Gravitational Effects on Light Propagation. Copyright 2009 Joseph A. Rybczyk

Transcription

1 Gravitational Effects on Light Propagation Copyright 2009 Joseph A. Rybczyk Abstract An examination of the theoretical relationship between gravity and light propagation is presented that focuses on the role played by gravitational time dilation. Included is a fundamental model of the photon showing the interrelationship between its internal periodic behavior and a changing gravitational potential during propagation. Also presented is a fundamental discovery involving the direct relationship between the gravitational time dilation based variance in light speed and the variance in gravitational potential between two gravitational bodies or points in a gravitational field. 1

2 Gravitational Effects on Light Propagation Copyright 2009 Joseph A. Rybczyk 1. Introduction According to the evidence light propagation appears to be affected by the intensity of the gravitational field through which it takes place. This behavior was first postulated by Albert Einstein in an article on light propagation 1 in Later, in 1923, an article on wave mechanics 2 by Louis de Broglie addressed the Special Relativity 3 principle of the time dilation related redshift of a particle and its supposed contradictory nature to the principle of relativistic energy increase. It is primarily these two works that have influenced the author s development of the ideas to be presented here. Most important among them are those dealing directly with the influence that gravitational time dilation has on the propagation process, an aspect of the behavior that has been inadequately treated up to the present day. Demystification of this process including its relationship to the internal behavior of the photon and the gravitational variance in light speed will elevate our understanding of light propagation in general and pave the way for a more accurate application of the gravitational equivalence principle. 2. Gravitational Potential and its Relationship to Light Propagation As with most aspects of relativistic theory the fundamental nature of gravitational potential and its influence on light propagation is far different from what might initially be expected. In fact it is the author s opinion that a correct understanding of this process is not possible without including the role played by gravitational time dilation. First, however, a quick review of the basic principles of gravitational potential and its effects on light propagation is necessary. To begin, let us consider the effects of the gravitational field of a source star such as the Sun and the gravitational field of the receiving body such as the Earth. Applying our understanding of gravity in general to the propagation of light, it would seem reasonable to conclude that light leaving the Sun will lose energy in the process of escaping its gravitational field and gain energy from the Earth s gravitational field upon being received on Earth. In accordance with the conservation of energy principle it is only necessary to consider the two end-points, (the Sun and Earth) to define this behavior mathematically. To convince ourselves of the correctness of this approach, let us first use integral calculus to determine the mathematical result. For the net gravitational potential between the surfaces of the Sun and Earth we derive

3 where Φ is the net gravitational potential, G is the gravitational constant, M s is the mass of the Sun, M e is the mass of the Earth, R s is the radius of the Sun, R e is the radius of the Earth, R o is the orbital distance, and R = R o R s R e. This then gives for the displacement amount where c is the speed of light. This result is in close agreement with the result obtained by the standard method using only the end point gravitational potentials as in 3 where 4 and 5 are the gravitational potentials on the surfaces of the Sun and Earth respectively. In this case we obtain for the final result, which as can be seen, is in close agreement with the previous result. (Refer to Appendixes A and B for the complete analysis.) (As a side note of interest, the gravitational acceleration rates are used in the integral equation (1) and not in the end-point definitions of equations (4) and (5). I.e. the radius distances are squared in equation (1) as opposed to not being squared in the end-point definitions of equations (4) and (5).) To derive the final factor needed for determining the frequency of light received on Earth from the Sun we simply subtract the Φ/c 2 fraction from the unit value of 1 to obtain 1 7 that then gives 3

