connection and curvature, when the metric is replaced by the area metric. We will refer to these quantities as area connection and area curvature, or just connection and curvature when there is no risk of confusion.

In Riemannian geometry, the Levi—Civita connection is uniquely fixed by the metricity condition and the torsion-free condition, and the Einstein equation and Hilbert—Einstein action can be expressed in terms of its curvature. In other words, it is the only tensor that one needs to fix in order to uniquely determine the connection for a given metric. The difference between metric and area metric is less significant. But there are more canonical forms of the area metric, categorized into 23 meta-classes [ 9 ]. Natural World - Hardcover / Mathematics / Science, Nature & Math: Books

Another condition that one may wish to impose on the area connection is its compatibility with the generalized geodesic equation, namely the equation of motion for a string to extremize its world-volume. In this subsection, we derive the generalized geodesic equation and find its compatibility condition with the area connection. We will show in this section that similarly one can add a tensor to any area connection such that the new area connection is torsion-free 67 , and satisfies the same area metricity condition 63 and 64 , although the algebra involved is more complicated.

There are in fact only 9 independent variables because of the constraint Thus the task is to solve a reduced set of 9 linear relations for 9 independent variables. In the discussions on physical theories of gravity the dynamics of space-time geometry , the choice of a connection is in some sense a convention. If a connection is used in a gravitational theory to write down its field equation, one can always rewrite it in terms of another connection, merely by a field redefinition.

For example, the Levi—Civita connection is conventionally adopted in Einstein's theory. The choice of connection depends on the choice of formulation of the physical theory of gravity. Nevertheless, the Levi—Civita connection has its conceptual advantage. With torsion viewed as the covariant information contained in a connection, the Levi—Civita connection is the special connection for which this information is empty.

It can be used as a reference connection.

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In this section, we aim at defining the counterpart of Levi—Civita connection for the geometry of the area metric. This will provide a reference area connection for the convenience of future discussions. More importantly, this task helps us understand better the content of the area connection. This understanding will be useful when we construct a dynamical theory for the area metric. As a generalization of the Levi—Civita connection, the reference area connection should satisfy both the area metricity condition 63 and the area torsion-free condition It was noted in Sect.

The problem is to find suitable constraints just enough to fix the area connection without inconsistencies with the area metricity and the area torsion-free conditions. The basic idea is to introduce just enough degrees of freedom in a simple way in the area connection to satisfy the constraints, i. There are, of course, infinitely many area connections satisfying the area metricity condition. Our task is to find one such connection that is simple, and can be easily related to the Levi—Civita connection when the area metric is defined from a Riemannian metric through 5.

The calculation of the area Levi—Civita connection for a given area metric can proceed as follows. We have no intention of claiming that this is the only way to uniquely determine the area connection for a given area metric. The simple fact is that, in the case of area geometry, there are a lot more covariant tensorial degrees of freedom in the area connection while in Riemannian geometry the torsion is the only one , and as a result there are a lot of different ways to establish a connection between the area metric and the area connection.