Précis of epistemology/The truth of relativistic principles

The principle of general relativity

The laws of physics must not depend on the coordinate system with which they are formulated. (Einstein 1916)

What does it mean ?

A coordinate system allows us to name physical objects. A point of space for example can be named with three numbers, its three coordinates, as soon as a system of identification is given. When we change the coordinate system, we change the names of the physical objects, but we do not change the objects themselves or their relationships. Truths should not depend on how objects are named. The same truths can be said regardless of how we name objects, they are only formulated differently. The principle of general relativity, proposed by Einstein as a major theoretical advance, because it is the foundation of his theory of gravitation, thus resembles a triviality: the truth remains when one changes its formulation.

A more modern formulation of the principle of general relativity does not change this appearance of triviality:

Every law of physics must be expressible as a geometric, frame-independent relationship among geometric, frame-independent objects. (Thorne & Blandford, Modern Classical Physics, p.1154, abbreviated thereafter by MCP)

More simply, the laws of physics must state truths about relationships among objects.

But if the principle of general relativity is only a triviality, why is it the greatest principle of all classical physics? Why does it allow us to understand in a unified way both Newton's physics, the theory of special relativity, and the theory of general relativity? Why did it lead Einstein to the discovery of the fundamental law of gravitation?

The way in which we apply the principle of general relativity gives it a physical meaning, that is, it leads us to formulate laws that can be confronted with observation. More precisely, we postulate that all physical objects are determined with scalars, vectors and tensors and that physical laws are always equalities between scalars, or between vectors, or between tensors. Three space-times are of prime importance for defining physical objects with scalars, vectors, and tensors: Newton's classical space-time, Minkowski's space-time, for the theory of special relativity, and the curved space-time of general relativity. The geometric properties of space-time impose constraints on scalars, vectors and tensors that can be defined there and on the physical laws that can be applied to them. The principle of general relativity alone has no physical meaning, but it acquires one as soon as it is accompanied by the principles of geometry of space-time.

When applying the principle of general relativity, the principle of locality is generally respected:

The laws of physics must be equalities between scalars, or between vectors, or between tensors, defined at the same point of space-time .

Newton's law of universal gravitation does not respect the principle of locality, because it postulates an instant action at a distance, but it is a weakness of the theory and an exception in Newtonian physics. In general, the physics of Newton and his successors respect the locality principle. The dynamics of solids and incompressible fluids is also an exception, but it is not a fundamental objection, because solids and incompressible fluids are only approximate models of reality. All solids and fluids are always compressible, at least a little.

The principle of general relativity has been curiously named because it asserts that the laws of physics are not relative to a particular coordinate system or observer. It is therefore a principle that affirms the absolute character of the physical truth. The theories of special and general relativity that it allows to found are also theories of absolute truth. The principle of relativity would therefore be better named if it were called the principle of absoluteness, but Einstein decided otherwise, and his use became usual.

What is a tensor?

A rank-n tensor is by definition a real-valued linear function of n vectors (MCP, p.11). It associates a scalar, i.e. a frame-independent real number, with n vectors.

In a euclidean vector space $E$ the scalar product of two vectors is a rank-2 tensor of utmost importance. In particular, it allows to identify a vector $v$ with the rank-1 tensor defined by $F(u)=u.v$. Similarly, a linear function $A$ from $E$ into $E$ can be identified with a rank-2 tensor defined by $T(u,v)=A(u).v$

Scalars and vectors are frame-independent, geometric objects. Since tensors are functions that assign scalars to vectors, they are also frame-independent geometric objects.

A misunderstanding about the relativity of truth

Einstein's theory of relativity is different from Newton's physics on one main point, the relativity of simultaneity. According to Einstein, two simultaneous events for one observer are not necessarily simultaneous for another, if he is moving relative to the first, while Newton implicitly postulates that the simultaneity of events is absolute, the same for all observers.