4 1 8 where f R is the redshifted frequency received on Earth and f S is the source frequency of the light emitted on the Sun. Before going on to the next subject it is important to note that the just derived factor (7) can also be used to determine the difference in light speed between the Sun and the Earth resulting from time dilation. In the context being discussed this is not a violation of the constancy of light speed principle of relativity. What is meant here is an effective difference in light speed resulting from the difference in gravitational potential between the Sun and the Earth. In conformance to the constancy principle the value of light speed is the same in all gravitational potentials from 0 to the maximum possible. This principle involving effective light speed variance due to time dilation is an important part of this present work and will be discussed in detail as we progress. For now it is only necessary to show that the factor just derived gives where v in m/s is the effective difference in light speed between the higher gravitational potential of the Sun and the lower gravitational potential of the Earth. Counter-intuitively the light arriving at the Earth is faster, by the amount shown, than the light emitted from the Sun. This seeming contradiction of the effects of gravity on light speed is not a contradiction at all when all of the factors are considered as will be done later in the analysis. For now let us independently verify the variance in speed c given by v using another integral that determines the cumulative effects of the changing gravitational intensity between the Sun and the Earth for a photon moving between them at speed c. Where T = s, is the time it takes light to travel from the Sun to the Earth at speed c, and s = T, we derive where v, the effective change in light speed, is the same as previously determined. (Refer to Appendixes A and B for additional correlations between the two different approaches used to determine the effects of gravity on light traveling from the Sun to the Earth.) Let us now discuss the rationale for light speed effectively increasing when light travels from a higher gravitational potential to a lower gravitational potential. 3. The Effects of Gravitational Time Dilation on Light Propagation In the earlier referenced article on light propagation 1, Einstein commented to the effect that if there is a constant transmission of light between two points that are not moving relative to 4

5 each other, how can any other number of periods per second arrive at the point of reception than is emitted at the point of the source? Yet, if we think about it that is exactly what it would seem is happening if the number of periods per second arriving are fewer than the number of periods emitted. For example, in referring to Figure 1, if the waves increase in length as the waves travel to Earth, and if the speed of light is constant, then a greater number of waves are being emitted by the Sun than are being received at Earth, which is, of course, without sense. Light Waves Earth FIGURE 1 Gravitational Redshift Sun to Earth Accepting the evidence that the gravitational redshift does in fact take place, the only reasonable answer to the question just raised is that the light from the Sun speeds up as it travels to Earth. This answer holds up under scrutiny without violating the light speed constancy principle when considered along with other evidence that shows that the gravitational time dilation on Earth is less than that on the Sun. Stated another way, time runs faster on Earth than on the Sun and it is in this faster time that the waves are observed at an increase in length. Yet, since the speed of the received light is measured on Earth in terms of this faster Earth time, it has the same value in terms of meters per second that it does on the Sun when measured using the slower time on the Sun. For example, let us say that a single period of emitted light has a wavelength of 400 nanometers on an exaggeratedly massive source star. And let us say that an Earth like planet in circular orbit around that star receives that light at 600 nanometers. The received light then has a wavelength of 1 ½ time the emitted wavelength. This means that the dilated time on the Earth like planet is 1 ½ times faster than the dilated time on the source star. It also means that the speed of the received light at that planet is 1 ½ times faster than the speed of that same light as it left the source star. Yet if measured on that source star using the slower time on that star, the speed of that light would have the same value c that it has if measured on the planet whose time is 1 ½ times faster than the time on the star. And since only 2/3 of a wavelength is received for each period of time as measured on the star, the energy of the light is only 2/3 of what it was on the star. All of these considerations are consistent with the evidence. 4. The Classical Doppler Effect Connection Once the fundamental nature of the gravitational effects on light propagation is properly understood it becomes apparent that the direction of light travel relative to the greater gravitational mass is also a consideration. Quite simply, the gravitational potential factor given previously as expression (7) should be rewritten as Sun 5

6 1 11 where the sign is + for the case where the light travels in the direction of approach relative to the greater of the two gravitational masses and is for the case where the light travels in the direction of recession relative to the greater of the two gravitational masses. Moreover, from the analysis conducted using the gravitational potential integral, Eq. (1), and the speed variance integral, Eq. (10), it is found that giving as an important equation for the fundamental relationship of variance v to light speed c and gravitational potential Φ. Applying this relationship to equation (11) gives and finally 14 as an alternative factor for that of equation (11) where as before, the sign is + for light approaching the greater gravitational mass, or higher potential, and for light receding from the greater gravitational mass, or higher potential. As is readily apparent, this is the same factor that is used in the classical longitudinal Doppler effect formula for electromagnetic propagation. This commonality between the two different effects provides insights that may have gone unnoticed before and deserves further examination at a future time. 5. Fundamental Photon Model Relationship If we wish to equate the wavelike behavior of Figure 1 to the particle-like behavior of a photon we need only to consider the insights gained in reference to de Broglie s wave mechanics article 2 referenced earlier. In the article de Broglie deduces that if the period of internal phenomenon assumed to be associated with a particle increases due to time dilation as the speed of light is approached, then the associated frequency along with its energy will decrease and this would violate the relativistic principle that energy should increase as light speed is approached. 6

