Classical Tests of General Relativity Part I: Looking to the Past to Understand the Present
CClassical Tests of General Relativity Part I: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
Einstein’s theory of general relativity (GR) provides the best available description of grav-ity. The recent detection of gravitational waves and the first picture of a black hole haveprovided spectacular confirmations of GR, as well as arousing substantial interest in top-ics related to gravitation. However, to understand present and future discoveries, it isconvenient to look to the past, to the classical tests of GR, namely, the deflection of lightby the Sun, the perihelion precession of Mercury, and the gravitational redshift of light.The objective of this work is to offer a non-technical introduction to the classical tests ofGR. In this first part of the work, some basic concepts of relativity are introduced and theprinciple of equivalence is analysed. The second part of the article examines the classicaltests.
Keywords : General relativity, classical test of general relativity, equivalence principle,undergraduate students.
The theory of general relativity (GR), proposed by Einstein in 1915 [1,2], is the best avail-able description of gravity. The first detection of gravitational waves in 2015 [3] and thefirst picture of a black hole obtained in 2019 [4–9] have provided spectacular confirmations ofEinstein’s theory, arousing significant interest among the general public in topics related togravitation. There is no doubt that in the next few years, we will witness new confirmationsof GR, but to understand the present and future discoveries, it is convenient to look to thepast, to the classical tests of GR, namely, the deflection of light by the Sun, the perihelionprecession of Mercury, and the gravitational redshift of light [10]. These tests were proposedby Einstein between 1907 and 1916 [1,2,11] and led to the first empirical confirmations of GR. A comment for experts: Strictly speaking, the gravitational redshift of light is a test of the Einsteinequivalence principle, while the other two are tests of the GR in the weak field limit. a r X i v : . [ phy s i c s . e d - ph ] A ug R is a fascinating but highly technical subject, with its detailed understanding requiringadvanced mathematical knowledge. The objective of this work is to offer a non-technicalintroduction to the classical tests of GR using simple mathematics. Therefore, the article canbe used, for example, as educational material in an undergraduate modern physics course orin an introductory astronomy course.In this first part of the work, some basic concepts of relativistic physics are presented as anintroduction to the detailed analysis of the classical tests, which we will address in Part II.Section 2 discusses the weak equivalence principle, while Section 3 addresses Einstein equiva-lence principle, which is the foundation of GR. Section 4 outlines a qualitative explanation ofthe concept of spacetime curvature, and discusses its relationship to the equivalence principle.The article ends with a few brief summarising comments.
This principle has been known since the time of Galileo, but no one until Einstein was ableto grasp its profound physical significance. The weak equivalence principle can be formulatedas follows: the motion of any free falling test particle is independent of its composition andstructure . To unravel the meaning of this principle, let us start by remembering that accordingto Newton’s law of gravitation, the magnitude of the attractive force F grav generated by aspherical body of gravitational mass M g on a test particle of gravitational mass m g is: F grav = GM g m g r , (1)where r is the distance that separates the particle from the centre of the spherical body. Onthe other hand, according to Newton’s second law, the net force acting on a particle of inertialmass m i , is related to acceleration as : F net = m i a. (2)If we assume that F grav = F net , we obtain: a = (cid:18) m g m i (cid:19) GM g r . (3)Numerous experiments have confirmed that m g = m i , and therefore: a = GM g r . (4)We see that all particles fall with the same acceleration, and consequently, the movement ofthe test particles is independent of their composition and structure, as stated by the principleof weak equivalence. Although this result may seem natural to us, to the point that manystudents and teachers automatically apply it in their calculations, it is important to note thatother forces do not satisfy it. The simplest example is Coulomb law. According to this law,the magnitude of the electrostatic force between a body with charge Q and a test charge q is: A test particle is an ideal object that has such a small mass that its gravity can be ignored and thereforedoes not affect the gravitational field in which it is immersed. Inertial mass is a measure of the resistance a body offers to be accelerated; for a given force, the greaterthe mass, the lower the acceleration. Instead, the gravitational mass is a measure of the attraction experiencedby two bodies separated by a certain distance; the greater the mass, the greater the attraction. elec = KQqr , (5)where r is the distance between the charges. Let m i be the inertial mass of the test charge.If F elec is the only force