But it would be wrong to believe that the theory of relativity forces us to renounce absolute truth. Einstein's theories, like all scientific theories, claim that truth is absolute, the same for all. Any truth that can be recognized as scientific knowledge by a rational being can be so recognized by all the others. The truth is the same for all or it is not scientific. A science that could only be known by a few is an absurdity. We are all equal before the scientific truth.

When truths are relative, to an observer, to a point of view, to hypotheses or principles, it is always absolutely true that they are. If a statement e is true from the point of view of A but not from the point of view of B, that it is true from point of view A is true for A as for B. e is a relative truth, but the statement that e is true from the point of view of A is an absolute truth.

One often believes to deny the absolute character of scientific truth when one observes that science hosts many theories that sometimes contradict one another, and that none can claim to be the absolute truth. But it is a mistake. When we are scientists, we know the absolute truth every day, since even when our truths are relative, it suffices to affirm that they are relative to tell the absolute truth. That no scientific theory can claim to be the absolute, complete, ultimate, and definitive truth does not prove that there is no absolute truth, only that we do not know it completely.

Newtonian physics and Galileo's principle of relativity

Newton taught us how to do physics by studying the solutions of a single equation:

$f=ma$

$f$ is the force exerted on a particle, $m$ its mass and $a$ its acceleration. This equation is the fundamental law of Newtonian dynamics.

The particles are either material points, or particles of solid or fluid. Material points are the elementary particles of classical physics. Solid or fluid particles are small regions near a point of a solid or a fluid.

In addition to material points, solids and fluids, classical physics welcomes the fields of forces exerted and undergone by matter.

For Newtonian physics, vectors are always vectors of three-dimensional Euclidean space. A vector can be identified as an arrow that separates two points of space. Two arrows that start from different points represent the same vector if they are parallel, of the same length and in the same direction.

With a reference frame, vectors are used to represent positions, velocities and accelerations of particles. A reference frame is a kind of rock from which geometric measurements can be made. If one chooses on the rock a point as origin, the position of any other point is defined by the vector which separates it from the origin. A velocity vector is the time derivative of a position vector. An acceleration vector is the time derivative of a velocity vector.

The position and velocity of a particle are always relative to the frame that allows to measure them. On the other hand, the existence of inertial frames gives an absolute character to acceleration. An inertial frame is a rock that is not subject to any force and does not turn on itself. The Earth is not exactly an inertial frame, because it turns on itself and because it is attracted to the Sun, but as a first approximation, it can often be considered as an inertial frame. If we are attached to a frame that is not inertial, such as a carousel, or a train that brakes, or that accelerates, or that changes direction, we feel the effects of rotation, or acceleration, while an inertial frame, or almost inertial, lets us rest quietly.

Galileo noticed that in a boat on a calm sea, one can make the same experiments as if one stays at the port. As long as the movement of the boat is not agitated, no experiment on the boat can tell if it is motionless or moving relative to the Earth. This is general. If a frame has a constant velocity with respect to an inertial frame then it is also inertial. All inertial frames have constant velocities relative to each other and they are all physically equivalent, in the sense that any experiment that can be done in one can be replicated in the other. This is Galileo's principle of relativity:

The results of an experiment do not depend on the inertial frame in which it is made.

Einstein called it the principle of relativity, because it shows that velocity and immobility are always relative, that there is no absolute space. Since no inertial frame is preferred, the laws of physics forbid the identification of an absolute state of rest. Movement and rest of a body are always relative to another body. To speak of rest or movement in relation to empty space has no physical meaning. But this principle of relativity can also be considered as a principle of absoluteness, since it asserts that the laws of physics are the same for all observers, whatever the inertial frame where they make their measurements.

Since all inertial frames have constant velocities relative to each other, the acceleration of a particle does not depend on the inertial frame in which it is measured. The fundamental law of the Newtonian dynamics is in agreement with Galileo's principle of relativity because it only mentions a force, which is an absolute, frame-independent magnitude, and an acceleration, which is also an absolute magnitude, independent of the frame, provided that it is inertial.