7 Actually, as stated earlier, this concern was directed at the principles of Special Relativity, and not General Relativity 4. Nonetheless, while according to the evidence it is true that a redshift (reduced frequency) is detected under the specified conditions, it is also true that the life of the particle increases and thus its energy is spread over a longer period of stationary frame time. Thus no conflict arises and any to attempt to stop such a particle would require the greater energy predicted by Special Relativity. Since the same rationale for a relativistic redshift is applicable to the principles of gravitational time dilation, its treatment here, though a bit more complicated, is justified because of the improved understanding it provides. Consider now a particle moving at the speed of light, (a photon) from the Sun to the Earth as shown in Figure 1. If the period of the photon and its subsequent frequency (as measured in the Sun s frame of reference) is unchanged by the change in gravitational potential along its path of travel, then as the photon speeds up due to reduced time dilation (faster time) on Earth, the periods of the photon will be spread over an increasingly greater distance as Earth is approached. This behavior is identical to that shown for the wavelike behavior illustrated in Figure 1. Subsequently, the frequency, as measured in the faster time of the Earth s reduced gravitational potential, will be reduced and a gravitational redshift will be observed for the photon on Earth. This is simply a more complicated way to explain the reason for the observed gravitational redshift than the standard explanation that states the photon loses energy in moving away from a greater gravitational mass such as the Sun to a lesser gravitational mass such as the Earth. Of greatest importance here is to remember the second postulate of Special Relativity that holds the laws of physics to the same form in all inertial systems in combination with the General Relativity premise to expand these laws to all systems. When applied to the mathematical relationships of frequency, and wavelength, to the speed of light as given by 15 where f is frequency, λ is wavelength and c is the speed of light, the received frequency will still depend on the wavelength of the received photon. And since the wavelength is spread over a greater distance in the Earth s gravitational potential, the frequency is reduced. The most important point to be taken from this is as follows: From one perspective the frequency of the photon is unchanged while from the other perspective it is changed. Both perspectives are correct and it is only important not to confuse the one with the other. 6. The Complete Frequency Formulas (Note: The following treatment is for the case involving two different gravitational bodies. For the case involving a single body refer to Section 12) When both directions of light propagation are considered in relation to two different bodies of gravitational potential it is apparent that the complete formula for frequency conversion is 7

8 1 16 where f R is the received frequency, f S is the source frequency, Φ is the net gravitational potential as defined by Eq. (18) below, c is the speed of light and the sign is + for the case where the light approaches the higher of the two potentials and if the light recedes from the higher of the two potentials. The reciprocal frequency shift formula is then where all of the definitions just given still apply. The net gravitational potential is then given by 18 where Φ is the net potential, Φ H is the higher potential, and Φ L is the lower potential. The higher and lower of the two gravitational potentials are then determined by 19 where Φ x is the specific gravitational potential for either of two different gravitational bodies, G is the gravitational constant, M x is the mass of the body, and R x is the radial distance from the point of emission or detection to the center of mass of the gravitational body. 7. Frequency to Wavelength Relationship and the Laws of Physics The second postulate of Special Relativity that holds the laws of physics to be the same in all inertial systems holds equally true for the case being discussed. Therefore the standard formula that relates frequency to wavelength 20 and its alternate form 21 still hold true, where f x and λ x denote the frequency and wavelength respectively, the subscript x denotes whether it is that being emitted or received, and c is the speed of light at the same 8

