Covariance. Covariance is the generalization of the theory of special relativity, where it is sought that the laws for nature have the same form in all reference systems, which is equivalent to all reference systems being indistinguishable. In other words, that whatever the movement of the observers, the equations will have the same form and contain the same terms. This was Einstein’s main motivation for him to study and postulate general relativity. The covariance principle suggested that the laws should be written in terms of tensors, whose covariant and countervariant transformation laws could provide the “invariance” as intended, satisfying the covariance principle.
Summary
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- 1 Introduction
- 2 General idea
- 3 Principle of Covariance
- 1 What’s New Einstein Introduces
- 4 Formulation
- 5 Application example
- 6 Informal use
- 7 Sources
Introduction
Between 1907 and 1916 Einstein spent much of his time and efforts in generalizing the theory of invariance to non-inertial frames of reference. Although the result of these works, which would later become known as general theory of relativity, is substantially more complex than special theory, in what follows we will use what we have already learned from special theory to understand some important aspects of general theory and then explore some of its implications.
We know that the special theory is based on two basic principles, the principle of relativity and the constancy of the speed of light . Similarly, the general theory is also based on two principles: the covariance principle and the equivalence principle. We are going to see that both are very easy to pose and understand, another much more complicated thing is to express their consequences mathematically, something we will not go into.
Newtonian mechanics equations assumed that space and time were absolute magnitudes, universal in character. However, this scheme was incompatible with special relativity, whose main axiom stated that each observer, depending on his speed, had a local time and a different spatial framework.
Hence, Poisson’s gravity equation had to be reformulated, since mass density is a concept that depends on two fundamental quantities: The first one is mass, which is a quantity whose measurement depends on the coordinate system we choose. and that it has to be replaced by the only preserved and invariant magnitude before the Lorentz transformations , the tetramomentum. The second of these magnitudes is space, which experiences a sensible contraction in those frames that move at high speeds. For this reason, mass density is not an invariant parameter, but its measurement gives different results as the speed of the observer changes.
The problem also arises in the framework of Maxwell’s equations , which also contain gradients and time derivatives, and therefore are not transformable.
Therefore, it is necessary to reformulate the main equations of classical mechanics and electromagnetic theory so that they are valid for all reference systems. To do this, these laws must be expressed tensorly: Their “ingredients” must be made up of elements that remain invariant in the face of Lorentz transformations, such as constants or scalars, or that are transformable according to them (this is the case of tensioners ).
General idea
Image showing texture based on Covalence
In general terms, duality exchanges covariance and countervariance; This is why these concepts are presented together. For purposes of practical matrix calculation, the transposed matrix is relative to two aspects (for example, two sets of simultaneous equations). The case in which the transposed matrix of any square matrix “A” coincides with the inverse matrix, that is, the matrix “A” is an orthogonal matrix, it is a case in which the covariance and the contravariance can be treated as same way. This is of utmost importance in the practical application of turnbuckles.
A cause of greater confusion is this covariance / countervariance duality , which intervenes each time in the discussion of whether a vector or tensor quantity is represented by its components. This causes discussions in the physical and mathematical literature by using apparently opposite conventions.
This is not the differing convention, but when an intrinsic or component description is the primary way of thinking about quantities. As the name suggests, covariant quantities are intended for forward motion or transformations, whereas countervariant quantities are transformed backward. So it depends on whether one is using any fixed fund — in fact, that changes the point of view.
Covariance principle
The covariance principle can be summarized by saying that the laws of physics are the same in all frames of reference. A statement that takes us back to the relativity principle of the special theory:
All the laws of physics are exactly the same for each observer in each frame of reference that is at rest or moving with uniform relative speed. This means that there is no experiment that can be performed within a frame of reference that reveals whether it is at rest or moving at a uniform speed.