Deformations are also absolute magnitudes, the same for all observers. They are represented by vectors that measure the difference between the deformed position and the initial position. It is natural to measure them in a frame where the material is at rest, but another frame is also suitable.

Newtonian physics is generally very well adapted to the study of movements of ordinary matter, solids and fluids. The measurement of a few parameters, density, elasticity, viscosity ... is often enough to calculate and predict the movements of a material. Some simple laws on the existence of the forces of elasticity or viscosity are sufficient to apply Newton's fundamental law, $f=ma$. The principle of general relativity and the locality principle are very powerful tools for finding the laws of motion, because the scalars, vectors and tensors that make it possible to state these laws are few. In fact, all Newtonian classical physics can be found as a consequence of the principle of general relativity when applied to Newtonian space-time (MCP, p.10).

On the other hand, Newtonian physics is not well suited to the study of fundamental interactions. The law of universal gravitation does not respect the principle of locality and the laws of electromagnetism, the Maxwell and Lorentz equations, formulated in a Newtonian framework, do not respect Galileo's principle of relativity. Einstein's theory of special relativity solves the second problem, while his theory of general relativity solves the first.

Minkowski space-time

The constancy of the velocity of light

Maxwell's equations predict that light has a constant speed $c$ in vacuum, and the same in all directions. This prediction seems incompatible with Galileo's principle of relativity. If the light is advancing at a speed $c$ relative to an inertial frame, then it should be motionless relative to another inertial frame that advances at the same speed $c$ relative to the first. If we were able to go as fast as light, it could be motionless in relation to us, and we could not even see ourselves in a mirror.

The constancy of the speed of light in vacuum was first interpreted as proof of the existence of absolute rest. An inertial frame would be absolutely at rest provided that it always attributes to light the velocity $c$ in vacuum. In any inertial frame moving with respect to it, the light would have a different velocity. It was also supposed that the light was a kind of vibration of a transparent and unknown material which was called ether. Absolute rest was then conceived as immobility with respect to this hypothetical ether. It was also necessary to renounce the universal scope of Galileo's principle of relativity, since it was contradicted by Maxwell's equations.

It is this last point that has challenged Einstein, because the laws of electromagnetism often seem on the contrary in very good agreement with Galileo's principle of relativity. This is why Einstein made a surprising choice to interpret Maxwell's equations while preserving Galileo's principle of relativity. This is a surprising choice because it leads to the following principle:

The velocity of light is the same, in all directions, regardless of the inertial frame where it is measured.

In particular, the light advances at 300000 km/s with respect to the Earth, but if I pursue it with a speed of 299000 km/s then it still advances at 300000 km/s with respect to me. It is an absurd conclusion only in appearance. Einstein has shown that the relativity of simultaneity with respect to the observer's movement is sufficient to remove the appearance of absurdity.

The relativity of simultaneity

Whether simultaneity is relative or absolute is an empirical question as soon as we have specified how we observe the simultaneity of two events. To do this, it suffices to measure distances on an inertial frame and to place there an emitter of spherical waves, sound waves for example, which propagate in a material medium attached to the frame. Two points located at the same distance from the transmitter receive a sound pulse simultaneously, by definition of simultaneity. Einstein used light itself, not sound waves, to define the simultaneity of events, but it weakened his reasoning, because he had to postulate that light has the same speed in all directions. This postulate is not necessary to define simultaneity.

The empirical results are clear. All experiments to date confirm the predictions of Einstein's theory. The simultaneity of the events is therefore relative to the movement of the observer and the speed of light is the same in all directions, regardless of the inertial frame where it is measured.

