General Theory Of Relativity - LogiLogi

The general theory of relativity

The following exposition of the general theory of relativity does not follow the historical chain of events, even though many historical events are discussed.

In this article, newtonian dynamics is presented in such a way that a transition to relativistic dynamics is prepared for as much as possible.

This article relies on the reader being comfortable with the content of the TheoryOfRelativity article

Preparation: Newtonian dynamics

Fields

In the course of the 19th century, it became customary to understand the laws governing motion of particles in terms of fields. Particles such as an electrically charged particle were thought of as the origin of a field, and other particles may interact with that field. Rather than assuming that particles exert a force directly on each other, the presence of a field is assumed, and the field acts as a mediator of the interaction.

Newton had been the first to show mathematically that if a force does not act instantaneously over distance, then the law of conservation of momentum will not hold good. The stability of the Solar system was strong evidence that momentum is always conserved, thus giving strong support to the case for the concept of instantaneous action at a distance.

It was found that a change of the origin of an electromagnetic field propagates at a finite speed away from the origin. That implies that electromagnetic interaction does not act instantaneously at a distance. To explain that conservation of momentum was observed to hold good, it had to be assumed that the electromagnetic field can carry momentum independently. This indicated that the electromagnetic field was not just a theoretical construct, the electromagnetic field has in common with matter that it can carry momentum. This indicates that a field and matter are not as distinct as one might expect.

Inertia field

Since in classical dynamics all aspects of motion are understood in terms of properties of particles interacting with fields, it is natural to postulate the existence of an inertia field.

To get a feel for the properties of the inertia field, reviewing the somewhat analogous phenomenon inductance is helpful. A coil of conducting wire with self-induction has the following property: it will offer little resistance to current, but it will oppose any change of current strength. (If the coil is cooled down to a temperature at which superconductance occurs, the coil wil offer zero resistance to current.) The mechanism of opposing change of current strength is as follows: change of current strength induces a changing magnetic field, which in turn induces an electric field that counteracts the change of current strength.

In a wire with zero resistance and zero self-induction, applying a voltage would result in an instantaneous jump to an infinitely strong current.
In a wire with some resistance and no self induction, applying a voltage results in a jump to a particular current strength (with the magnitude of the current strength described by Ohm's law: V=I*R)
In a wire with no resistance and some self-induction, and starting with zero current strength, applying a voltage results in a steadily increasing current strength. That is, if only self-induction is involved, the rate of change of current strength is proportional to the applied voltage.

In the case of the inertia field, no mechanism is known. All that is possible is to describe the properties of the interaction of particles with the inertia field. That is what Newton's laws of motion do.

Newton's laws of motion

Newton's laws of motion can be understood as the axioms of a theory of the inertia field.

Newton's first law:
The very act of opting to use euclidean geometry for representing space is in itself a theory of physics. Quite understandably, using euclidean geometry was not regarded as a theory of physics, for the obvious reasons that there was no alternative for euclidean geometry, and euclidean geometry is perfectly adequate for the purpose. Newtons first law (reinterpreted from a modern perspective) states that euclidean geometry is an appropriate physics model for the physics of the inertia field. When no force is acting on an object (or when the forces acting on it are perfectly balanced) the object will move along a euclidean straight line.

Newtons second law:
The inertia field opposes change of velocity. In order to change velocity with respect to the inertia field, a force must be exerted. The rate of change of velocity is proportional to the applied force.

Newton's third law:
The inertia field is symmetrical for all directions and for all uniform velocities.
The original formulation of the third law is that momentum is always conserved: if object A exerts a force on object B, and both objects are unattached, then both object A and B will accelerate, and in the process the change of momentum of object A and the change of momentum of object B will be the equal. Let object A and object B have mass m_a and m_b respectively, and acceleration a_a and a_b respectively. Then the following will be valid: m_a*a_a=m_b*a_b. Hence the statement F_ab=-F_ba (reaction force is always equal and opposite to the exerted force) is equivalent to stating that the inertia field is symmetrical for all directions and all uniform velocities.

A crucial aspect of Newton's laws of motion is that velocity with respect to the inertia field does not enter the axioms of the theory. Only rate of change of velocity enters the mathematical framework of the theory.

In summary:
Newton's three laws of motion can be understood as the axioms of a theory of the inertia field, and Newton's law of universal gravitation is an axiom of the theory of the gravitational field.

The equivalence class of inertial coordinate systems

The laws of the inertia field and the law of gravitation have in common that they hold good for a specific class of coordinate systems: the class of inertial coordinate systems. The criterium for distinguishing an inertial coordinate system from a non-inertial coordinate system is that in an inertial coordinate system the laws of motion hold good.

