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.

Section 7–7 What is gravity?

(Mechanisms of gravity / Strength of gravity / Test of gravity)

In this section, the three main points discussed are mechanisms of gravitational force, the relative strength of gravitational force, and a test of gravitational force.

1. Mechanisms of gravitation:

“Many mechanisms for gravitation have been suggested. It is interesting to consider one of these, which many people have thought of from time to time (Feynman et al., 1963, section 7.7 What is gravity?).”

Feynman explains that many mechanisms of gravitation have been suggested, but no one has given any satisfactory machinery. On the other hand, Newton made no hypotheses about gravitation and he was satisfied with mathematical laws for which no machinery is available. Thus, Feynman cites the law of conservation of energy as a theorem concerning quantities that can be calculated with no mention of the machinery. From a perspective of pragmatism, physicists may continue with this approach such that there are more discoveries or inventions. However, some physicists are deducing how Higgs mechanism may result in gravity. Importantly, we can now detect gravitational waves that are propagating in the speed of light.

Feynman briefly describes Le Sage’s kinetic theory of gravity in terms of tiny invisible particles that impact material objects from all directions. In this theory, any two bodies partially shield each other from the impinging corpuscles, resulting in a net pressure exerted by the impact of corpuscles on the bodies. Feynman reasons that the Earth would stop as a result of the continuous bombardments from these particles. Historically, Newton said that the only possible mechanical cause of gravity was conceptualized by Nicolas Fatio de Duillier in 1690 (Cohen & Whitman, 1999). At present, there are new perspectives on Le Sage’s theory of gravity (Edwards, 2002), and more important, the tiny invisible particles can now be known as gravitons.

2. Strengths of gravitation:

“…The relative strengths of electrical and gravitational interactions between two electrons (Feynman et al., 1963, section 7.7 What is gravity?).”

Feynman expresses the relative strengths of electrical and gravitational interactions between two electrons as 1 divided by 4.17×10⁴². This is similar to the ratio of the volume of the flea to the volume of the Earth. Interestingly, Dirac suggests that the gravitational constant is time dependent and it may be related to the age of the universe. However, Feynman raises an issue if one considers the age of the universe in years: years are not “natural” units of time. That is, years were arbitrarily defined by men depending on the rotational period of a planet. In Feynman’s lectures on gravitation for postgraduate students, he further criticizes Dirac and mentions that “he might just as well describe the situation by saying that the electric charge is time-dependent... (Feynman et al., 1995, p. 8).”

Feynman adds that there is no explanation of gravitation in terms of other forces at the present time. According to Wilczek, a possible mechanism of gravity may be a residual effect of fundamental forces. In his own words, “Gravity might be derived from the other fundamental forces. Because it is a small (feeble) effect, maybe gravity is a byproduct, a small residual after the near-cancellation of effects of opposite electric or color charges, or something more exotic (Wilczek, 2008, p. 149).” Nevertheless, one needs to justify why this force is universal. Moreover, Wilczek suggests that the question “Why is gravity so feeble?” may be rephrased as “Why are protons so light?” In essence, the gravitational force is relatively weak because the object is very light.

3. Tests of gravitation:

“The absence of such an effect has been checked with great accuracy by an experiment done first by Eötvös in 1909 and more recently by Dicke (Feynman et al., 1963, section 7.7 What is gravity?).”

Eötvös first investigated the proportionality (or equivalence) of gravitational mass and inertial mass in 1889 instead of 1909 (Bod et al., 1991). Feynman explains that the gravitational force is exactly proportional to the mass with great precision because there should be an observable effect if the inertia (inertial mass) and weight (gravitational mass) are numerically different. In other words, in his first Messenger lecture, Feynman (1965) states that “[o]ne other test of the law of gravity is very interesting, and that is the question whether the pull is exactly proportional to the mass (p. 29).” Importantly, the uncertainty of this experiment pertaining to the two kinds of mass is within 1 part in 1,000,000,000, or less based on all of the substances tried.