The covariance principle is thus a generalization of the relativity principle: while this is limited to inertial frames of reference, the covariance principle says that the laws of physics are the same in any frame of reference, regardless of how it is moving relative to another. It is in this sense that the theories of relativity are distinguished in special and general: whereas special theory applies when certain special circumstances occur (whenever we are dealing with inertial frames of reference), general theory lacks this restriction.
The other basic principle of general theory is the equivalence principle, which says that the effects due to acceleration and those due to gravity are indistinguishable. It may seem like a hollow statement, because we already knew that gravity causes an acceleration in Newtonian physics and the relationship was already known to Newton himself.
News that Einstein introduces
In physics of Newton acceleration and gravity they are treated as two separate phenomena and the relationship between them as a coincidence. But the equivalence principle states that there is essentially no difference between the two effects: we cannot distinguish between them.
Formulation
The general covariance principle states that the fundamental laws or equations of physics must have the same form for any observer regardless of the state of motion of the observer. The objectivity of the material world requires that the measurements made by different observers be relatable by means of fixed transformation laws:
Mathematically the principle of covariance implies that the laws of physics must be tensor laws in which the magnitudes measured by different observers are relatable according to the coordinate transformation of each observer.
Physically, the principle of covariance depends on the fact that for various coordinate reference systems there is no physical procedure to distinguish between them. Influenced by the equivalence principle and other observations, Einstein and others came to theorize that it was possible to construct a theory where all equations could be written in a general enough form to have the same form in any coordinate system.
Application example
An example of the requirements of the covariance principle is the relativistic equivalent of Newton’s second law that is written for any coordinate system xi, in terms of proper time (τ), the Christoffel symbols (Γ) of the coordinate system and the components of the quadriforce (F), as shown in the image.
Thus the apparent distinction between inertial and non-inertial systems of Newtonian mechanics was illusory and disappears in general relativity, since these are nothing more than systems in which the Christoffel symbols that appear in the previous expression are annulled, and therefore, inertial systems are only a particular case of a reference system, but not a privileged or in any way prominent type of reference system, once laws are formulated in the appropriate covariant form.
This essential equivalence between acceleration and gravity is usually illustrated by the elevator mental experiment:
Assuming that a person is in a windowless elevator shaft, and that the shaft is deposited on the static Earth’s surface , but it is not known. That person begins to do physical experiments to know what to do in such a situation. What you should do is measure the objects that you leave free at shoulder height and that move with uniformly accelerated movement towards the ground (it is known that it is the ground because there is a force that pushes you towards that surface and, therefore, it is called ground) with a constant acceleration of 9.8 m / s.
Now, assuming that again, without that person knowing anything, the elevator car moves through interstellar space (and is therefore not significantly influenced by any gravitational field ) with a uniform acceleration of 9.8 m / s in the direction perpendicular to what that person previously called the floor and the sense “from floor to ceiling”. Again, his experiments would lead him to exactly the same results under these circumstances. That is, the practical effects of acceleration and gravity are identical and you cannot distinguish one situation from another.
Informal use
In common use of physics, the adjective covariant can be used informally as a synonym for invariant (or equivalent), in mathematical terms. For example, the Schrödinger Equation does not keep its written form under the coordinate transformations of special relativity; thus one can say that it is non-covariant. In contrast, the Klein-Gordon Equation and the Dirac Equationthey take the same form in any coordinate framework of special relativity: thus, one can say that these equations are covariant or more formally, one could actually say that the Klein-Gordon and Dirac equations are invariant, that the equation of Schrödinger is not, but this is not the dominant usage. It should also be noted that neither of the two equations (Klein-Gordon and Dirac) are invariant before transformations of general relativity (neither in the covariant sense), and in formal use, it should be indicated that the invariance is with respect to the special relativity.
Similarly, informal usage is sometimes seen with respect to quantities such as mass and time in general relativity: mass is technically a component of the four-momentum or the energy-momentum tensor, but one may occasionally refer to the covariant mass. , which means that it is the length of the four-vector moment.