(Compare electric resistance: R. The definition of the concept of electric resistance is Ohm's law: R = V/I . There is no such thing as first defining the physics of electric resistance, and then proceed to discover Ohm's law R = V/I. Each law of physics serves both as law of physics and as operational definition of the specific context in which it is valid.)

From a physics point of view, this equivalence class of inertial coordinate systems and the inertia field and Minkowski spacetime are effectively one and the same concept. By comparison: the electromagnetic field is regarded as an occupant of space. The inertia field is not an occupant of space, from a physics point of view, the inertial field and spacetime are one and the same.

Unification: general relativity

The general theory of relativity is both a theory of gravitation and a theory of motion. Instead of postulating two fields, an inertia field and a gravitational field, general relativity postulates the existence of a single field: the inertio-gravitational field. The newtonian conceptual distinction between inertial mass and gravitational mass does not apply in general relativity. Given the postulate of the inertio-gravitational field any distinction between inertial mass and gravitational mass does not enter the general theory of relativity.

To see how it is possible to formulate a theory of gravitation starting from that postulate it is helpful to explore acceleration in Minkowski spacetime.

Acceleration in Minkowski space-time

Animation 1.
Accelerating spaceships, the red ship chasing the blue ship.

Animation 1 represents two accelerating spaceships with the "red ship" following the "blue ship". The two ships are exchanging light signals (not shown) to maintain a constant separation. The red ship adjusts it acceleration in such a way that the transit time of a round trip of the light signal remains the same. When the roundtrip time of the light signal remains the same, the following applies:

The red ship is accelerating harder than the blue ship, that is: the red ship is pulling more G's.
Signals emitted by the blue ship are on reception by the reds blueshifted as measured by the reds.
Signals emitted by the red ship are on reception by the blues redshifted as measurd by the blues.
For the blues a larger amount of proper time elapses than for the reds.
Since a smaller amount of proper time elapses for the reds it follows that the reds are travelling a longer distance than the blues, and they are travelling faster.

Animation 1 does not represent a Minkowski spacetime diagram, for in the animation there is no global coordinate time, and no global standard of length. The vertical bars that move from left to right in the animation represent how the motion of inertially moving objects would be perceived by the red and the blue ship.

Animation 2. Circular motion in Minkowski space-time.

Animation 2 shows two spaceships in circular motion. In this animation the diameter of the circle of circular motion is rather small. Now consider circular motion along a circle with an extremely large diameter. The situation is then very close to being indistinguishable from the situation depicted in animation 1, which depicts linear acceleration. This indistinguishability can be seen as a form of the principle of relativity of inertial motion. In the case of motion along a circular path there is a large sideways velocity, a uniform velocity perpendicular to the direction of acceleration. But in Minkowski space-time uniform velocity is relative.

An object that is released by the blue ship will from that moment on be moving inertially. It will then accelerate with respect to the blue and red ship, accelerating (while moving inertially) towards the red ship.

Animation 3.
An infinitely large slab of matter generates a uniform gravitational field.

Animation 3 shows two spaceships and the grey area represents a sideways view on a slab of matter that is infinite in size. Such an infinite slab of matter would alter the physical properties of the inertio-gravitational field in a uniform way. In order to keep their distance to the slab the same, the spaceships must use thrusters. As is the case with animation 1, the animation does not represent a Minkowski spacetime diagram, for there is no global coordinate time, nor a global standard of length.

The assumption of just a single field, the inertio-gravitational field, implies that the situation depicted in animation 3 must be equivalent to the situation of animation 1 in all aspects of physics..

Gravitational time dilation

Figure 1.
All over the world, clocks located at sealevel will count the same amount of proper time.

Figure 1 shows the Earth, with its equatorial bulge very much exaggerated. For clocks located on Earth there are two factors that determine the amount of proper time that elapses. One factor is gravitational time dilation. Clocks located near the poles are closer to the Earth's center of mass; they are located deeper in the potential well, which corresponds to a smaller amount of proper time that elapses, as compared to objects located less close to the Earth's center of gravitation. The other factor is velocity time dilation. Objects located away from the poles are circumnavigating the Earth's axis, which corresponds to more velocity time dilation (smaller amount of proper time elapsing) than for object located at the poles. The closer to the equator, the more velocity time dilation.

