### 物理代写|宇宙学代写cosmology代考|Reference frames and relativity

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## 物理代写|宇宙学代写cosmology代考|Reference frames and relativity

Let us now pass on to the notion of a reference frame (RF). By definition, a reference frame is an imaginary, massless, in general, arbitrarily (but smoothly) deformable body (the reference body), existing in a certain region of space-time and equipped at each point with perfect rulers, allowing for length measurements, and perfect clocks, allowing for time measurements.

Thus, unlike coordinate systems, which notion is purely mathematical, a RF is a physical notion, quite necessary for connecting the theory with measurements. As is well known, real prototypes of perfect clocks are atomic clocks, while real prototypes of perfect rulers are the length standards attached to the wavelengths of certain spectral lines.

The General Relativity Principle asserts the equivalence of all RFs in formulations of the laws of nature. It is a consistent physical principle, from which follows the absence of preferred RFs in the nature. Let us note that even theories in which there are preferred RFs admit a generally covariant formulation of their equations. An evident example is SR, where inertial RFs (IRFs) play a distinguished role.

## 物理代写|宇宙学代写cosmology代考|Reference frames and chronometric invariants

In principle, it is possible to use coordinate systems quite independently from RFs. For example, when describing the life of a city in its natural RF connected with its streets and houses, nothing and nobody forbids one from using coordinates connected with a certain system of regularly moving trams, or with the shadows of clouds floating in the sky.

In practice, however, it is much simpler and more convenient to attach the coordinates to a certain RF, assuming that the reference body is covered by a three-dimensional coordinate net $x^{i}$ while the time coordinate $x^{0}$ changes along the world lines of fixed points of the reference body (they are naturally called time lines). It is said in this case that the coordinate system belongs to a reference frame. The world lines of particles at rest in a given RF are described by the equation $x^{i}=$ const (are $x^{0}$-independent). If one makes a coordinate transformation (2.24), then the condition that the new coordinates $y^{\mu}$ belong to the same RF as $x^{\mu}$ is that the new spatial coordinates $y^{i}$ are $x^{0}$-independent.
Thus the transformations
\begin{aligned} y^{i} &=y^{i}\left(x^{1}, x^{2}, x^{3}\right) \ y^{0} &=y^{0}\left(x^{0}, x^{1}, x^{2}, x^{3}\right) \end{aligned}
are the most general transformations between coordinate systems that belong to the same RF. Equation (2.26) describes threedimensional coordinate transformations which change the spatial coordinate net. Equation $(2.27)$ describes arbitrary chronometric transformations that change the course of arbitrary (coordinate) clocks as well as their synchronization from one spatial point to another.

## 物理代写|宇宙学代写cosmology代考|Covariance and relativity

As we have seen, the general covariance principle is nothing else but an opportunity to use any coordinates for describing the space-time and events in it.

On the contrary, various forms of the relativity principle are important postulates of physical theories. Thus, the Galilean relativity principle that acts in Newtonian mechanics asserts that the laws of nature are independent of the choice of an IRF in Newtonian absolute space. The special relativity principle asserts quite the same but in Minkowski space, where IRFs are related by Lorentz transformations. In both cases, the theory contains a class of privileged RFs, and the transformations in question occur inside this class.

The Cartesian coordinates in Newtonian (Euclidean) space and the Minkowski coordinates in Minkowski space are privileged in two respects. They are distinguished mathematically since their coordinate lines coincide with the orbits of isometry groups of the corresponding spaces (though, this choice is not unique: spherical and cylindrical coordinates also possess this property). On the other hand, they are distinguished physically by the convenience of describing IRFs with their aid. It is well known, however, that, in many problems of classical mechanics, curvilinear coordinates are used, and physical phenomena in various noninertial RFs are studied. In $\mathrm{SR}$, there is also a necessity to study accelerated RFs and to use coordinates belonging to them (e.g., uniformly accelerated frames and Rindler coordinates [467]). In such cases, quite helpful are the mathematical methods elaborated in GR and related to generally covariant formulations of all equations.

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