Chapter 15


Einstein's Geometric Theory of Gravity

I. Introduction

The major conclusion of Chapter 14 was that the Minkowski spacetime structure is incompatible with the Principle of Equivalence. That is to say, gravity can not be imposed upon a Minkowski spacetime background without creating anomalies. Since the Principle of Equivalence is considered to be verified by experiment, we suggested that Minkowski spacetime must be changed in some manner. In this chapter, we explore in more detail just how Minkowski spacetime might be changed. We survey the main qualitative features of Einstein's 1915 geometric theory of gravity, and also give an example of the theory, the black-hole solution. Einstein's general theory of relativity is not the only theory in which Minkowski spacetime can be made compatible with the principle of equivalence. We discuss some of the solar system experiments that have been performed to test the extent to which Einstein's theory has more empirical support than other gravitational theories.

II. Two Roads to the Geometric Theory of Gravity

How should Minkowski spacetime be changed to incorporate gravitation? This is a fundamental question, one that Einstein worked on from arolmd 1908 until 1915 when he completed the geometric theory of gravitation. By 1912 Einstein was convinced that the equivalence of the laws of physics in any pair of accelerated reference frames necessitates a special brand of mathematical language - the language of "tensor analysis and differential geometry". Initially Einstein did not know this kind of mathematics. He learned about it from his close friend Grossman sometime in 1912. From then until 1915 he explored several alternative ways of representing the


gravitational field geometrically. It was a long, hard road littered with false starts and dead ends. What emerged in the end was a theory of gravity that produced another revolution, perhaps more radical in content than the 1905 Special Theory of Relativity. Our task is to describe this theory. The manner in which we do this will deviate from the historical record. We believe that we can give a rational reconstruction, which, while not reflecting the historical sequence of events, nevertheless has the advantage of making the 1915 theory seem to be a more or less natural step in the development of spacetime theories.

In 1922, some seven years after the general theory was published by Einstein, a paper by the famous mathematician Elie Cartan showed how one could start with Absolute Space, Universal Time, and the Newtonian theory of gravity (the inverse square force law) and generate a type of spacetime model in which gravity was interpreted as a manifestation of some type of geometric structure. The spacetime model which emerges in this "geometrization" of gravity is not the same as the geometric theory of gravity that Einstein finally generated. Cartan's spacetime model does not include the constraint that the speed of light is a constant for all observers. When it is included, the result is the same as Einstein's. In what follows we will describe the Cartan path in qualitative terms.

Einstein's procedure was to start with Minkowski spacetime and generalize it. (Both the Einstein and the Cartan paths ultimately lead to the same destination if Cartan's results is modified to include the truths of Special Relativity in some way.) These alternatives are diagramed in Fig. 15.1.


Absolute Space

Universal Time

Newtonian Gravity


E Universality of Light Geometrize only Newtonian C


- ' t R

S Minkowski Spacetime Newtonian Spacetime TA

T _ . _

I ~ ~ N

N |Principle of Equivalence I rUniversality of light I S


"Geometrize Gravity"
* "Geometrize" Special

R by modifying Minkowski Relativity by Modifying R

deg. Spacetime Newtonian Spacetime to O

A ~ make it locally con- A

D \ consistent with the D

~ universality of light

Einstein's Geometric Theory of Gravity

Locally: Minkowski Spacetime

Space: in general not Euclidean

Time: not universally constant

Matter affects geometry and vice versa

Spacetime structure dynamic ~ ,9 I h I

III. Geometrical Digression: Geodesics, Connection, and Curvature

We pause in this section to give a qualitative discussion of several geometrical concepts which are useful in our treatment of Newtonian and Einsteinian gravitation in the sections which follow the present. This interlude is presented now in the hope that the reader will digest the conceptual essence of the concepts and then be ready to witness applications of them to gravity.


Geodesics and the Connection

In our discussion we will focus on some of the geometrical features of two examples: the two-dimensional plane and the two-dimensional surface of a sphere of given radius.