For all terrestrial clocks the two effects, gravitational time dilation and velocity time dilation cancel out exactly, and for all terrestrial clocks located at sealevel the same amount of proper time elapses. This state of the same amount of proper time elapsing all over the Earth's surface is a state towards which the system naturally evolves. If the Earth would be spinning too fast, so that instead of equilibrium a smaller amount of proper time elapses at the equator than at the poles, then there is a shear stress. Over time there will be a redistribution of the Earth's mass toward a smaller equatorial bulge, until the amount of lapse of proper time is the same all over the Earth's surface.

In the discussion above, only gravitational time dilation is discussed. Gravitational time dilation is one aspect of curvature of spacetime. A theory of gravitation that would only incorporate gravitational time dilation would in particular yield wrong predictions for very fast motion.
General relativity describes that gravitational deflection of light that just grazes the sun will be 1.75 arcseconds. A theory of gravitation that would only incorporate gravitational time dilation would predict half that value: 0.83 arcsecond. In fact, in a 1912 paper by Einstein on an early exploratory theory of gravitation that value was stated: 0.83 arcsecond.
A theory of gravitation that would only incorporate gravitational time dilation would predict planetary orbits that are very close to orbits described by newtonian dynamics and general relativity, but it would not correctly predict the precession of the planet Mercury.

Motion along a geodesic

In special relativity it is invariably seen that the trajectory of an object that is moving inertially is (out of all possible trajectories) the trajectory that is seen to have the largest amount of lapse of proper time. Why this law holds good is not known, it is only known that taking that as a law turns out to be very fruitful for a theory of motion and gravitation.

The general theory of relativity describes that in the presence of matter physical properties of space-time are changed away from uniformness. This change of the properties of space-time away from uniformness, which is referred to as 'curvature of space-time' acts as the mediator of gravitational interaction. A trajectory that from afar is seen to be non-uniform can be the trajectory with the largest lapse of proper time.

Animation 4.
The blue object follows the path with the largest lapse of proper time.

John Wheeler has introduced the example of a corridor drilled through a planet to illustrate the concept of moving along a geodesic.

A tunnel is constructed that connects in a straight line two point of a planet that are on opposite locations on that planet. The tunnel diameter is so small compared to the size of the planet that the difference in mass distribution is negligable.

When a capsule is released at one entrance to the corridor, it will from that moment on be oscillating in that corridor. In the case of the Earth the period of one oscillation would be about 90 minutes, the same amount of time as a circular orbit that circumnavigates the Earth at very low altitude. Given the properties of the space-time that the capsule is in, the oscillating trajectory is the path that corresponds to the path with the largest possible amount of lapse of proper time. If there would also be some friction, then the capsule would eventually come to rest at the midpoint of the corridor.

By analogy with euclidean geometry, this path that corresponds to an extremum is called a 'geodesic'. In euclidean geometry, a geodesic is the path with the shortest possible spatial length. In the general theory of relativity, the expression 'geodesic' refers to the path with the largest amount of lapse of proper time.

An accelerometer onboard the capsule would at all times measure zero acceleration; the capsule that is oscillating in the corridor is in inertial motion. On the other hand, objects that are at rest on the surface of the Earth are not in inertial motion. For an object at rest on the surface of the Earth the local inertial frame is accelerating towards the center of the Earth. An object at rest on the surface of the Earth is (due to the normal force exerted by the Earth's surface) being accelerated with respect to the local inertial frame.

In Minkowski space-time, the equivalence class of inertial coordinate systems is globally valid. In space-time as described by general relativity, the equivalence class of inertial systems at one point in space can be accelerating with respect to the equivalence class of inertial systems at another point. In the case of the solar system one can distinguish a hierarchy. A spacecraft that is in orbit around the Moon is in inertial motion. The center of mass of the Moon in in inertial motion around the common center of mass of the Earth-Moon system. The common center of mass of the Earth-Moon system is in inertial motion around the Sun. The center of mass of the solar system is in inertial motion around the center of the galaxy.

Analogies between two fundamental unifications

Figure 2.
Each plane of simultaneity can be seen as cutting a different cross-section of Minkowski spacetime.

Figure 2 is an illustration that was used in the article about special relativity. Every relative velocity has a corresponding distribution of coordinate time and coordinate distance. For every object/observer in spacetime there is a particular plane of simultaneity that is co-progressing in time with the object/observer. What each observer measures is a projection of spacetime in his proper plane of simultaneity. Each observer can only observe a particular slice of spacetime; the slice that corresponds to his plane of simultaneity.

