Basic Relativity pp 75-87 | Cite as

# General Relativity

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## Abstract

The idea of geometrization of the gravity is elucidated. The place of the inertial frames in the theory is explained. GP-B experiment, light deflection, perihelion precession and gravitational red shift are discussed.

## Keywords

Weight Mass Gravity Geodetic Local frames## 3.1 Gravitational and Inertial Mass

The mass on the left hand side of the Newtonian equation of motion \(ma = F\) and the masses in the expression \(F_G = GmM/r^{2}\) of the inverse-square law of universal gravitation share a physical dimension but their physical meanings are in a sense opposite to each other. The mass which multiplies acceleration is the measure of the body’s resistance to any type of accelerating influence (including gravity). It is called the *inertial mass* of the body and will be denoted simply by \(m\). The masses participating in the gravitational force formula express, on the contrary, the willingness of either body to subject itself to the accelerating power of the other in this special (gravitational) type of interaction. Masses of this kind are known as \(gravitational\;masses\) and will be distinguished by an asterisk from their inertial counterparts: \(F_G = Gm^{*}M^{*}/r^{2}\).

The inertial mass of a body can be measured by e.g. comparing the centrifugal force acting on it on a rotating disc with the centrifugal force operating on the international kilogram prototype under similar conditions. The essential point is that since in such a procedure the weight of the bodies plays no role it is the inertial mass which is measured. Elastic collision can also be employed to compare the inertial mass of a body with that of the prototype.

On the contrary, any measurement of the gravitational mass must utilize the weight of the object. Therefore, the gravitational mass of a body is equal to the gravitational mass of the international kilogram prototype if under identical conditions they produce identical stretches of a spring-balance. From these definitions it is clear that both the inertial and gravitational masses of the prototype are by definition equal to 1 kg. Moreover, any body made of the same material as the prototype has equal inertial and gravitational mass (but not necessarily equal to 1 kg). But in Newtonian physics no compelling reason is found in favour of the equality of the two types of mass of a body whose material composition is different from that of the prototype.

Experience, however, is strongly in favour of the equality of the inertial and gravitational masses for all existing bodies. Experiments designed to determine the ratio \(m^{*}/m\) (experiments of Loránd Eötvös using a torsion balance invented by him and their modern versions) prove this equality with extremely high precision but the equality \(m^{*} = m\) finds its natural place only in general relativity theory.

- (1)The equation of motion for a free-falling body under normal earth-bound conditions would take the form($$ ma = - m^{*}\frac{GM^{*}}{R^{2}} = -m^{*}g $$
*g*is the gravitational acceleration of the prototype and \(R\) is the radius of the Earth). Hence, gravitational accelerationwould no longer be the same for all bodies.$$ a = \frac{m^*}{m}g $$ - (2)The formula for the angular frequency of the mathematical pendulum would be of the formand, therefore, its period would depend, beside its length \(l\), on the material composition of the swinging body also (before Eötvös this method was employed to measure the ratio \(m^{*}/m\)).$$ \omega = \sqrt{\frac{m^{*}}{m}\frac{g}{l}} $$
- (3)
If \(m^{*}/m\) was different from 1 no weightlessness would be experienced in freely orbiting spaceships. A satellite, performing free rotationless motion on a circular orbit, is not an inertial frame because of the centripetal acceleration \(a_c\), acting within. Objects of inertial mass \(m\) experience, therefore, an outward directed radial inertial force \(ma_c\) in it. But these same objects are subjected to the gravitational attraction \(Gm^{*}M^{*}/r^{2}\) of the Earth too. When the inertial and gravitational masses of all bodies (including the spacecraft capsule itself) are exactly equal, the two forces acting on them compensate each other since the radius of revolution is defined by the equation \(a_c = GM/r^{2}\) common to all bodies belonging to the spacecraft. If, however, the ratio of \(m^{*}\) to \(m\) had diverse magnitude depending on the material composition then bodies would be acted upon by forces of various strengths directed either toward or opposite the Earth.

The circular shape of the orbit is by no means a necessary condition of apparent weihtlessness. This phenomenon should be observed in spacecrafts orbiting freely (i.e. with engines switched off) on trajectories of any shape. This is the direct consequence of the fact that, when \(m = m^{*}\), under these circumstances mass drops out of the Newtonian equations of motion.