~ne S~lr~ce S~ S(~a~e

F-g ~

We choose these two surfaces because they are both familiar and because they exhibit significant differences in their intrinsic geometrical properties. Probably the most obvious difference is that the plane is "flat" while the sphere is "curved". This is particularly clear from our vantage point "up" in three-dimensional space. However, it is less clear as to how one should characterize this difference if one were confined to "live" in these surfaces and could not look out and around to see that the sphere "curves out" from under us while the plane just keeps its steady "flatness" no matter how far we look out. We want to discuss, then, how can we stay within the given surface and make measurements and comparisons of the properties of these surfaces from point to point.

Let us consider the plane first. Suppose we identify two points Pl and P2 on the plane and ask what is the shortest path in the plane connecting

Pl to P2



F;9 . IS.3

As we have indicated~ the shortest path is a straight line, since the other, non-straight paths, take more turns and consequently require a larger number of steps to traverse. Thus the problem of going from Pl to P2 in the plane along the shortest path is simply solved by constructing the straight line from one point to the other.

How is it that one recognizes whether the path one is traversing is straight or not given that such recognition must be accomplished by making measurement in the plane? The criterion of straightness may be quantified by focussing on a local property of the path: its local tangent vector. Thls refers to a directed line segment which is tangent to the given point on the path. For example consider the following path in the plane.

~g. l5. "

Imagine traversing the path from A to B along some arbitrary curve. At the point P we construct a line segment which is tangent to P. Since the motion over the path is from A to B we assign an arrow head as shown. This directed line segment is what we mean by the tangent vector at P.


Suppose now that we have two vector V and W defined at the same point P. How can we compare these two vectors? Since they are at the same point we compare them by determining whether they have the same direction or the same length: ~

F,g. ~S. 5

The situation is not the same when the vectors are based at different points in the plane, as illustrated above for the vector Z at point Q and the vector S at point R. In order to compare the tangent vectors at different points on the given curve we need to specify a rule by means of which a tangent vector, defined at one point can be compared to a tangent vector defined at another point. Such a rule allows quantities at nearby but distinct points to be connected. It is as though we are wearing "blinders" so that we can only "inch" our way along from one point to a slightly different point. The rule which allows us to transport a tangent vector from one point to the slightly different point is thus a "locally" defined rule. We use the name connection for this rule which tells on how to take a tangent vector and move it over to the other nearby point. Once there we can compare the two vectors by asking whether the vector which is naturally there points in the same direction, e.g., as the one which we transported over via our rule of connection.

The rule which geometers have adopted for the plane is one which allows the comparison of tangent vectors at only slightly different points. We start at a given point in the plane and move to another nearby point by "pushing along" the tangent vector, defined at the original point, keeping


it as parallel as possible to its original direction. This pushing along of the tangent vector is called parallel propagation because the vector is

moved (propagated) parallel to itself. It serves as the basic criterion


that we have for insuring the straightest line in the plane. In the case of the point in the plane we would have an initial tangent vector and a "final" tangent vector obtained by the rule: Keep the original direction unchanged. We illustrate this idea in the following diagram.

or jg ~ po,r ~ nspor ~ Qc~or

~ ~4i.. ~ t

O ~ a~ bO;n~

Fig ~

Notice in the above diagram that by preserving the original direction of the tangent vector we get two tangent vectors which could be "strung together" by drawing a short, straight line between Pl and P2.

Let us emphasize that there is a natural sense in which this idea of connection is already present in the Euclidean two-dimensional plane. We are making it explicit now because in the less familiar context of curved two-dimensional or higher dimensional spaces (and spacetimes) to be discussed later in this chapter, the connection plays a significant role.

Returning to our question, we can determine which curve is straight as follows. At the starting point Pl we choose a direction tangent to curve through Pl. Note that there are an infinite number of distinct tangents issuing from Pl.