acting on q , according to Eq. (2), we will have: a = (cid:18) qm i (cid:19) KQr . (6)Unlike Eq. (4), in this expression the acceleration depends on the q/m i ratio, that is, itdepends on the composition and structure of the particles involved.In the weak equivalence principle, expressed through Eq. (4), Einstein discovered somethingfundamental: if the movement of the test particle is independent of its composition and struc-ture, then said movement is determined by a property that resides solely in the gravitationalfield. This property is the curvature of spacetime, since in relativity, space and time are in-extricably linked, and what happens with spatial coordinates determines what happens withtime intervals and vice versa.Curvature is the central concept of GR and it conceives of gravity as a purely geometricproperty of the spacetime structure, where the mass induces curvature. That is, in theEinstein universe, the force of gravity does not exist: what moves planets, stars and galaxiesis the curvature of spacetime. There are two formulations of the theory of relativity. The first, known as special relativity,was published by Einstein in 1905 and applies to phenomena where gravity is absent or sosmall that it can be ignored [2,12]. The second formulation, GR, was published by Einsteinin 1916, and is a generalisation of special relativity that proposes a revolutionary descriptionof gravity, understood as a purely geometric phenomenon [1,2]. The foundation of GR isthe
Einstein equivalence principle : Experiments made locally in a reference frame uniformlyaccelerating with acceleration (cid:126)a whit respect to an inertial frame, produces the exact same ex-perimental results as an inertial frame of reference in a uniform gravitational field − (cid:126)a [13,14].We will use an example to clarify the meaning of the Einstein equivalence principle. Let usimagine a spacecraft traveling with constant acceleration g , in a region of the universe withoutgravity (see Fig. 1). An astronaut A inside the ship has mass m and is standing on a scale.What value does the scale record? Since the only force acting on A is the normal N exertedby the scale in the direction of g , it follows from Newton’s second law that the recorded valueis N = mg .If we consider the same spacecraft at rest on the surface of a planet where the gravitationalacceleration is – g , the scale will also record the value mg . Furthermore, it can be shown thatno experiment carried out inside the spacecraft will allow A to determine if it is at rest onthe surface of a planet with gravity – g , or if it is in a place in the universe without gravitymoving with uniform acceleration g [15]. However, this conclusion is only valid locally, sincethe gravity of a celestial body is not uniform, it varies with position and height. The gravi-tational effects can then only be reproduced locally, in small regions of space where gravity3 igure 1: Left: An astronaut is standing on a scale inside a spacecraft traveling with gravitationalacceleration g through space without gravity. Right: The astronaut is at rest on the surface of a planetwith a surface acceleration of gravity – g . In both cases, the scale shows mg . can be considered uniform.However, the Einstein equivalence principle not only allows the local effects of gravity to bereproduced, but also allows us to eliminate these effects locally for a free falling referenceframe under the action of a gravitational field. The latter leads to an alternative formulationof the Einstein equivalence principle: A free-falling reference frame in a gravitational field islocally equivalent to an inertial frame (without gravity) [13,14].To understand this new formulation, let us go back to the image of the spacecraft that wewill now suppose encounters with its engines off and free falling toward a massive celestialbody. Let us imagine that A has a portable video camera. As the spacecraft falls, A activatesthe camera and releases it from his or her hands (see Fig. 2). If the camera is pointed at A ,what will it record? An image will be seen where A floats motionless and stationary, as if inempty space, far from any source of gravity. Furthermore, no experiment carried out insidethe spacecraft will allow A to determine if it is in free falling under the action of the celestialbody’s gravity, or if it is in some remote corner of the universe, far from any source of gravity.However, again, due to the non-uniformity of the gravity field, the above conclusions are onlyvalid locally in comparatively small regions.The Einstein equivalence principle is based on the weak equivalence principle. Indeed, if themovement of a body in a gravitational field depended on its composition and structure, ashappens with electrical charges, we would see that the camera and the astronaut in the lastexample would fall differently, and we could not locally cancel the effect of gravity for a freefalling reference frame. As noted in the previous section, it was this property of gravity thatled Einstein to the conclusion that the motion of an object in a gravitational field depends ona property that resides solely in the field, and that corresponds to the spacetime curvature.4 igure 2: A free-falling astronaut appears to float inside the spacecraft, as if there is no gravity.
