We are here to consider an extraordinary topic, one that ranges from the infinitely large to the infinitely small. We want to find what general relativity and gravitation physics have to do with the description of nature. This task imposes a heavy burden of judgement and courage on us, for never before has theoretical physics had to face such wide subject matter, assisted by so comprehensive a theory but so little tested by experiment.
Our problems fall under three heads. First: What checks do we have today of Einstein's general relativity theory, and what can be done to improve our tests? Second: What ways can we see to draw new richness out of the theory? And third: What connection can we foresee between this theory and the quantum world of the elementary particles?
Without going over all the evidence from observational tests of general relativity, it is perhaps worthwhile to set down a few numbers relating to the three traditional tests of the theory.
Planet  Observed Value  Prediction from General Relativity 

Mercury  43.15  
Earth  3.84  
Mars  —  1.35 
Tab. 3.1: Precession in seconds of arc per century
Table 3.1 compares the predicted and observed values of the precession of the perihelion of Mercury and the Earth; for Mars there are not yet data to give a check. The predicted effects are well within the experimental error.
The second wellknown test is the deflection of light in passing the sun, which shows itself as the difference between photographs of the star field taken when the sun is in its midst and those taken when the sun is elsewhere. The expected effect is 1.751 seconds of arc deflection for a ray which passes the sun at grazing incidence. Although there has been some disagreement among astronomers, the careful review of the evidence by McVittie leads to acceptance of Mitchell's result, based on the 1919, 1922, and 1929 eclipses, of seconds of arc, which value has since been confirmed by observations of the 1952 eclipse. The theoretical prediction is thus in agreement with the data.
The third observational test of general relativity, the gravitational red shift of spectral lines emitted on the surface of a massive star, presents insuperable difficulties in the case of the sun, owing to the fact that spectral shifts originating in atmospheric currents and turbulence completely mask the effect. On the other hand, white dwarf stars have mass to radius ratios large enough to make these Doppler effects negligible in comparison with the gravitational red shift. Observations are available on two stars, and are summarized in Table 3.2.
40 Eri B  
Sirius B  1  0.008 
Tab. 3.2: Observed and Predicted Gravitational Red Shifts
It appears from these three traditional tests that general relativity is not in disagreement with the observations.
The chief possibilities of improvement exist in the second and third effects  that is, the bending of light and the gravitational red shift. We hope to hear from Professor Dicke later what prospects there are for new experiments. In particular, the development of the “atomic clock” may begin to make feasible detection of gravitational red shift on the earth itself.
To go on to the second general problem, namely, how to draw new richness out of the theory on the classical level: there is a farreaching analogy
We can, however, go on to explore some of these analogies on a deeper level. First among the problems is the “initial value problem.”
Now this is a local condition; but boundary conditions in the large must be considered, especially when considering systems with closed topology  and to this problem we may assign a score of, say, 25.
A third problem is that of translating
To pass on from these issues, let us consider some further analogies with electromagnetic theory.
in which
and
have been reduced to parallelism. We then have a canonical situation in which, if we chose the
axis as the direction of
and
, any rotation in the

plane leaves the situation unchanged, and any Lorentz transformation in the

plane also leaves the situation unchanged. Thus what Schouten has called a “twobladed” structure of the spacetime continuum at any point is defined by the electromagnetic field. This is then a geometrical interpretation of what we have to deal with in the electromagnetic case, and we would like to understand what the analogous situation is in the gravitational case. Here we have to deal with the quantities
that measure the curvature of space  geometrical quantities much more elaborate than the electromagnetic field tensor, of course. We would like to understand what kinds of invariance this quantity defines. We have the remarkable work of Géhéniau
This in turn leads us to another issue. Einstein has taught us to think in terms of closed continua, rather than the open continua such as Euclidean space; but since the differential equations are purely local and say nothing about the topology in the large, we are obliged to consider what the possibilities are. Recently we have become aware, for example, of one way of understanding, in terms of topological notions, such a concept as that of electric charge. Figure 3.1 below shows one intriguing kind of topological connectedness in the small.
Fig. 3.1
If one thinks of the upper region
as a twodimensional space, there exists the possibility of connecting two regions
and
by a “wormhole”
Fig. 3.2
An objection one hears raised against the general theory of relativity is that the equations are nonlinear, and hence too difficult to correspond to reality. I would like to apply that argument to
The nonlinear couplings between gravitation and electromagnetism also give rise to new structures such as “geons,”
The last topic to be covered under the classical aspects of general relativity is the topic of
The result is that one can write the equations of both electromagnetism and gravitation in a form in which only purely metric quantities enter, as in Einstein's theory. It is a system of fourth order equations which contain all the content of the two coupled second order equations of electromagnetism and gravity. Further investigation of theories of this kind deserve a score of, say, 25 points on our “shuffleboard” court.
In summarizing, let me remark that we have here, in the