## 3.2 The Equivalence Principle

If the equality \(m = m^{*}\) is indeed generally true then, as we have just noticed, the mass of the body, moving freely or under the action of gravity alone, is absent from the Newtonian equation of motion. For free motion this equation is simply \(m\ddot{\bf r} = 0\) and the mass can obviously be omitted. When gravitation is also present, the zero on the right hand side must be replaced by the expression of the gravitational force proportional to the gravitational mass \(m^{*}\) of the moving body and the masses are not cancelled out. But in the case of their equality cancellation becomes possible and again *no parameter remains in the equation which would be characteristic of the moving body*.

From this point of view, therefore, gravitational motion is similar to free motion. But world lines in the latter case (the solutions of the equation \(\ddot{\bf r} = 0\)) are straight lines which are *pure geometrical objects*. This almost banal consequence of the equation \(ma = F\) prompted Einstein to pose the following surprizing question
^{1}: is it not possible that the world lines of freely gravitating bodies are, from a suitable point of view, also purely geometric objects i.e. ‘straight lines’? Is it conceivable that flying tennis-balls, the Moon, the planets and artificial satellites are all moving on ‘straight lines’?

Einstein gave affirmative answer to this perplexing question which he based on the assumption that domains of spacetime where gravitation is present are \(curved\) four dimensional manifolds and freely moving bodies follow the ‘straightest path’ compatible with their curvature. Since for human perception even three dimensional curved space presents an insurmountable obstacle, the elucidation of the nature of curved manifolds must be based on the example of curved two-dimensional surfaces. ‘Straight lines’ do indeed exist on them since between pairs of not very distant points there is always found a unique shortest path called *geodesic*.
^{2} On a sphere, for example, the geodesics are the main circles. Though sphere is a surface of very regular shape, even its geodesics posses properties uncommon with ordinary straight lines on a plane: they cross twice each other. The mathematically elaborated form of the theory permits in real spacetime geodesics as diverse as belonging to a tennis-ball or a planet.

According to general theory of relativity, therefore, no such physical entity as ‘gravitational force field’ exists. Gravitational phenomena are rather manifestations of a hitherto unsuspected geometrical property of spacetime: its deformation around massive heavenly bodies.
^{3} The world lines of bodies (e.g. planets) moving freely around them are the geodesics in this deformed spacetime (*geodesic hypothesis*). This is a free inertial motion in the same sense as the motion along the straight world lines in flat (free from gravity) spacetime of special relativity.

There is, however, a problem here. At the end of the previous section we have explained that the weightlessness in a freely orbiting spacecraft is only apparent, caused by the compensation of the weight of bodies in it by the outward directed inertial force \(ma_c\). But if gravitational attraction was indeed absent then, contrary to experience, this inertial force would remain unbalanced.

The solution of general relativity to this puzzle is that objects belonging to spacecrafts in free motion are \(not\) in fact exposed to the action of either gravitation (weight) or inertial forces because such spacecrafts are *true inertial frames*. This is a perfectly logical corollary to the geodesic hypothesis since in Newtonian physics too objects in free motion are at the same time inertial frames.

Therefore, the conclusion drawn by Einstein from the equality \(m = m^{*}\) is in fact the \(generalization\) of the two basic Newtonian properties of free motion to gravitation: the trajectories are rectilinear (geodesics) in both cases and reference frames, following them (without rotation), are inertial frames. This last principle is known as the *equivalence principle*.

These ideas were transformed from a program into a theory when, after eight years of painful reflexion, Einstein discovered that system of nonlinear partial differential equations which describe the deformation of spacetime around celestial bodies such as the Sun (\(Einstein{\text{-}}equations\)). By about this time mathematicians had already clarified how to calculate geodesics in manifolds of given deformation. It then became possible to calculate the orbits of planets within the framework of the Einstein’s concepts, without resorting to the Newtonian law of universal gravity. This totally new approach confirmed the correctness of Keplerian orbits but only as accurate first approximations. The corrections to them are small but fully accessible to observation. Their verification proved unequivocally the superiority of general relativity over the Newtonian law of universal gravity.

