Section 7–8 Gravity and relativity

(Einstein’s law of gravitation / Quantum gravity / Consistency)

In this section, the three interesting points mentioned are Einstein’s law of gravitation, quantum-mechanical aspects of gravitation, and consistency in our physical theories.

1. Einstein’s law of gravitation:

“By correcting it to take the delays into account, we have a new law, called Einstein’s (Feynman et al., 1963, section 7.8 Gravity and relativity).”

Einstein’s modification of Newton’s law of gravitation is worth some discussions here. Essentially, Newton’s law of gravitation is incorrect and it was modified by Einstein by incorporating his theory of relativity. First, Newton’s law implies the gravitational effect is instantaneous. In other words, gravitational signals travel at infinite speed. On the other hand, Einstein suggests that we cannot send signals faster than the speed of light, and thus, Newton’s law of gravitation must be wrong. Importantly, Einstein’s suggestion is proved correct with the detection of gravitational waves on 14 Sep 2015.

Based on Einstein’s special theory of relativity, anything which has energy has mass in the sense that it can be attracted gravitationally. For example, light, which has an energy, has a “mass.” Feynman explains that there is a gravitational attraction on a light beam by the Sun because the light beam has energy (and mass). Thus, the light does not go straight and its deflection can be observed during an eclipse of the sun. As a result, the stars which are around the sun should appear displaced from where they would be as if the sun were not there. However, particle physicists define light as massless and may not agree with Feynman on the principle of mass-energy equivalence. To be precise, some physicists prefer to state that “rest energy” and mass are equivalent.

In short, gravity is geometry. In Volume II of The Feynman Lectures, he elaborates that “Einstein said that space is curved and that matter is the source of the curvature. Matter is also the source of gravitation, so gravity is related to the curvature (Feynman et al., 1964, section 42–3 Our space is curved).” For example, two ants at the equator are getting closer to each other when they are moving towards the north. We can explain this by using spherical geometry or the shape of the Earth instead of an attractive force. Similarly, gravity is not a “force” in general relativity, but a manifestation of the curvature of space-time.

2. Quantum gravity:

“…The quantum-mechanical aspects of nature have not yet been carried over to gravitation (Feynman et al., 1963, section 7.8 Gravity and relativity).”

According to Feynman, we have discovered that all mass is made of tiny particles (e.g., protons, neutrons, or electrons) and that there are interactions, such as nuclear forces. However, there is no satisfactory theory that explains gravitation in terms of nuclear or electromagnetic forces. In his first Messenger lecture, Feynman gives an elaboration on the quantum-mechanical aspects of gravity: “the question is, how does gravity look on a small scale? That is called the Quantum Theory of Gravity. There is no Quantum Theory of Gravity today. People have not succeeded completely in making a theory which is consistent with the uncertainty principles and the quantum mechanical principles (Feynman, 1965, p. 33).”

Although Feynman mentions that the gravitational effects are so weak that the need for a quantum theory of gravitation has not yet developed, he was one of the first physicists who try to develop a consistent theory. Interestingly, he develops the concept of “ghost particles” by using the Yang-Mills theory. During an interview, Feynman reveals that “I feel I have solved the [problem of the] quantum theory of gravity in the sense that I figured out how to get the quantum principles into gravity. The result is a nonrenormalizable theory, showing it is an incomplete theory in the sense that you cannot compute anything. But I am not dissatisfied with my attempt to put gravity and quantum mechanics together (Mehra, 1994, p. 507).”

It is worth mentioning that Feynman investigated a possibility of gravitation is due to neutrino exchange. In Feynman words, “[w]e might consider whether gravitational forces might not come from the virtual exchange of a particle which is already known, such as the neutrino… (Feynman et al., 1995, p. 16).”

3. Consistency in physical theories

… for consistency in our physical theories, it would be important to see whether Newton’s law modified to Einstein’s law can be further modified to be consistent with the uncertainty principle (Feynman et al., 1963, section 7.8 Gravity and relativity).”

Some physicists attempt to unify general relativity and quantum mechanics by using string theory. However, Feynman does not have a good opinion of string theory. He raises the following questions: “We can say that haven’t got a consistent quantum theory of gravitation, except perhaps for the string theory, maybe! Who knows? It has got eleven dimensions. The world doesn’t have eleven dimensions, so it rolls out seven. Why not six, why not four? It’s a hell of a theory, isn’t it? One can’t even check the number of dimensions. I don’t think we know anything very much (Mehra, 1994, p. 507).” Thus, it may be more appropriate to describe this attempt as “string hypothesis” instead of string theory.

During Feynman’s 1962 lecture at the Conference on Relativistic Theories of Gravitation, he humorously mentions that meson physicists who had been fooling around the Yang-Mills theory did not investigate the case of zero mass carefully. Furthermore, he adds that “[t]he present theory is not a theory as it is incomplete. I do not give a rule on how to do all problems. I expect of course that if I spend more time on figuring out how to untangle the pretzels I shall be able to make it into such a theory. So let’s suppose I did. Now you can ask the question would the completed job, assuming it exists, be of any interest to an esoteric question about the quantization of gravity. Of course, it would be, because it would be the expression of the quantum theory; there is today no expression of the quantum theory which is consistent. You say: but it’s perturbation theory. But it isn’t (pp. 721-722).”

Questions for discussion:

1. What are the differences between Einstein’s theory of gravitation and Newton’s theory of gravitation?

2. How is the quantum theory of gravitation different from Einstein’s theory of gravitation?

3. Do we have a consistent physical theory of gravitation that incorporates the uncertainty principle?

The moral of the lesson: physicists need to develop a consistent physical theory of gravitation that unifies Einstein’s theory of relativity and quantum theory.

References:

1. Feynman, R. P. (1963). Quantum theory of gravitation. Acta Phys. Polonica, 24, 697–722.

2. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

5. Feynman, R. P., Morinigo, F. B., Wagner, W. G. (1995). Feynman Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

6. Mehra, J. (1994). The Beat of a Different Drum: The life and science of Richard Feynman. Oxford: Oxford University Press.