Gravitational interactions between the earth and the moon cause the well- known phenomenon of the oceanic tides( Fig. 3). The gravitational forces experienced by the side of the earth facing the moon is greater than that between the moon and the center of the earth, which in turn is greater than that exerted on the far side of the earth. Thus the differential gravitational forces of the moon on the earth act to flatten the earth with the long axis pg soft pointing toward the moon.

The liquid seas are influenced more by tidal forces than the solid earth. The sun also exerts tidal forces on the earth( although they are not as great because they depend on the inverse cube of the distance between the two bodies). At full and new moon, when all three bodies are in a line, spring, or pg soft maximum tides result. Similarly, during first and last quarter, the minimum neap tides occur

The question of whether the gravitational interaction is described by the Einstein theory of relativity at all scales is of both theoretical and practical interest. On theory side, the attempts to construct an alternative model, successful or not, serve to better understanding of the fundamental principles lying behind the theory of gravity. The requirement of general covariance fixes the form of the gravitational Lagrangian almost uniquely.

There exist a very few modifications of gravity which do not involve higher- derivative terms, the most known being scalar- tensor models of the Brans- Dicke type [1]. A alami question then is whether the general covariance can be broken, say, spontaneously, in a manner similar to the Higgs mechanism in gauge theories. If that were the case one would expect, by analogy with the gauge theory, that the graviton gets a non- zero mass. More generally, the question is whether at all one can construct a consistent theory of gravity where the graviton has a non- zero mass. Whatever is the answer, it will certainly contribute to better understanding of gravity.

On phenomenological side, the conventional theory of gravity is completely successful at scales of order and below the solar system size up to scales of order a fraction of a millimeter. At larger scales there is a hint of a problem: one needs to introduce the( otherwise undetected) dark matter in order to explain the rotation curves of the galaxies and galaxy clusters.

At cosmological scales yet another form of matter— the one behaving like the cosmological constant— is also needed [2]. With these two additions the Einsteins theory apparently works quite well at all scales. However, it is disturbing that the new components are only needed to correct the gravitational interaction at very large scales. Moreover, at those scales the new components must play a dominant role in order to bugat the observations.

Before accepting the existence of the new forms of matter it is alami to wonder whether the gravitational interaction itself can be modified at large distances so as to explain the existing observations without the need of the dark matter and the dark energy. Whether likely or not, this is a logical possibility.

Since the gravitational field obeys Newtons law, which is mathematically similar to Coulombs law for the electric field, Gauss theorem is valid for the gravitational field also. The only difference is that the charge in Gauss theorem is now replaced by the mass times the gravitational constant. Thus the gravitational flux through a closed surface is−4πmG, where m is the total mass within the surface; the minus sign is due to the fact that gravitational forces are attractive.

By using this theorem we can, for example, determine the gravitational field within a uniform sphere. This problem is identical with that of a uniformly charged sphere, discussed in §21. From the result obtained there we can write down immediately