Spatio-temporal measurement devices

A rigid ruler, or a compass, makes it possible to measure the distance between its extremities by comparing it to other distances. A rigid ruler is therefore a device for measuring a length, or a spatial interval. The important point is that the ruler is rigid, so that the measured distance is always the same. Spatio-temporal measurement devices can be designed on the same model. A simple clock is such a device. It allows to point two events which are always separated by the same duration. One can also use two clocks fixed to the same rigid support, each being used to point a certain event. If the device that triggers the two clocks is regular, the spatio-temporal interval between the two pointed events can always be the same. Such devices make it possible to measure spacetime in the same way that rigid rulers make it possible to measure space.

An interval is timelike if it is on the path of a massive material point. It is lightlike if it is on the path of a ray of light in vacuum. All other intervals are spacelike. When an interval is spacelike, there is always an inertial frame for which its extremities are simultaneous events. Spatio-temporal measurement devices measure the three kinds of intervals. A single clock is sufficient to measure timelike intervals. Two clocks fixed on a rigid ruler and suitably synchronized make it possible to measure spacelike or lightlike intervals.

All lightlike intervals are equal

The theory makes it possible to assign a real number to all spatio-temporal intervals, timelike, lightlike and spacelike, or more precisely to their square. Timelike intervals in particular are all equal to zero and therefore all equal to each other. This is a priori surprising. This means, for example, that the spatio-temporal interval between the emission of a photon and its reception three meters away is equal to the interval between its emission and its reception three light-years away, as if the advancing photon never departed from its starting point. Isn't it wonderland?

Spatio-temporal measurement devices and the relativity of simultaneity allow to understand this counterintuitive result. Imagine a rocket launched towards a star three light-years from Earth. A photon is emitted at the rear of the rocket and received at the front, three meters away, from the point of view of the rocket, so a tiny fraction of a second later. But because of the relativity of simultaneity, what lasts a tiny fraction of a second from the point of view of the rocket can last three years from the point of view of the Earth, provided that the rocket is fast enough. The same spatio-temporal device measures both a lightlike interval of three light-years and a lightlike interval of three meters. It thus establishes their equality.

Minkowski's metric and the tensors of space-time

In Euclidean space, the distance can be defined from the scalar product, because the length of a vector is the square root of its scalar square:

$||v||={\sqrt {v.v}}$

This is why the scalar product is said to define the metric of Euclidean space.

Conversely, the scalar product can be defined from the distance, with the formula:

$u.v={\frac {1}{4}}(||u+v||^{2}-||u-v||^{2})$

When we introduce Cartesian coordinates $v=(x,y,z)$ the scalar square is the formula of Pythagoras:

$||v||^{2}=x^{2}+y^{2}+z^{2}$

In Minkowski's space-time, we can also define a "scalar" product from interval measurements, but we sometimes call it pseudo-scalar because it does not have all the properties of the usual scalar product.

As in Euclidean space space-time vectors can be identified as intervals, but they are intervals between events and not intervals between points in space. Vectors therefore have four components, one temporal component and three spatial components.

Timelike intervals can be measured in seconds and spacelike intervals in meters. The "scalar" product distinguishes them by attributing to the ones a positive scalar square and to the others a negative scalar square. The scalar square of the lightlike intervals is zero.

If we choose that timelike intervals have a negative scalar square, the scalar square of an interval $v=(t,x,y,z)$ is:

$||v||^{2}=-c^{2}t^{2}+x^{2}+y^{2}+z^{2}$

If one chooses as unit of distance the light-second, approximately 300000 km, $c=1$ and the scalar square is written more simply:

$||v||^{2}=-t^{2}+x^{2}+y^{2}+z^{2}$

From the scalar square and thus from the measurement of space-time intervals we can define the spatio-temporal "scalar" product as in Euclidean space.

Tensors are defined as in Euclidean space, except that the fundamental tensor is Minkowski's "scalar" product.

Why is Newtonian physics nevertheless true?

Newtonian physics implicitly postulates that the simultaneity of events is absolute and applies the principle of general relativity to the three-dimensional vectors of Euclidean space and to their tensors. Now the theory of relativity has shown that the simultaneity of events is relative and that the principle of general relativity must be applied to the four-dimensional vectors of space-time and to their tensors.