Although the reading of this section is not strictly necessary to understand the ideas developedin Part II of this work, this section complements and enriches some of the topics developedbefore.The central idea of GR is that gravity is a manifestation of the curvature of spacetime. Unlikein Newtonian physics, in GR, space and time are dynamic and flexible entities that respond tothe presence of mass, or its energy equivalent. Thus, mass-energy curves spacetime and saidcurvature determines the movement of bodies. Therefore, in GR, there is no force of gravity.Bodies move freely following the trajectories dictated by the curved geometry of spacetime.Due to the curvature, these trajectories are not straight but geodetic . A geodesic generalisesthe notion of the straight line of the flat Euclidean geometry; that is, a geodesic is a curverepresenting the shortest path between two points.The American theoretical physicist John Archibald Wheeler beautifully described the ideabehind GR, noting that ”mass tells spacetime how to curve, and spacetime tells mass howto move” [16]. To understand the basic ideas of GR, it is useful to use some analogies withcurved surfaces. In general terms, a two-dimensional surface can have only three kinds ofcurvature: null , positive and negative curvature , as illustrated in Fig. 3.The best example of zero curvature is the surface of a stretched sheet of paper. Imagine twoparticles that can only move on the surface of the sheet. For particles, the paper surfaceis their entire universe. On this surface, the classic results of flat Euclidean geometry arefulfilled. For example, (R1) if the particles move following parallel (straight) geodesics, theywill keep their mutual distance constant (their trajectories do not intersect) (see Fig. 3, left).It is usual to take this result as a criterion to define the zero curvature .The best example of positive curvature is the surface of a sphere. If we think about this sur-face in geographical terms, we conclude that the meridians are geodetic, but not the parallelones, except for the equator, which is a geodetic. In this case, R1 undergoes an importantmodification, namely, (R2) if the particles move following initially parallel geodesics, theirpaths intersect (see Fig. 3, centre). Following the example of the sheet of paper, this result Another criterion to define the zero curvature is the Euclidean theorem that states that the sum of theinterior angles of a triangle is 180 o igure 3: Three types of curvature and the surfaces that best illustrate them.
Figure 4:
Two particles at rest separated by a distance ∆ x in a region of the universe without gravity,describe parallel and straight geodesics in a spacetime diagram, indicating zero curvature. can be used as a criterion to define the positive curvature.The best example of negative curvature is the surface of a saddle. In this case, it can beshown that, (R3) two initially parallel geodesics diverge (Fig. 3, right). We take this resultas a criterion to define negative curvature.Now, if we consider a sufficiently small region of the surface of the sphere or the saddle, wediscover that the Euclidean geometry is fulfilled with a high degree of approximation, sincelocally, the curvature is negligible, the geodesics are almost straight and the surface is almostflat. Then, in a very small region on the surface of the sphere or on the saddle, two initiallyparallel geodesics seem to keep their mutual distance constant.The analogy between a curved surface and gravity is remarkable. To properly understandthis analogy, it is necessary to keep in mind that a particle that moves freely in spacetime,without any force acting on it, will describe a spacetime geodetic. If there is only gravity, themovement of the particles will also be free, since we know that gravity is not a force, but amanifestation of the curvature of spacetime.With the above ideas in mind, imagine two test particles that are left to stand at a distance∆ x . Since by definition we can assume that the gravity between the test particles is zero, theseparation ∆ x will remain constant through time, and if we draw their geodetic trajectorieson a position x versus time t diagram (spacetime diagram), we will see that they move alongstraight lines that remain parallel, as shown in Fig. 4. In analogy with the surface of a sheetof paper, we will say that spacetime is flat, and as will be made clear shortly, we can attributethis zero curvature to the absence of gravity. 6 igure 5: Left: Two particles in free fall toward Earth approach each other in the direction transverseto the radius. Centre: Spacetime diagram for the particles in the left picture, where it is observed thattheir geodesics are approaching, indicating positive curvature.
