BaVa b RabV aVb

The term Rab is the Ricci curvature term, which is a measure of how spacetime is curved. An indication of what is curvature is given below. For those acquainted with the most elementary of physics things are indeed starting to become strange! Yet this is illustrated for some small degree of education, and to indicate that this leads to the famous Einstein field equation.

The third law of motion is:

• For one mass acting upon a second mass with a momentum divergence Pa;b will experience a momentum divergence Pa'b so that

However, this is not the full form of this third law. The Einstein field equation is

2 c4

for R = gabRab the Ricci curvature scalar. The conservation law analogous to the third law of motion is that Tab;b = 0. The term Tab is the momentum-energy tensor, which contains all the source fields for a spacetime configuration. If Tab is zero spacetime can still be curved with Rab = ^Rgab, which is called an Einstein space. This will describe spacetime physics without any sources, such as gravity waves in free spacetime. Gravity waves are analogous to electromagnetic waves in a region free of charges.

This is an enormous jump in conceptual and mathematical abstraction, which is presented here for one's ability to say, "I have seen Einstein's field equation," and to see that general relativity has much the same outline as special relativity and Newtonian mechanics.

So how is curvature defined? Above we see that there is this object called the Ricci curvature, which is some measure of how a space is curved. Consider a ball with a vector at the north pole tangent to the surface there. One can do this with a pencil on a ball. First slide the pencil down the ball keeping it tangent to the sphere. Then slide the vector along the equator, without changing the pointing direction, until it has gone one fourth the way around the ball. Now slide the pencil back to the pole. This is illustrated in Figure 6.5. The vector has been rotated by an angle 0 = n/2, and the area of the sphere enclosed by this path is A = nr2 /2, or an eighth of the spherical surface area. The Ricci curvature scalar, R = 0/A, is this angle

Fig. 6.5. Parallel transport of a vector on a two dimensional sphere.

divided by the area enclosed by this curve,

This is the simplest definition of curvature, where in general curvature is defined by a tensor. This tensor involves components which "project" out of the area enclosed by how a vector is carried along this closed loop. A vector carried this way, called parallel translation, around a curve defines curvature components, which when summed together gives the curvature scalar defined by how its angle changes divided by the enclosed area.

One might ask, "What about quantum gravity?" Issues of quantum mechanics are largely avoided here, and quantum gravity in any depth is outside the scope of this book. However, it can be said that the pattern is clear. A successful theory of quantum gravity is likely to be an effective deformation of the three laws of motion in a manner that incorporates quantum theory in the appropriate geometric content.

This look at general relativity is meant for the reader's enlightenment and for some background for the discussion on the prospects for exotic methods of star travel in Chapter 11. This sets up some of the ideas that will be used to discuss these matters. The Einstein field equation do predict solutions such as wormholes and even warp drives. However, there are problematic issues associated with these. These solutions may in fact not really exist, but are probably mathematical artifacts.

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