## 3.3 The Meaning of the Relation *m** = *m*

In the Newtonian framework this equality means that every body possesses both an inertial and a gravitational mass and, according to the experience, they are always equal to each other to a high degree of accuracy. From the point of view of general relativity, however, the correct interpretation of this empirical equality is that objects posses only inertial mass \(m\) which automatically appears wherever the Newtonian concept of gravity requires the presence of the gravitational mass \(m^{*}\).

The appearance of the gravitational constant \(G\) and the mass \(M\) of the Sun in this equation is the consequence of the fact that the Einstein-equation for the spacetime domain around the Sun contains these parameters and, therefore, the deformation of spacetime also depends on them. Since the geodesic equation (3.3.1) is specific to this domain these parameters appear necessarily in it. This equation would be identical to the Newtonian equation of motion of a test body in the gravitational field of the Sun if its left and right hand sides contained the body’s inertial and gravitational mass \(m\) and \(m^{*}\) respectively. But if the latters were of different magnitude this equation of motion would not be the consequence of (3.3.1) since the validity of an equation is destroyed when its sides are multiplied by different numbers. Since the acceleration on the left hand side is multiplied always by the inertial mass \(m\) independently of the nature of the accelerating force, the right hand side should also be multiplied by \(m\). This is how in general relativity the place of the gravitational mass \(m^{*}\) is automatically taken up by the inertial mass \(m\).

## 3.4 Locality of the Inertial Frames

In Newtonian physics and special relativity the spatial extension of the inertial frames may in principle be arbitrarily large but actually they are real physical objects of finite dimensions. The coordinate system, however, attached to them can in principle be extended up to infinity. The spatial part of Minkowski coordinates is a Cartesian system whose axes are infinitely long in both directions. In this sense in Newtonian physics and special relativity the inertial frames are \(global\).

But from what had so far been said about general relativity it must be clear that the inertial frames in a deformed spacetime should be \(local\). A freely gravitating spacecraft in rotationless motion can be taken to a good approximation for an inertial frame only if its dimensions are of limited magnitude. Mental extension of local inertial frames to global ones is an illegitimate intellectual operation. The satellites of the Earth or the planets of the Sun, the stone which is thrown up perform free inertial motion. If to think of an inertial frame as large as the Solar System was a permissible abstraction then all these objects could be related to it and should perform uniform rectilinear motion. If general relativity theory is correct, nothing in Nature corresponds to the notion of such global inertial frames.

As a matter of fact, the inertial frames, occurring in Newtonian physics, are of very limited extension with only one exception which is, however, of paramount importance: it is the inertial frame implicit in the Newtonian theory of planetary motion. The theory’s basic equation is (3.3.1) on whose right hand side no inertial force is present. Hence, the equation is only valid in an inertial frame whether or not this is stated explicitly. Therefore, Newtonian theory of gravitation assumes that an inertial frame however large is a meaningful concept. On the contrary, in general relativity planetary orbits are calculated without such an assumption. We have outlined above the steps of this calculation which requires only the choice of a coordinate system adapted to the spherical symmetry and stationarity of the Sun without assuming the existence of a reference frame of any kind.

The local inertial frames of general relativity posses all the genuine properties of the inertial frames as discussed in Sect. 1.1. In particular, it is true with respect to them (and only them) that light speed is of equal magnitude \(c\) in any direction. Owing to the lack of global inertial frames, no \(general\) statement can be formulated concerning the propagation of light in the interstellar space. It must be calculated in each particular case in essentially the same manner as calculations of planetary orbits are performed. Therefore, in the deformed spacetime of general relativity it is impossible to synchronize distant clocks by means of light signals for the same reason as in the accelerating reference frames in the flat spacetime of special relativity. As a consequence, no global Minkowski coordinates are compatible with such spacetimes (just as no Cartesian coordinates exist on a sphere). As in special relativity, distant simultaneity remains contingent on the choice of the coordinate system. Judged by the equality of their coordinate times, with an event *here now* a whole segment of events can be simultaneous \(there\) in a distant point. This is the same ‘interval of simultaneity’ which we have already met with in the example of the Mars rover ( Sect. 1.3), but when gravity is taken into consideration, the deformation of spacetime makes to apprehend its structure more difficult.