Let us keep the particles at a distance ∆ x , but now let us introduce a massive celestial bodylike Earth. Let the particles free fall toward the centre of the Earth in the radial direction.Under these conditions, the particles will move along geodesics and the distance ∆ x will de-crease as time passes. If we extend the paths of the particles to the centre, they intersect (Fig.5, left). By drawing the geodesics of the particles on a spacetime diagram (Fig. 5, centre),we see that they initially appear parallel, but then begin to approach until they intersect ,describing curved paths. This is analogous to what happens on the surface of the sphere.According to Einstein, we can interpret the approach of the particles as a consequence of thepositive curvature of spacetime around the Earth in the transverse direction. The effect ofgravity is to then curve spacetime.If we now assume that the particles are located in the same radial direction while free falling,they will move away from each other and the distance ∆ x will increase with time, since theparticle closest to Earth will be attracted with more force than the farthest (Fig. 6, left).If we again draw the geodesics of the particles on a spacetime diagram (Fig. 6, centre), wesee that they initially appear parallel but then deviate and begin to move away , describingcurved paths. This is analogous to what happens on the surface of the saddle. Accordingto Einstein, we can interpret the separation of the particles as a consequence of the negativecurvature of spacetime in the radial direction. Once again, we see that the effect of gravity isto curve spacetime.How are the above conclusions related to the Einstein equivalence principle? Remember thatthis principle establishes that locally gravity is annulled in a free falling reference frame in agravitational field. However, a region with zero gravity is a region with zero curvature. Thismeans that the trajectory of an object (like the astronaut in Fig. 2) that is free falling is alocally straight geodesic, where special relativity is locally valid. Then, adding on the locallystraight geodesics, we recover the spacetime curvature and therefore the gravity, which is thedomain of GR.In summary, the Einstein equivalence principle ensures that GR contains special relativity as In Fig. 5, it is true that ∆ x ∝ t n ( n (cid:54) = 1), which explains why the geodesic trajectory of each particle inthe spacetime diagram is curved. For example, for small distances, the approximate relationship is ∆ x ∝ t ,which implies that in this case the geodesics are parabola segments. In Fig. 6, it is again true that ∆ x ∝ t n and therefore the geodetic of each particle is curved. igure 6: Left: Two particles that free fall in the direction of the Earth’s radius and move away fromeach other. Centre: Spacetime diagram for the particles in the left picture, where it is observed thattheir geodesics separate, indicating negative curvature. a particular case, just as geometry on the surface of the sphere or saddle contains flat Euclidgeometry as a particular case.
Perhaps the example of the astronaut and the analogies between gravity and curved surfacescould leave the idea that Einstein’s notion of curvature is nothing more than a sophisticatedinterpretation of the Newtonian concept of gravitational force. If this idea were correct,employing GR or the law of universal gravitation would be a matter of personal preference.However, Newton and Einstein’s perspectives are not equivalent, as they lead to differentpredictions, and one of the main objectives of the classical tests proposed by Einstein wasto determine which of the perspectives is correct. In Part II of this article, we will analysein detail the classical tests and we will discover that the theory that best describes physicalreality is not Newton’s law of gravitation but GR.
Acknowledgments
I would like to thank to Daniela Balieiro for their valuable comments in the writing of thispaper.