9 standard universal value of m/s regardless of whether it is used in relation to the emitted or received variables f x and λ x. In the intended use of equations (20) and (21) x cannot have two different meanings, i.e. emitted and received, within the same equation. 8. The Complete Wavelength Formulas (Note: The following treatment is for the case involving two different gravitational bodies. For the case involving a single body refer to Section 12) Corresponding to the case for frequency in Section 6, when both directions of light propagation are considered in relation to two different bodies of gravitational potential, the complete formula for wavelength conversion is where λ R is the received wavelength, λ S is the source wavelength, Φ is the net gravitational potential as defined by Eq. (18), c is the speed of light and the sign is + for the case where the light approaches the higher of the two potentials and if the light recedes from the higher of the two potentials. The reciprocal wavelength shift formula is then 1 23 where all of the definitions just given still apply. 9. The Gravitational Time Dilation Formula Given the standard relationship of time to distance and speed, and the facts that wavelengths are distances traveled at speed c where c as a constant has the same value in all physical systems we can state 24 and 25 where λ S and t S are the distance and time at the point of light emission and λ R and t R are the corresponding distance and time at the point of light reception. Since the value of c is constant we then obtain 9

10 26 from which we derive 27 and by way of substitution using Eq. (23) further derive 1 28 that simplifies to 1 29 where t S is the time at the gravitational potential of the light source and t R is the corresponding time at the gravitational potential of the reception point. Obviously then gives the reverse relationship where the variables have the definitions just given and where in all cases the sign is + for approach and for recession in regard to the greater gravitational potential. Experimenting with formula (29), using the sign as in Sun to Earth light travel, shows that if time interval t R is assigned the value t R = 1s then as variable v as given by formula (13) increases from 0 to c, time interval t S goes from 1 to 0. Then, as variable v is increased from c to 1.883x c, time interval t S decreases from 0 to ( )1.88x s. Of curious theoretical interest here is the mathematical consideration that time goes backward if speed c is exceeded. There is no real intent here, however, to show that speed c can actually be exceeded. 10. The Effect of Gravity on Radiation Energy Probably the most confusing aspect of light propagation is the treatment of its energy as it travels between two different gravitational potentials. In Sections 3 and 5 we acknowledge that time runs faster in a lower gravitational potential and slower in a higher gravitational potential. 10

11 Subsequently light slows down as it travels toward a gravitational body and speeds up as it travels away from a gravitational body. This is exactly the opposite of what we experience in regard to other material objects as they travel toward or away from a gravitational body. Perhaps the easiest way to think of light is in the form of the photon, and in that regard accept that its period is unaffected by a change in gravitational potential. Thus, the photon may be thought of as a master clock whose time is unaffected by the external time in the gravitational environment that it travels through. By comparison then, if the external time slows down as the photon travels toward a gravitational body, the photon will have increased energy in that direction. Conversely, if the photon travels away from a gravitational body, the photon will have reduced energy in that direction because of the speeding up of external time. The formula for determining this relative change in energy can be written as or conversely as where E R is the energy at the point of reception, E S is the energy at the point of emission, Φ is the net gravitational potential as given by Eq. (18) and the sign is + for a direction of approach with regard to the greater gravitational potential, and for a direction of recession with regard to the greater gravitational potential. Note: It will help with the sign convention if we make it a point to remember that it refers to the greater gravitational potential and not the light source and it is opposite to the sign convention used for the Doppler effect. 11. Comparison of Light Speed Variance Treatments In the previously referenced article by Einstein on the influence of gravity on light, the formula 1 33 is given for the velocity of light c at a place with the gravitational potential Φ in relation to the velocity of light c o at the origin of coordinates. Whereas solving this formula for c leads to a very large and complex formula, solving for c o is as simple as

12 where it is assumed that c has the standard value. This assumption appears to be confirmed by the finding that when we equate the difference between c and c o as in 35 where V is the general relativity variance in light speed, V has the same value as v in Eq. (13) given earlier, and repeated below for convenience. 13. (Refer to the bottom of Appendixes A and B for a detailed comparison of the two different treatments.) Based on these findings and others involving the complex formula mentioned above, it is reasonable to assume that the variance v in light speed given by Eq. (13) is a valid substitution for the value given by the General Relativity variance V of Eq. (35). It is further assumed that the simpler relationship given by Eq. (13) can be used in place of the General Relativity treatment in general. 12. The Case Involving a Single Gravitational Body For the case of (approach) involving light traveling directly toward a single gravitational body, such as the Earth for example, we derive 36 where Φ is the gravitational potential, G is the gravitational constant, M e is the mass of the Earth, R e is the radius of the Earth, and L is a distance above the Earth s surface. For the case of (recession) where the light travels directly away from the gravitational potential, again using the Earth as an example, we derive 37 where the variables are the same as previously given. As can be seen, the only difference between the two formulas is the reversal of the integral limits. Although these formulas appear to have unlimited validity in the sense that L can be any distance from 0 to, the endpoint formulas of General Relativity used in the Harvard Tower experiments 5 seem to lose accuracy in their midrange as will be discussed next. One of the formulas typically associated with the Harvard Tower experiments is 12