In Sect. 1.1 we have already hinted at the curious property of inertial frames in general relativity that their relative motion is different from being uniform rectilinear. We can now see with more clarity the content of this statement and that one of the main tasks of general relativity is to clarify how local inertial frames are moving. General relativity is, in a sense, *the general theory of the inertial frames*.

## 3.5 The Weight

According to general relativity gravitation is not a force but the curvature of spacetime. Where then the weight of bodies comes from?

When standing on the ground, we are not at rest in any inertial frame. The latters are freely falling objects and, therefore, we would only be at rest with respect to one of them if we were standing in a freely falling lift (*Einstein lift*)—but then we wouldn’t experience the weight of a body held in our hands. When we are standing firmly on the ground our rest frame is accelerating \(upward\) with respect to our instantaneous rest frame of inertia with an acceleration of \(g=9.81\,\hbox{m}/\hbox{s}^{2}\) and, consequently, bodies of mass \(m\) in our hands experience a *downward directed* inertial force of magnitude \(mg\). This force is called weight.

The name of the equivalence principle is rooted in this interpretation of the weight. It is meant to express that ‘weight is equivalent to the inertial force’.

## 3.6 The GP-B Experiment

Thanks to the GP-B (Gravity Probe B) experiment of NASA we are now in possession of an important experimental evidence to the effect that an inertial frame of the dimensions of the Earth is indeed an empty abstraction. In this experiment the orientation of the spin axes of four rotating balls, serving as flywheels of gyroscopes fixed in a satellite, were followed up during a year. Owing to cardan suspension applied in gyroscopes, the behaviour of the balls from the point of view of the orientation of their spin axis is the same as if they were floating in the satellite and revolving freely round the Earth.

But what if experiment contradicts this expectation and a slow precession of the spin axis is observed? If experimental errors can be safely excluded the only possibility is to admit that it is not permissible to view the motion in its relation to an inertial frame because nothing in Nature presumably corresponds to this concept.

The conclusion drawn from GP-B experiment is that the spin axis of the rotating ball does not remain parallel to itself but performs indeed very slow rotation (precession) in the plane of the orbit in positive direction (i.e. in the same sense as it revolves around the Earth). But even this minute deviation from the Newtonian prediction presents strong \(direct\) evidence against global inertial frames.

*geodesic precession*whose angular velocity is given by the formula

The spin axis should, however, be confined to the orbital plane only if the angular momentum of the Earth is neglected. Formula (3.6.1) refers to this approximation. Under the influence of the angular momentum of the central body the spin axis departs the orbital plane but the angular velocity of this motion called *drag*^{4} is 170 times slower then that of the geodesic precession. For the separation of the drag from geodesic precession the most convenient orbit is that whose plane contains the angular momentum vector of the Earth. The polar orbit in the GP-B experiment was selected to meet this requirement. Since, however, the accuracy of the measurement is estimated about 1% no conclusion can be drawn from it about the magnitude of the drag.

## 3.7 Light Deflection

An unexpected direct consequence of the principle of equivalence is that a light beam should be deflected under the influence of gravitation. If equivalence principle is true then a freely falling Einstein lift is a local inertial frame with respect to which light propagates along straight line. Consider a light ray which is horizontal with respect to the lift. Since the latter is accelerating toward the center of the Earth, the ray under consideration will be deflected downward with respect to an observer standing on the ground. This deflection is the consequence of the gravitational acceleration \(g\) of the lift, therefore, light deflection must be attributed to the same origin: the gravitation exerted by the Earth.

This conclusion from the equivalence principle was made by Einstein as early as in 1907 in a paper where outlines of general relativity were first envisaged. But only eight years later was he able to calculate the angle of deflection in the vicinity of the Sun. It turned out equal to 1.6 s for the ray grazing the Sun and even smaller for rays farther apart from it. This angle can be measured by observing the radio waves of radio stars when they disappear behind the Sun and reappear again. Observations of this type verify the theory’s prediction with an accuracy of about 1%.