Newtonian physics is false but only a little false, and often its falsity is negligible, undetectable given the imprecision of measurements. Three-dimensional vectors are like shadows of four-dimensional vectors. As long as the velocities are small compared to the velocity of light, these shadows are little deformed, and one makes a small mistake if one reasons on the shadows rather than on the four-dimensional vectors. It is only when velocities approach the velocity of light that the falsity of Newtonian physics becomes detectable.

The curvature of space-time and gravitation

Free fall and the orbits of planets

Newton's theory explains free fall at the surface of the Earth and the trajectory of the Moon with the same force of gravitational attraction:

Everything happens as if the Moon did not stop falling on the Earth, but it always falls beside, and finally it only turns around the body on which it falls. In general, satellites and projectiles all obey the same law. The orbits of the planets and the trajectories of all the satellites can be considered as free fall trajectories, except that it is a fall which has no end.

Newton's theory thus shows, against Aristotle, that the same laws explain the movement of terrestrial and celestial bodies.

Einstein's great idea

Einstein was led to the theory of general relativity by understanding that free fall is identical to zero gravity. If we are in an elevator in free fall, it is like being in a zero gravity elevator, until it crashes to the ground. This enables to produce zero gravity at the surface of the Earth. Just use an airplane that simulates a free fall path. As long as it stays on such a trajectory, all passengers experiment weightlessness:

Stephen Hawking aboard a Boeing 727 during a zero gravity experiment

For Newtonian physics, being in free fall is to be subject to the force of gravity. For Einsteinian physics, being in free fall is to be not subject to any force, because a free-falling body behaves in the same way as if it were weightless. According to Einstein, the force of gravity does not exist, gravitation is not a force.

For Newtonian physics, a body at rest on the Earth's surface has zero acceleration and is therefore not subject to any force, or more precisely it is subject to two forces that compensate each other exactly, the gravitational attraction of the Earth and the reaction force of the ground. For Einsteinian physics, a body at rest on the surface of the Earth is permanently subject to the ground reaction force. This is not compensated for by the force of gravity, which does not exist. If no force acted on the body, it would remain on a free fall path and therefore would not rest at the surface of the Earth.

In the space-time of Newton, and in that of Minkowski, the bodies which are not subject to any force always go in a straight line and at constant velocities with respect to each other. These rectilinear trajectories are the geodesics, i.e. the lines of shortest path, of these space-times. In a curved space-time, all free-falling bodies and all satellites in orbit are not subject to any force, and their trajectories are all geodesics.

In a flat space, such as a sheet of paper, the shortest path lines are always straight lines, but a curved space can accommodate other geodesic lines. For example, on a sphere, geodesics are large circles. The space-times of Newton and Minkowski are flat, in the sense that the geodesics are always straight lines, but the space-time of the general relativity is a curved space-time, because the geodesics there are free to adopt more forms.

In Newton's theory, massive bodies exert gravitational forces on each other. In Einstein's theory, these forces do not exist, but the bodies affect the curvature of space-time where they are present. The Sun, for example, curves the space-time by its presence and thus makes appear in its neighborhood geodesics which come back on themselves. The trajectories of the planets are precisely such geodesics.

The equality of inertial and gravitational mass

The inertial mass $m_{i}$ is by definition the coefficient of inertia that appears in the fundamental law of Newtonian dynamics:

$f=m_{i}a$

It is a coefficient of inertia because it measures the ability of a body to resist the action of a force. The higher it is, the smaller the effect $a={\frac {1}{m_{i}}}f$ of a force $f$.

The inertial mass measures how a body experiences the forces exerted by other bodies. The gravitational mass $m_{g}$ measures the way a body acts on other bodies. It is the coefficient that appears in the law of universal gravitation:

Two bodies of masses $m_{g}$ and $M_{g}$ separated by a distance $d$ each exert on the other an attractive force equal to $G{\frac {m_{g}M_{g}}{d^{2}}}$

$G$ is the constant of universal gravitation.