Eötvös’s apparatus consists of two objects which have different materials and are connected by a rod that is suspended horizontally by a thin fiber or torsion balance. Due to Earth’s rotation, the two strings that support the weights of the two objects are not exactly vertical. Based on the Earth’s reference frame, the sum of the string’s tension and the weight of the object are equal to the inertial (centrifugal) force. Note that the weight of the object is dependent on the gravitational mass of the object, whereas the centrifugal force is related to the inertial mass of the object. Essentially, if the gravitational mass and inertial mass are not exactly the same, the rod will rotate.

Feynman explains that the experiment was first done by Eötvös and more recently by Dicke. To give a better picture, Roll, Krotkov, and Dicke (1964) clarify that Eötvös experiment has a precision of 3 ´ 10^-9 and thus, it only supports a weaker form of Einstein’s principle of equivalence (in short, weak equivalence principle). More significantly, Einstein’s general theory of relativity is based on the assumption of “strong equivalence principle.” Furthermore, Roll, Krotkov, and Dicke (1964) argue that the accuracy of the experiment is reduced by effects such as gravitational field gradients, varying magnetic fields, variable electrostatic forces, temperature variation effects, and ground vibration disturbances. By improving their experimental design, they reported that this experiment achieves an accuracy of 1 ´ 10^-11.

Questions for discussion:

1. Should there be a mechanism of gravity?

2. Why is the gravitational force relatively weak?

3. Why should the test of gravity be more precise?

The moral of the lesson: the gravitational force is relatively weak and there should be more precise tests of gravity.

References:

1. Bod, L., Fischbach, E., Marx, G., & Náray-Ziegler, M. (1991). One hundred years of the Eötvös experiment. Acta Physica Hungarica, 69(3-4), 335-355.

2. Edwards, M. R. (ed.) (2002). Pushing Gravity: New Perspectives on Le Sage's Theory of Gravitation. Montreal: Apeiron.

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

5. Newton, I. (1999). The Principia: Mathematical Principles of Natural Philosophy (Translated by, I. B. Cohen & A. Whitman). Berkeley: University of California Press.

6. Roll, P. G., Krotkov, R., & Dicke, R. H. (1964). The equivalence of inertial and passive gravitational mass. Annals of Physics, 26(3), 442-517.

7. Wilczek, F. (2008). The lightness of being: Mass, ether, and the unification of forces. New York: Basic Books.

Section 7–6 Cavendish experiment

(Cavendish’s apparatus / Weighing the earth / Gravitational constant)

In this section, the three interesting points discussed are Cavendish’s apparatus, weighing the earth, and gravitational constant.

1. Cavendish’s apparatus:

“…It was first measured by Cavendish with an apparatus which is schematically indicated in Fig. 7–13 (Feynman et al., 1963, section 7.6 Cavendish experiment).”

The heart of Cavendish’s apparatus is a torsion balance that is made of a very fine fiber. The strength of the gravitational force is provided by two large, fixed balls of lead and two smaller balls of lead on the ends of an arm. By measuring the extent the fiber is twisted, one can measure the strength of the force that it is inversely proportional to the square of the distance between the two objects. One difficulty of this experiment is the weakness of the force that may be measured inaccurately if there are disturbances such as electrical forces or a draught. Thus, there is a need to cover the apparatus to keep the air out and make sure it is not electrically charged. Cavendish placed the whole apparatus in a room which remained shut and he observed the motion of the arm from the outside using a telescope.

The apparatus was originally designed by Rev. John Michell. To recognize Michell’s design and work, it should be renamed as Michell-Cavendish apparatus. In his seminal paper, Cavendish (1798) writes that “Many years ago, the late Rev. John Michell, of this Society, contrived a method of determining the density of the earth, by rendering sensible the attraction of small quantities of matter; but, as he was engaged in other pursuits, he did not complete the apparatus till a short time before his death, and did not live to make any experiments with it. After his death, the apparatus came to the Rev. Francis John Hyde Wollaston, Jacksonian Professor at Cambridge, who, not having conveniences for making experiments with it, in the manner he could wish, was so good as to give it to me (p. 469).”