13 1 38 where f R is the received frequency, f S is the source frequency, g is the acceleration of gravity at the Earth s surface, and L = 22.6 m is the distance to the detector at the top of the Tower. A variation of this formula is 1 39 where a is the rate of acceleration at distance L above the Earth s surface as given by 40 where distance L has been added to the surface distance R e. In both cases involving Eq s. (38) and (39), the sign is + for light approaching the Earth and for light receding from it. The problem with both of the frequency formulas just given is that each of them uses only a single point of gravitational potential as opposed to the integral treatment of Eq s. (36) and (37) where the whole range of acceleration rates can be computed from L = 0 to L = virtually. Although all of the formulas, the integrals and endpoint versions give the same results for a range of L values from 0 to 22.6 x 10 2 m, approximately 2.3 km; the results of the single point formulas vary from the results of the integral formulas at greater distances and thus have limited use. Nonetheless, the single point formulas give valid results for the actual distance of 22.6 m used in the Harvard Tower experiments. Appendixes C and D are included at the end of the paper for providing a detailed analyses of the computed results for the experiments. 13. Conclusion The primary purpose of this research effort was to explore the underlying nature of gravitational effects on light propagation at the most fundamental level. It is the author s belief that only through such processes of demystification, can the process of scientific discovery be advanced. The principles presented herein can be expanded upon in future research efforts to further evaluate their validity and overall applicability regarding gravitational influences on all physical phenomena. Only in such manner can we determine the true scope of such influences and thereby determine their inevitable limitations. Of particular interest in that regard are the limitations applicable to the gravitational equivalence principle. It is the author s contention that such principle is valid in only the most restricted sense and not to the almost unlimited degree presently assumed by most scientists in the field of relativistic physics. A final point of interest involves the fundamental relationship given by equation (13) in this present work with regard to 13

14 the variance of light speed between different points of gravitational potential. On the widely held belief that simplicity in mathematical relationships in itself offers a form of proof that the finding must be true, the relationship for light speed variance given by that equation cannot be ignored. Appendixes The following Appendixes are included at the end of the paper: Appendix A Dual Gravitational Bodies Approach Appendix B Dual Gravitational Bodies Recession Appendix C Single Gravitational Body Approach Appendix D Single Gravitational Body Recession REFERENCES 1 Albert Einstein, On the Influence of Gravitation on the Propagation of Light, (1911), As reprinted in, The Principle of Relativity, Dover Publications, Inc., 31 East 2 nd Street, Mineola, N.Y Louis de Broglie, RADIATION Waves and Quanta, Note of Louis de Broglie, presented by Jean Perrin. (Translated from Comptes rendus, Vol. 177, 1923, pp ) 3 Albert Einstein, Special Theory of Relativity, (1905); Originally: On the Electrodynamics of Moving Bodies 4 Albert Einstein, General Theory of Relativity, (1915), Einstein was not satisfied with his Special Theory because it identifies inertial frames as different from other frames. Thus, during the ten years following his 1905 paper, he developed a new theory that we call his General Theory of Relativity. 5 In 1959 physicists Robert V. Pound, and G. A. Rebka Jr. conducted experiments in a tower at the Jefferson Physical Laboratory at Harvard University and accurately measured the frequency shift in light traveling up and down the tower. These experiments were followed by more accurate experiments at the same location in 1964 by Robert V. Pound, and J. L. Snider. 14

15 Appendix A Dual Gravitational Bodies Approach 15

16 Appendix B Dual Gravitational Bodies Recession 16

17 Appendix C Single Gravitational Body Approach 17

18 Appendix D Single Gravitational Body Recession 18

19 Gravitational Effects on Light Propagation Copyright 2009 Joseph A. Rybczyk All rights reserved including the right of reproduction in whole or in part in any form without permission. Note: If this document was accessed directly during a search, you can visit the Millennium Relativity web site by clicking on the Home link below: Home 19