Classical Tests of General Relativity Part II: Looking to the Past to Understand the Present
CClassical Tests of General Relativity Part II:Looking to the Past to Understand the Present
Jorge Pinochet ∗ ∗ Departamento de F´ısica, Universidad Metropolitana de Ciencias de la Educaci´on,Av. Jos´e Pedro Alessandri 774, ˜Nu˜noa, Santiago, Chile. e-mail: [email protected]
Abstract
The objective of this second part of the work is to present heuristic derivations of thethree classical tests of general relativity. These derivations are based on the Einsteinequivalence principle and use Newtonian physics as a theoretical framework. The resultsobtained are close to Einstein’s original predictions. Historical and anecdotal aspects ofthe subject are also discussed.
Keywords : General relativity, classical test of general relativity, undergraduate students.
The main objective of this second part of the work is to present heuristic derivations of thethree classical tests of general relativity (GR). The first derivation is a variant of an argumentpresented by the author in another work [1], the second derivation is original, and the third isa variant of an argument that appears in some texts of modern physics [2]. These derivationsare based on the Einstein equivalence principle and use Newtonian physics as a theoreticalframework. Despite these simplifications, the results obtained hardly differ from Einstein’soriginal predictions. Although we will not delve into the notion of space-time curvature, itis important to keep in mind that a detailed explanation of the classical tests requires thenotion of curvature, as discussed shallowly in Part I.The article is organised as follows. We first analyse the deflection of light by the Sun. We willthen examine the perihelion precession of Mercury and finally tackle the gravitational redshiftof light. Although this sequence does not coincide with the historical order of events (which isnot linear), in the author’s opinion, it is the most convenient order from a pedagogical pointof view. The article ends with some comments on the current and future scientific status ofGR. 1 a r X i v : . [ phy s i c s . h i s t - ph ] A ug Deflection of light by the Sun
The basic idea behind this classical test is illustrated in Fig. 1. A ray of light that passesvery close to the surface of a massive celestial body, such as the Sun, suffers a deflection fromits straight path (this deflection is a manifestation of the curvature of spacetime around theSun). If the light comes from a distant star, it will show an apparent position different fromits actual position.From the Einstein equivalence principle, it is possible to predict the deflection of light by theSun and derive the approximate equation that describes this effect. To achieve this, we canconsider a spacecraft moving with constant acceleration g in a region of the universe withoutgravity. A photon enters through a window on the left side wall and reaches the other endof the spacecraft. Fig. 2 shows two perspectives of the photon (drawn as a red dot) at fourequidistant instants of time. The left image shows the point of view of an observer who isoutside the spacecraft, in the reference system where the photon is emitted. For this observer,the photon follows a straight path. The right image shows the point of view of an observerlocated inside the spacecraft. For this observer the photon ”falls”, describing a parabolic path.As Fig. 2 shows, during its journey within the spacecraft, the photon travels a horizontaldistance l and descends or falls a vertical distance s . For an observer inside the spacecraft,the distance the photon falls is: s = 12 gt . (1)We can estimate the time it takes for the photon to fall this distance as: t = lc , (2)where c = 3 × m · s − is the speed of light in vacuum. By introducing this result in Eq.(1), we obtain: s = gl c . (3)We will use this equation again when we discuss the perihelion precession of Mercury in thenext section. Due to the enormous value of c , in general, the angle of deviation will beextremely small. Said angle (in radians) can be obtained by deriving Eq. (3): α ∼ = dsdl = glc . (4)By virtue of the Einstein equivalence principle, we conclude that if the photon undergoes adeviation α within the spacecraft with acceleration g , it will undergo the same deviation ifthe spacecraft is on the surface of the Sun, where the acceleration of gravity is – g (see Fig. 3).Then, Eq. (4) must be valid in the vicinity of the Sun (and of any spherical celestial body)and we can express g in the form: g = GMR , (5)where M is the mass of the Sun and R is its radius. By eliminating g between Eqs. (4) and(5), we get: 2 igure 1: Deflection of light by the Sun. An observer on Earth perceives the star to be in an apparentposition other than the actual one.