2. Weighing the Earth:

“This experiment has been called ‘weighing the earth’. Cavendish claimed he was weighing the earth… (Feynman et al., 1963, section 7.6 Cavendish experiment).”

Feynman mentions that Cavendish’s experiment has been called “weighing the earth.” In his first Messenger lecture, he explains that “[w]ith pedantic and careful teaching today we would not let our students say that; we would have to say ‘measuring the mass of the earth’ (p. 28).” Essentially, we can calculate the mass of the earth by determining the Newtonian gravitational constant (G), the attractive gravitational force, and the distances between the two objects from this experiment. However, Cavendish did not determine either G through this experiment. In fact, he determined the average density of the earth, which he expressed as a ratio of the earth’s density to the density of water.

The title of Cavendish’s (1798) paper is “Experiments to determine the density of the Earth.” Cavendish’s main purpose in this paper was to solve an eighteenth-century dispute: what is the mean density of the Earth? In other words, Cavendish’s aim was not to determine the gravitational constant, but rather to deduce the Earth’s density. This experiment was motivated by criticisms of astronomical investigations which ignored the gravitational attraction of mountains.

Note: In the Feynman lectures websites, it is recently changed to “This experiment has been called ‘weighing the earth’ by some people.”

3. Gravitational constant:

“This is the only way in which the mass of the earth can be determined. G turns out to be 6.670×10⁻¹¹ newton⋅m²/kg² (Feynman et al., 1963, section 7.6 Cavendish experiment).”

Feynman explains that Cavendish’s apparatus can be used to determine the coefficient G of the gravity law instead of only weighing the Earth. Interestingly, in his first Messenger lecture, he adds that Cavendish “was weighing the sun and everything else at the same time, because the pull of the sun is known in the same manner (Feynman, 1965, p. 29).” Simply put, one may accurately determine the gravitational constant from the formula, F = Gmm′/r². According to Feynman, Newtonian gravitational constant turns out to be 6.670 × 10⁻¹¹ newton⋅m²/kg². However, the 2014 Committee on Data for Science and Technology (CODATA) recommended the value of Newtonian gravitational constant to be 6.67408(31) × 10⁻¹¹ m³ kg^-1 s^-2.

Newtonian gravitational constant (G) is extremely difficult to measure precisely and it has no definitive relationship with other fundamental constants. Furthermore, variations of gravitational constant measured by different laboratories suggest that the uncertainty in G could be much larger. Recently, Kuroda (1995) suggested that internal friction in the torsion fiber may have caused errors in the previous measurements. One way to resolve this problem is to include a turntable and adjust its rotation rate such the torsion fiber is not twisted anymore. Alternatively, physicists have determined G by using an atomic fountain in which atoms are allowed to rise and fall (Peters, Chung, & Chu, 1999).

Note: One may argue that Feynman has goofed again when he says that “[t]his is the only way in which the mass of the earth can be determined.” This is because there are many different ways to determine the mass of earth, for example, by using a turntable or atomic fountain to deduce the gravitational constant.

Questions for discussion:

1. What are the main features of Cavendish’s apparatus?

2. Should Cavendish’s experiment be known as “weighing the Earth”?

3. What are the other ways of determining Newtonian gravitational constant?

The moral of the lesson: Newtonian gravitational constant can be determined by many ways such as the use of Cavendish’s apparatus.

References:

1. Cavendish, H. (1798). Experiments to determine the density of the Earth. By Henry Cavendish, Esq. FRS and AS. Philosophical Transactions of the Royal Society of London, 88, 469-52.

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. Kuroda, K. (1995). Does the time-of-swing method give a correct value of the newtonian gravitational constant?. Physical Review Letters, 75(15), 2796-2798.

5. Peters, A., Chung, K. Y., & Chu, S. (1999). Measurement of gravitational acceleration by dropping atoms. Nature, 400(6747), 849-852.