According to theory the effect of spacetime deformation on light rays around Sun is mathematically equivalent to the effect of a \(condenser\) with varying in space refractive index \(n = 1 + 2GM/c^{2}r\): the closer light travels to the Sun the more it deflects toward it. Astronomical objects of larger dimension (e.g. clusters of galaxies) ‘bend’ light rays similarly (\(gravitational\;lensing\)).

It is tempting to interpret light deflection on the basis of the corpuscular theory of light as the effect on ‘corpuscles’ of Sun’s gravitational attraction. As a matter of fact, at the beginning of nineteenth century the german naturalist J. G. von Soldner predicted light deflection in the framework of Newton’s universal law of gravity based on this physical picture. For the angle of deflection he obtained precisely the half of the relativistic value calculated by Einstein in 1915 (Soldner shouldn’t have known the mass of the corpuscules since it drops out of the equations of motion if their inertial and gravitational masses are equal, as it was tacitly assumed by him). Believers of the popular view on mass–energy relation ( Sect. 1.12) often bring forward essentially this same interpretation asserting that it is only ‘the rest mass’ of the photon which is equal to zero, its ‘relativistic mass’ being given by the relation \(h\nu /c^{2}\) where \(h\nu\) is its energy. This ‘explanation’ is, however, grossly misleading since, as it has been clarified in Sect. 2.20, the mass–energy relation is not applicable to massless particles and, even if it were, this picture can account only for half of the value of the deflection. The interpretation based on the analogy with light refraction by a spherically symmetric medium is much closer to the truth.

## 3.8 Perihelion Precession

Keplerian planetary orbits are ellipses of definite orientation with respect to fixed stars. The most convincing verification of Newton’s universal law of gravity is that the laws of Kepler follow from it \(provided\) mutual gravitation attraction of planets is neglected as compared to the much stronger attraction from the side of the Sun. It is in this approximation that planetary orbits are stationary ellipses whose perihelion, which is their point nearest to Sun, is seen from the center of the Sun at a fixed point on the sky. However, owing to mutual attraction of the planets, Keplerian ellipses undergo slow continuous change of their shape and orientation. In particular, their perihelion is shifting continuously on the sky (*perihelion precession*). This phenomenon is best seen on Mercury’s orbit which is the most elongated one among planetary orbits.

In the middle of nineteenth century French astronomer Urbain Jean Joseph Le Verrier investigated the action of the other planets on Mercury. According to the principle of additivity of small perturbations, the perihelion precession of Mercury’s orbit is determined by the sum of planets’ individual contributions. He found that on the basis of Newtonian gravitation the rate of precession should be equal to 527 s in a century, but the observed shift actually exceeded this value by an amount of 38 s in a century. Later S. Newcomb on the basis of an improved analysis concluded that Newtonian theory accounts for only 534 s in a century of the total observed shift which is equal to 575 s in a century.
^{5} The discrepancy between the predicted and observed values is far beyond experimental uncertainties. To appreciate the precision of these observations remember that the apparent diameter of the Sun and the Moon is about 30’ which is 45 times larger then the anomaly of 41 seconds accumulated in a century.

## 3.9 Gravitational Red Shift

*Gravitational red shift* is a phenomenon akin to transverse Doppler-effect. In Sect. 1.2 this latter phenomenon was discussed for the special case when the emitter was at rest and the receiver revolved around it on a circular orbit. Suppose now that both the emitter end the receiver are revolving on concentric circles with identical angular velocities. Let the radius and rotation velocity of the emitter and receiver be \(r_e,\;V_e\) and \(r_r,\;V_r\) respectively. The equality of their angular velocity can be expressed as \(V_r/V_e = r_r/r_e\) and for the sake of definiteness assume that \(r_r > r_e\).

For the first time the existence of the gravitational red shift was demonstrated by R. Pound and G. Rebka in 1960, in a laboratory experiment with the help of the Mössbauer-effect. The distance between the emitter and the receiver was equal to 22.5 m in which case, according to (3.9.3), the expected relative frequency shift is only about \(2\cdot 10^{-15}\). Twenty-first century technology permits one to observe the effect at a height difference less than 1 m.