In the case of the electromagnetic field the observer's velocity relative to the electromagnetic field is the determining factor for what the observer will measure. In the four-dimensional world of Minkowski spacetime, the electromagnetic field is a single field. To any observer, there are apparently two fields: an electric field and a magnetic field. For observers at different velocities, the electromagnetic field "decomposes" differently in an electric component and a magnetic component.

In the case of the inertio-gravitational field, the determining factor for the what an observer will measure is the acceleration of the observer with respect to the field. For observers that are accelerating at different uniform accelerations with respect to the local inertio-gravitational field the field manifests itself differently.

In the case of classical electrodynamics it is technically possible to formulate a theory in which it is assumed that there is an actually present background of newtonian absolute space and absolute time. In such an ether theory, having a velocity with respect to the ether results in time dilation and length contraction effects that act in such a way that the assumed background is rendered unobservable. Any theory that assumes separate space and time needs additional hypotheses to account for the fact that no experiment ever detects uniform velocity with respect to the presumed background. Special relativity has no such need because velocity with respect to spacetime does not enter special relativity.

In the case of special relativity it is technically possible to formulate a theory of motion and gravitation in which it is assumed that there is an actually present absolute Minkowski spacetime. That is: a theory that assumes a separate inertia field and gravitational field. Such a theory needs to find a way to accomodate the equivalence of gravitational and inertial mass. It turns out that a theory that assumes an absolute Minkowski spacetime needs to be amended by assuming time dilation and length contraction effects that act in such a way that the assumed Minkowski background is rendered unobservable. Any theory that assumes a separate inertia field and gravitational field would need additional hypotheses to account for the fact that no experiment ever detects uniform acceleration with respect to the assumed Minkowskian background. General relativity has no such need because assumption of a fixed background does not enter general relativity.

No fixed background

John Wheeler has coined the phrase: "spacetime is telling matter/energy how to move, matter/energy is telling spacetime how to curve." That is, in general relativity, the curvature of spacetime is a dynamic variable.

In Newtonian dynamics the purpose of writing down and solving the equation of motion is to find the motion of material objects with respect to the background. Solving equations of motion in Newtonian dynamics can be hard at times, but at least there is a fixed background: inertia. In general relativity, the purpose of writing down and solving the field equations is to find expressions of how the shape of the inertio-gravitational field evolves over time, and how the motion of objects evolves over time. Also, gravitational potential energy contributes to the total energy that must be taken into account. That is, like all forms of energy that are confined to a certain volume of space gravitational potential energy has inertial mass, hence it has gravitational mass, so gravitational potential energy contributes to space-time curvature. Given these difficulties, it is just astonishing that it has been possible to formulate a theory at all.

The equations of general relativity rely on a sophisticated mathematical apparatus that handles coordinate transformations, thus allowing equations to be expressed in a way that is not committed to a particular choice of a coordinate system. The class of coordinate mappings that is supported by this apparatus is called the diffeomorphism class. Members of the diffeomorphism class can be transformed into one another by a transformation that does not "tear" or "cut" the "fabric" of space-time. That is, two points in space-time that are adjacent in one member of the diffeomorphism class are also adjacent points in all other members of the diffeomorphism class. Other than that, the mathematical apparatus accommodates a vast range of transformations: translation, uniform acceleration, any acceleration, rotation with constant rate, rotation with variable rate, any "flexing" deformation, etc.

<p>Employing the mathematical apparatus that allows equations to be expressed in a form that is not committed to any choice of a particular coordinate system (among the diffeomorphism class) is called 'using coordinate-independent equations'. General relativity uses coordinate-independent equations to express certain physical properties. The physics of "matter/energy telling space-time how to curve, and curved spacetime telling matter/energy how to move" can be expressed in an coordinate-independent way. That is, the mathematical power of coordinate-independent representation makes it possible to formulate equations even with the background itself being a dynamic variable. When a solution to the equations is obtained, the final step is to choose a coordinate system and map the solution to that coordinate system.

Matter, Energy, Fields and Spacetime

In the introduction it was mentioned that the electromagnetic field can carry momentum, indicating that fields and matter are not as different as is usually assumed.

Electrodynamics describes that an oscillating electrically charged particle will radiate electromagnetic waves. The process of emitting electomagnetic waves decreases the kinetic energy of the oscillating particle.

General relativity describes that when two masses are orbiting each other in non-circular orbits, then that system will radiate gravitational waves. That is: gravitational waves (propagating undulations of the inertio-gravitational field) can carry away energy and momentum from a system of orbiting masses. General relativity indicates that Matter, Energy, Fields and Spacetime are not as different as one might expect.