It is observed that all bodies subject to gravitation undergo all the same acceleration, whatever their mass and the material of which they are made. This observation can be explained by postulating that the inertial mass is always equal to the gravitational mass $m_{i}=m_{g}$, because a body with inertial mass $m_{i}$ and gravitational mass $m_{g}$ subject to the gravitation of a body with gravitational mass $M_{g}$ has an acceleration equal to

$a=G{\frac {m_{g}}{m_{i}}}{\frac {M_{g}}{d^{2}}}=G{\frac {M_{g}}{d^{2}}}$ if $m_{i}=m_{g}$

Newton's theory does not impose the equality $m_{i}=m_{p}$. In principle these two coefficients could be different, and the acceleration of a body subject to gravitation could depend on the material that constitutes it. But the more precise measurements have never enabled us to observe such a variation.

Einstein's theory requires that the trajectory of a body subject to gravitation does not depend on its material, because all bodies, whatever their material, must follow the same geodesics. It imposes therefore that inertial and gravitational masses be equal, which is exactly what we observe.

Special relativity and general relativity

The metric of Euclidean space is described with the Euclidean metric tensor, i.e. the scalar product, which can be defined from the length of vectors and hence from the measurement of lengths in space.

The Minkowski space-time metric is described with the Minkowskian metric tensor, that is, the Minkowski "scalar" product, which can be defined from the measurement of space-time intervals.

The metric of a curved space-time is described with a Riemannian metric tensor. This is the analog for a curved space-time of the Minkowskian metric tensor for a flat space-time.

Differential geometry is the mathematical tool adapted to define curved space-times and their metric tensors. It thus makes it possible to define all the vectors and tensors that can exist in these space-times. To allow all conceivable forms of space-time, it allows all coordinate systems. The same object can therefore always be represented even after an arbitrary change in the coordinate system. More precisely, we are only interested in changes of coordinates that respect the differential structure of space-time, that is to say the diffeomorphisms.

Arbitrary changes in coordinate systems are necessary to define curved space-times in complete freedom. For flat space-times, Newton's or Minkowski's, such coordinate changes are not forbidden, but they are not necessary. We can always limit ourselves to Cartesian coordinates, defined with orthonormal reference frames. This is why Minkowski's space-time can be defined with a principle of special relativity:

The laws of physics must not depend on the orthonormal coordinate system with which they are formulated.

The curved space-time of general relativity must be defined with a general principle of relativity:

The laws of physics must not depend on the coordinate system with which they are formulated.

But the principle of general relativity can also be applied to the space-time of Minkowski, or that of Newton, and we do not hesitate, since we do not always use Cartesian frames to study them.

Relativistic principles are confirmed by their fruits

The successes of physics show that truths can be stated about the objects and relationships that we define. The principle of general relativity, which we should call the principle of absoluteness, is thus confirmed whenever a physical theory makes us discover truths.

Newtonian physics applies the principle of general relativity to Euclidean space. The vectors and tensors of Euclidean space suffice to represent all the motions of ordinary matter, provided their velocities are small in comparison with that of light. The principle of general relativity applied to the Euclidean space is thus confirmed whenever the theories it allows to formulate are in agreement with the observations, that is to say always, or almost always.

The theory of special relativity applies the principle of general relativity to Minkowski space-time. The vectors and tensors of Minkowski space-time suffice to represent all the motions of matter, so long as one can ignore gravitation and quantum effects. The principle of general relativity applied to Minkowski space-time is thus confirmed by all classical physics experiments, except those involving gravitation.

The theory of general relativity applies the principle of general relativity to curved space-time. The vectors and tensors we can define are sufficient to represent all the movements of matter, so long as we can ignore the quantum effects. The principle of general relativity applied to curved space-time has always been confirmed by all the experiments that could have refuted it and it has led to remarkable predictions: the Big Bang, black holes, gravitational waves ...

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