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Together with the deviation of light rays near the Sun the change of the orientation of Mercury's orbit century after century is one of the first dramatic confirmations of the correctness of the general relativity theory conceived by Einstein. But it is quite difficult to provide the reader with the details of the calculation as there is no simple way to do it. In fact, to get the result it is necessary to dive into the full theory of general relativity. Here is a good opportunity to discover this theory while illustrating it on an example!
The derivation of the equations given here closely follows the presentation of Edwin F. Taylor and John Archibald Wheeler in their delightful work "Exploring Black Holes, Introduction to General Relativity" (Addison Wesley Longman, 2000). The calculation of the integral giving the value of the precession of the orbit is adapted from the one developped by Steven Weinberg (Gravitation and Cosmology, Principles and Applications of the General Theory of Relativity, John Wiley & Sons, 1972).
General relativity teaches us that by propagating into space a free particle (that is a particle not submitted to the acceleration of a motor) follows what is called a "geodesic" of spacetime. Thus we have to examine the following points:
Newtonian mechanics describes the motion of a particle in an absolute space with respect to an absolute time. The position of the moving object is located by its coordinates with respect to some frame of reference and is given as a function of time . General relativity affirms that there exists no absolute time ant that time cannot be dissociated from space. The theory bases its reasoning on events, each event being characterized by a point M (where it happens!) and a time (when it happens!). The events attached to a moving particle constitute what is called a worldline.
Let us consider for example a spaceship moving freely through space, which means that all his motors are turned off. Let us imagine that regular flashes are emitted in accordance with a clock located inside the rocket and beating the time. The time interval between two successive flashes will be denoted by (this quantity is thus measured with respect to the proper time of the spaceship). Think now of another frame of reference as constituted by an ensemble of space beacons, also free from acceleration, at constant mutual distances from one another (each free beacon stays at the same distance from its neighbours). Every signal bears the indication of its position in space (for instance by showing its distance from some origin) and holds its own clock. The clocks of this second frame are synchronized between them. Then in that frame the interval between two flashes (i.e. two events) is characterized by two numbers: the space interval and the time interval . To determine those two quantities it suffices to record which beacon faces Flash #1 and which beacon faces Flash #2 while noting the times of those events.
Special relativity is based on the following principle. The proper time interval between Event #1 et Event #2 is given by the formula
Be careful: unless otherwise indicated, distances will be measured in units of time, as is often done in astronomy. We have chosen to do so in writing Equation (1). On the contrary if distances s are expressed in conventional units, for instance in centimeters, then one should pass from the latter to our distance s expressed in seconds via the formula
In general relativity the property of invariance of the proper interval with respect to a change of the coordinates remains valid but only locally, i.e. under the condition of staying in a sufficiently small region of spacetime (its size depends on the accuracy of the measurements). The main novelty concerns the expression of the proper time as given by Formula (1). The coefficients entering this formula depend now on the point of spacetime under consideration and the resulting expression takes the name of metrics. In fact the whole structure of spacetime, and especially its curvature, is included in the local expression of and in the form of its coefficients.
We are interested here in the structure of spacetime around the sun. In order to describe the physics locally we consider two nearby events separated by infinitesimal amounts of the time and space coordinates , , and . If space were flat, the metrics would have the form
where r denotes the distance to the center and an azimutal angle in the plane of the orbit (see the figure below).
But spacetime around a center of attraction of mass (for instance a black hole or the vicinity of the sun) is not flat. It is characterized by the Schwarzschild metric
The story is really fantastic: the whole structure of spacetime is embodied in this "simple" formula (3).Even the famous black hole lurks behind those apparently innocuous symbols.
One question: in which units is expressed the mass in that formula? It is seen that has the dimension of a length, a quantity that we measure here in seconds. Therefore will also be measured in seconds. The formula allowing to transform grams in seconds is
The metric, that is (exactly) the formula expressing at a givent point of spacetime the temporal interval between two nearby events, reveals the presence of curvature as soon as the expression deviates from Formula (2) corresponding to flat euclidian space. That metric will allow us to find the properties of the motion of a test particle free from acceleration. Actually both special and general relativity teach us that between two given events and a freely moving body follows the path for which the time interval is maximum. Equivalently one can say that a freely moving particle follows a geodesic of spacetime as a geodesic is precisely defined by this property of maximazing the time interval.
Definition of a geodesic: the geodesic between two events and is the wordline for which the interval of proper time between and is maximum.
That property of maximazing the proper time will allow us to derive the equations of a geodesic. It will also yield the expressions of the energy and angular momentum of a particle in orbit around the center of attraction.
In order to avoid varying all quantities at the same time, we assume in this experiment that the locations of the radii
are fixed and that only the time
, at which the second flash is emitted, is allowed to change. According to Formula (3) the interval of proper time over the first segment is given by its square
The lapse of time over Segment
between the events
, and therefore the proper time duration
is given by
To make the total time interval
maximum with respect to a variation
of the time
We have discovered in Equation (10) a quantity that is the same for both segment. This quantity is thus a constant of the motion for the free particle under consideration. For good physical reasons (especially to recover the formulae of special relativity), one is led to identify that constant of motion as the ratio of the energy of the particle to its mass. We write this very important result under the form
Incidentally we may notice that with the units we have chosen, energy and mass are expressed in the same unit (for instance the centimeter).
We have applied the principle of maximazing the proper time interval by varying the time of the intermediate event E2. We now perform the same operation but this time we vary the angle of that intermediate event. We recall that measures the direction of the moving particle with respect to some direction chosen as the origin. We call it the azimuth.
We consider again three events consisting in the emission of flashes inside a spaceship floating freely in space. The first segment connects Event to Event . The second segment connects to . The azimutal angle of the first event is fixed at . The angle of the last one is fixed at . The intermediate azimuth is taken as the variable . Again in order not to vary everything at the same time, we assume that the radius at which the second flash is emitted stays constant.
We follow the same chain of reasoning as in the previous section. From the metric (3), the time interval
over the first segment is given by its square
Technically speaking in order to determine the trajectory of a moving body free from acceleration we apply the following strategy. Knowing the energy
and the angular momentum
of the particle of mass
depend on the initial conditions) we can follow the position of that particle by computing the increments of its spacetime coordinates
as the proper time
itself advances. Algebraically for each increment
of the proper time we compute (or the computer calculates) the corresponding increments
of the coordinate of the mobile body. The squares of the increments
are extracted from Equations (11) and (17) in the following form:
We notice that the expression of
is missing. We get it by transporting the values of
into the metric equation (3) and solving it for
. This yields
By dividing both sides of Equations (20) and (19) we directly arrive to the equation of the orbit in polar coordinates as
It is easy to extract
from those equations as
Multiplying both sides by the factor
and developing that term up to the second order in M
The trick to simplify the apparently quite complicated right side consists in noticing that we are dealing with a quadratic function of
which vanishes at
(if we neglect the terms of order
Therefore it has the form
By taking the square root of both sides of that equation and by neglecting terms of order
or higher the trajectory of the particle around the sun can be computed by integrating the expression
The integration is trivial if one makes the change of variable
The first term inside the brackets, which is equal to unity, leads to the classical newtonian ellipse.
Actually, if, then Expression (31) yields the equation of the trajectory in polar coordinates
in the form
Identifying Formulae (34) and (35) we see that
The third sinusoidal term in in the integral (33) adds only a periodical perturbation which produces a kind of noise.
Let us calculate the numerical value of the angle of precession. The semimajor axis of the elliptical orbit of Mercure is
and its eccentricity is
. The mass of the sun in centimeters is
, which yields (with
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