Figure 2:
Left: Path of the photon as seen by an inertial observer in four instants of time. Right:Path of the photon seen from inside the spacecraft, in the same instants of time. α ∼ = GM lc R . (6)Although, strictly speaking, Eq. (6) only has local validity, that is, it is only applicable in asmall region on the solar surface, where g is uniform, we can estimate the non-local deflectionof a photon that skims the surface of the Sun assuming that the spacecraft is wide enough sothat l = 2 R : α ∼ = 2 GMc R . (7)The equation found by Einstein is: α E = 4 GMc R . (8)We see that α E = 2 α , so that Eq. (7) is a good approximation, especially considering thesimplicity of the derivation. Eq. (8) was first verified by two English astronomical expedi-tions conducted during the total eclipse of the Sun on May 29, 1919. The expeditions soughtto determine the deflection caused by the Sun over the light of stars located behind the so-lar disk (the stars were visible thanks to the eclipse). One of the expeditions was led byArthur Eddington and Frank Dyson, and was conducted on Prince Island, off the east coastof Africa. The other expedition was led by Andrew Crommelin and Charles Davidson and3 igure 3: Spacecraft at rest on the surface of the Sun. A photon that enters through the lateralwindow experiences a deflection similar to that illustrated in Fig. 2, right. was conducted in Sobral, Brazil. The plan was to compare images of the stars taken duringthe eclipse (apparent position) with images of the same stars taken six months later (actualposition), when the Sun does not come between the Earth and stars.If in Eq. (8) we consider M = 1 . × kg (solar mass) and R = 6 . × m (solar radius),and we introduce the other constants, we obtain α = 4 . × − rad . As 1 rad = 360 o / π and 1 o = 3600 (cid:48)(cid:48) , we get: α = 4 . × − × o π × (cid:48)(cid:48) o = 1 . (cid:48)(cid:48) . (9)Within observational uncertainties, the value found by astronomical expeditions coincideswith Eq. (9) [3]. Newton’s law of gravitation does not predict any deflection for light in agravitational field , so the results obtained by the expeditions not only meant a triumph forGR, but also the decline of the Newtonian worldview. According to the Newton-Kepler laws, an isolated planet that revolves around a star, such asthe Sun, will describe an elliptical orbit with the star in one of the foci. This means that theangle described by the radius vector (the line connecting the planet with the star) betweenone perihelion and the next is zero. That is, after completing one lap, the perihelion returnsto the starting point.However, when calculations are made using GR, a perihelion precession or perihelion advanceof the planet is found, which means that between one perihelion and the next, the periheliondoes not return to its initial point, and consequently the angle described by the radio vectoris slightly greater than zero, as illustrated in Fig. 4. The closer a planet is to its star, thegreater this effect is. Einstein applied GR to calculate the perihelion advance of Mercury, the If we assume that light is a material particle beam, as Newton believed, the deviation of light can becalculated, obtaining a figure that is half the value calculated in Eq. (8), and therefore coincides with Eq. (7).However, we know that light is not made up of material particles. igure 4: Mercury (black circle) in two consecutive positions of its perihelion. For each revolution, aperihelion advance occurs at an angle ϕ , which corresponds to an additional orbital displacement ∆ l . planet closest to the Sun.We can make an approximate calculation of the perihelion advance of Mercury using theEinstein equivalence principle. However, in this case we will use it implicitly, drawing on theideas developed in the previous section. To do this, suppose that Eq. (3) allows us to describeboth the angular deviation of a photon and the angular deviation of Mercury (or of anothersmall celestial body in relation to the Sun) in the solar gravitational field. Suppose also thatMercury is the only planet that revolves around the Sun and that its elliptical orbit does notdiffer much from a circumference.By applying Eq. (3) to the perihelion advance, we see that l plays the role of a circumferenceof perimeter 2 πr , where r is the radius of Mercury’s orbit. From Eq. (3), we find that ineach revolution, Mercury advances (deviates angularly) between one perihelion and the nextan approximate distance: s ∼ = (2 πr ) c g = 2 π c GM, (10)where we introduce the value of g given by Eq. (5). Furthermore, Kepler’s third law forcircumferential orbits states that: GM π = r T , (11)where T is the Mercury orbital period and r is the radius of its orbit. By eliminating GM between Eqs. (10) and (11), we get: s ∼ = 8 π r c T . (12)Dividing by r , we obtain the perihelion advance of Mercury per revolution, expressed inradians (see Fig. 4): ϕ ∼ = sr = 8 π r c T . (13)The equation originally found by Einstein is: 5 E = 24 π a c T (1 − e ) , (14)where e is the eccentricity of the elliptical orbit and a is the semi-major axis. For r ≈ a and e (cid:28) ϕ/ϕ E = π/ ∼ = 1 .