As explained above, formula (3.9.2) for the gravitational red shift comes from ( 2.11.2) which for \(\Upphi = 0\) describes the transverse Doppler-effect in special relativity and for \(v = 0\) represents gravitational red shift. However, applying (3.9.2) to gravitational red shift, we make the tacit assumption that, while travelling from the emitter to the receiver in curved spacetime, the frequency \(\nu\) of a light beam remains unchanged, as was the case in the pure Doppler-effect (see above). In the framework of general relativity rigorous proof can be given to the effect that in spacetimes which are constant in time, this is always the case.
^{6} The frequency \(\nu\) refers to the coordinate time which is the time, occurring in the equations of trajectories as e.g. in \(h = gt^{2}/2\) for free falling.

Having this property of light propagation in mind, gravitational red shift can be viewed as the consequence of ‘slowing down of the speed of time by gravitation’. Indeed, the inequality \(\nu_r < \nu_e\) means that while at a height \(z\) the time elapsed between two consecutive signals is equal to, say, 2 s, on the ground level the time passed by the same pair is only 1 s. Since on the way between the two points the time \(T\) (the frequency \(\nu = 1/T\)) remains unchanged, this can only mean that at the ground level, where gravitation is stronger, clocks go slower than at the height \(z\). This is the basis for the variant of the twin paradox sometimes called tower effect: If Alice and Bob are twins and Bob goes upstairs a high tower and remains there for a time, while Alice is staying at rest downstairs, he turns out, when comes down, older than Alice.

Doppler-effect and gravitational red shift are very general phenomena not restricted to light signals. When, for example, the emitter is replaced by a machine gun, firing at a rate \(\nu_e\), bullets will be hitting the target at the place of the receiver at the rate \(\nu_r\) determined by (3.9.2). This generality is the direct consequence of the fact that it is *the speed of the flow of time* which is responsible for the observed effect.

Advocates of the popular view on the mass–energy relation claim that gravitational red shift is the consequence of this relation. Their argumentation starts with the assertion that, while the rest mass of the photon is zero, its relativistic mass is equal to the *kinetic* energy of the photon divided by \(c^{2}\). At the ground level, the reasoning continues, where the photon is born, its potential energy is zero, therefore its energy \(h\nu_e\) is a purely kinetical one. Hence its mass is equal to \(h\nu_e/c^{2}\). At a height \(z\), where it is detected, its potential energy is then equal to \(h\nu_e/c^{2}\times gz\), and so its kinetic energy should have decreased to \(h\nu_e - h\nu_e/c^{2}\times gz = h\nu_e(1 - gz/c^{2})\). Since this energy is by definition equal to \(h\nu_r\), equating them we obtain the relation \(\nu_r = \nu_e(1 - gz/c^{2})\) which is identical to (3.9.2).

This argumentation is, however, unacceptable. Already its starting point is untenable since the mass–energy relation is inapplicable to massless particles (see Sect. 2.20). Details of the argumentation (separation of the photon energy into kinetic and potential part, identification of \(h\nu\) with the former) need justification in the framework of quantum electrodynamics. The scope of the ‘explanation’ is, moreover, extremely narrow. It has nothing to say about the example with the machine gun which is the mechanical analogue of the gravitational red shift. This indicates clearly that it hopelessly misses the point.

## Footnotes

- 1.
That was of course not as simple as it seems. It took Einstein several years of painstaking efforts to cast the problem of gravitation into this geometrical form.

- 2.
Straight lines on curved manifolds are called geodesics. The term ‘straight line’ is reserved to the geodesics on a plane (or on multidimensional analogues of the plane).

- 3.
The source of the spacetime curvature around these bodies is in fact not their mass but the more complex ten-component quantity known as

*energy-momentum tensor*. In the case of a star at rest its main component is star’s rest energy which is proportional to its mass. - 4.
The meaning of this term is that rotation of the Earth drags the spin axis with it out of the plane of revolution.

- 5.
Expressed in ecliptic longitude the rate of perihelion precession contains the additional constant 5037 s in a century which is the rate of displacement of the origin of the ecliptic longitude (of the vernal equinox) in opposite direction.

- 6.
The analogue of this theorem in geometric optics is that when light

*in static conditions*passes from one medium into another with a different index of refraction its frequency remains constant.