05, meaning that Eq. (13) is an excellentapproximation.Since the mid-19th century, astronomers have observed a perihelion advance of Mercury of57 (cid:48)(cid:48) per century, which they attributed to the gravitational influence of the rest of the planets.By applying Newton’s law of gravitation, it was possible to predict an advance of 531 (cid:48)(cid:48) percentury, so that there were 574 (cid:48)(cid:48) –531 (cid:48) = 43 (cid:48)(cid:48) per century without explanation [3]. Using GR,Einstein was able to explain this difference. Indeed, in the case of Mercury, we know that a = 57 . × m , T = 10 s and e = 0 . ϕ E = 5 × − rad/revolution , or: ϕ E = (cid:18) × − rad × o πrad × (cid:48)(cid:48) o (cid:19) /revolution = 0 . (cid:48)(cid:48) /revolution. (15)This is the perihelion advance of Mercury without considering the gravitational influence ofthe other planets, that is, subtracting said influence. Since Mercury makes 415 revolutions ina century, we conclude that the perihelion advance is 415 × . ∼ = 43 (cid:48)(cid:48) per century, whichis exactly the advance that Newtonian gravitation cannot explain. This figure was the firstempirical confirmation of GR. According to GR, the light that moves away from a celestial body, like the Earth, experiencesa decrease in its energy, so that a distant observer detects the light with a redshift, that is,the wavelength presents a systematic shift towards the low frequency range. This is similarto what happens to an object that is thrown up from the Earth’s surface. As the objectrises, moving against gravity, it loses speed and kinetic energy. However, since light alwaysmoves with constant speed c , the only way that energy loss can occur is through an increasein wavelength or a decrease in frequency.We can use the Einstein equivalence principle to predict the gravitational redshift of lightand derive the equation that describes this effect. To do this, let us go back to the spacecraftdescribed in Section 2 and suppose it is at rest in a region of space without gravity. On thefloor of the spacecraft there is a light source F that emits a photon of wavelength λ verticallyupwards. At the time of emission, the spacecraft begins to move with constant acceleration g . On the ceiling is a detector D at a height H (see Fig. 5).As the spacecraft moves away from the emission point, a Doppler effect will occur, that is, D will perceive that the photon wavelength has increased to a value λ > λ . As a firstapproximation, we can determine the relationship between λ and λ using the equation forthe non-relativistic Doppler effect: Strictly speaking, the gravitational redshift of light is a test of the Einstein equivalence principle, whilethe other two are tests of the GR (Schwarzschild solution) in the weak field limit. igure 5: As the spacecraft moves with uniform acceleration g , the astronaut uses a detector D tomeasure the wavelength of a photon emitted at F . λλ = 1 + vc , (16)where v is the speed of the spacecraft when D detects the photon. We can estimate the timeit takes for the photon to reach D as: t = Hc . (17)Since the spacecraft is initially at rest, its speed when D detects the photon is v = gt , so that: v = g Hc . (18)By introducing this value of v in Eq. (16), we obtain: λλ = 1 + gHc . (19)We define redshift z as the fractional change in wavelength: z = λλ − λ − λ λ = ∆ λλ = gHc . (20)By virtue of the Einstein equivalence principle, we conclude that if the photon undergoes red-shift within the spacecraft with acceleration g , it will also undergo offset if the spacecraft isat rest on Earth’s surface (or on the surface of any celestial body), where the acceleration dueto gravity is – g (see Fig. 6). Then, Eqs. (19) and (20) must also be valid in a gravitationalfield.Einstein’s equation for gravitational redshift is equivalent to Eq. (19), which is a valid ap-proximation when H is very small compared to the Earth’s radius.The first experimental confirmation of the gravitational redshift of light was obtained 40years after the astronomical expedition that verified the deflection of light by the Sun. In1959, Robert Pound and Glen Rebka conducted an experiment at Harvard University, wherethey observed the redshift from photons that rose to a height of 22 . m . If in Eq. (20), we7 igure 6: Spacecraft at rest on Earth’s surface. A photon traveling towards D undergoes a redshiftanalogous to that illustrated in Fig. 5. take H = 22 . m and g = 9 . m · s − , it is found that z = 2 . × − , which within theexperimental uncertainties, agrees very well with the value found by Pound and Rebka [4]. After the 1919 astronomical expedition, scientific interest in GR rapidly waned, and overthe following decades, physicists turned their attention to other topics. This can be ex-plained mainly by two factors: (1) GR is a mathematically very complex theory, and inEinstein’s time only an exact solution of astronomical interest was known to the equations ofGR (Schwarzschild solution), thus, there was little incentive to investigate the subject, sinceit seemed difficult to find new solutions of interest; (2) The empirical corroboration of GRwas extremely difficult with the technological resources of that time.Fortunately, in the 1960s the situation changed dramatically and GR experienced a renais-sance. This is the time that Kip Thorne calls the golden age [5], during which new mathe-matical techniques were developed that made calculations easier. In addition, technology hadadvanced enough to allow accurate tests of GR to be carried out, such as the Glen and Re-bka experiment, or other tests carried out shortly thereafter, such as the confirmation of theShapiro time delay effect or the Hafele-Keating experiment. Furthermore, astronomical obser-vations were beginning to reveal extreme phenomena that could only be adequately explainedin the framework of GR (such as pulsars and quasars), so that physicists and astronomersbegan to pay attention to Einstein’s theory. The reader who wants to delve into these topicscan turn to the excellent Thorne’s book mentioned earlier. Another highly recommendedpopular science book is [3].With ups and downs, since the golden age, interest in GR has not waned. In fact, over thepast few years, interest has boomed, where new astronomical observations have made GR takeon an importance it probably never had before, even in the golden age. Two milestones in thisregard are the detection of gravitational waves in 2015 and the first picture of a black hole,obtained in 2019. In both cases, and in many others that we cannot analyse here, GR hasbeen successfully confirmed. Everything seems to indicate that this is just the beginning of aperiod where GR will experience a second golden age. In this scenario, it is to be hoped thatlooking to the past will contribute to making physics teachers and students more prepared to8nderstand and value current and future advances in GR.
Acknowledgments
I would like to thank to Daniela Balieiro for their valuable comments in the writing of thispaper.