Various Problems of General Relativity with and without a Gravitational Field

Abstract

A consistent principle of equivalence is formulated. An example of the necessity of strict observance of Einstein’s rule for the transformation of tensor coordinates is given. The problem of a homogeneous stationary gravitational field is being solved. All metrics with non-zero curvature. A metric for the space of constant curvature, different from the De Sitter space, is found. A stationary space with a cosmological redshift has been discovered, which creates the effect of accelerated expansion. It is shown that the plane-parallel motion of gas in a force field obeys the law of compression-expansion of the shock adiabat.

1. Introduction

Einstein’s theory of relativity is based on strict definitions and postulates. Neglecting them can lead to misunderstandings or even catastrophic errors.

In Einstein’s special theory of relativity, the speed of light is independent of the frame of reference. This makes it impossible to describe the movement of bodies using coordinates and a single time, independent of the speed of these bodies. In this case, the spatial and temporal scales change when moving from one uniformly moving (inertial) frame of reference to another.

Already in the special theory of relativity, we see that this is a law of nature that describes the ratio of the scales of space and time, but not “weighty bodies.” Time is on a par with space here. Hermann Minkowski took advantage of this and introduced time t as the Euclidean three-dimensional space with coordinates {x, y, z} as an additional, fourth dimension. In Minkowski’s [1, 2] space, the interval between events

$$s^2={\left(x_1-x_0\right)}^2+{\left(y_1-y_0\right)}^2+{\left(z_1-z_0\right)}^2-{с}^2{\left(t_1-t_0\right)}^2$$E1

is a generalization of the Pythagorean theorem. The value of s does not change if the coordinates are changed according to the Lorentz transformation recipe. Here, for the first time, space is considered an independent physical object. Minkowski owns the words:

“From now on, time itself and space itself become an empty fiction, and only their unity preserves the chance for reality.”

The next step was taken by Einstein in 1907 [3]. The connection between time and acceleration in accelerated reference systems is found and the principle of equivalence of accelerated reference systems and systems with a gravitational field is established. For several years, Einstein accumulated separate results on the relativistic properties of gravity. Perhaps the most important result was the conclusion that the presence of a gravitational field changes the speed of light, i.e. the speed of light is not a constant depends on the gravitational potential [4]. This makes it impossible to describe gravity in Minkowski’s space. The natural way out of this situation is the transition to the curvilinear Riemann space. Fortunately, by that time, the creation of a mathematical apparatus for calculating the parameters of such a space, tensor analysis (absolute differential calculus), had been completed. Einstein resorts to the help of his friend, the mathematician Grossman, and they publish the basic principles for constructing a relativistic theory of gravity (general relativity). It also outlines the basics of tensor analysis.

Here it is already necessary to quote the words of Einstein [5], formulating the subject of the theory of relativity as the study of the physical properties of space:

“When it is said here about the ether, then, of course, it does not mean the corporeal ether of the mechanical wave theory, which obeys the laws of Newtonian mechanics and whose individual points are assigned speed. This theoretical notion with the creation of the special theory of relativity, in my opinion, finally disappeared from the scene. On the contrary, we are talking about those physically conceivable real things that, along with weighty matter, consisting of electrical elementary particles, play a role in the structure of the causal relationships of physics. Therefore, instead of the word “ether“ one can just as well say “the physical properties of space“. In this case, of course, it would be possible to express the opinion that all objects of physics fall under this concept, since, according to consistent field theory, weighty matter or its constituent elementary particles should also be considered a special kind of “field”, or special “states of space”. However, one has to admit that in the present state of physics, such an idea is premature, since until now all the efforts of theoretical physicists directed towards this goal have failed. Thus, we are now effectively forced to distinguish between “matter” and “fields”, although we can hope that future generations will overcome this dualistic view and replace it with a single concept, as the field theory of our days has tried in vain to do.”

Modern evidence confirms that space is indeed not flat. The most compelling evidence for this is the experiment to detect the delay of an electromagnetic signal in a gravitational field, discovered by the group of Irwin Shapiro [6]. Subsequent more accurate observations of nearby relativistic star systems proved that space is not flat - in the presence of a gravitational field, the Pythagorean theorem does not work.

It should be added that the well-known unshieldability of the gravitational field makes the physical space so unusual that attempts to attribute properties of ordinary matter or (quantum) fields to gravity do not look convincing.

General relativity can be compared to a capricious lady trying on outfits and trying to pull on fashionable outfits from someone else’s shoulder. I will allow myself to express surprise at the consideration of quantum fields of particles endowed with the ability to interact with matter. This contradicts the observed transparency of matter for the gravitational field. The shading of the gravitational field by the Moon would lead to catastrophes on Earth. In addition, we would observe daily morning and evening tsunamis if such shading took place. In general, attributing the properties of ordinary matter to the gravitational field does not seem reasonable. Speaking of the ether as a degree of the materiality of space, Einstein says [5]:

“According to the general theory of relativity, space is inconceivable without ether; indeed, in such a space, not only would the propagation of light be impossible, but scales and clocks could not exist, and there would be no space-time distances in the physical sense of the word. However, this ether cannot be imagined as consisting of parts traceable in time; only weighty matter has this property; in the same way, the concept of motion cannot be applied to it.”

It is noteworthy that almost immediately after Einstein’s death, in 1958, David Finkelstein identified the Schwarzschild surface as an event horizon, “an ideal unidirectional membrane: causal influences can cross it in only one direction.” At the same time, from the point of view of pure mathematics, the Schwarzschild surface does not belong to the original metric space - the metric on this surface and inside it simply does not exist, i.e. is not a solution to the Einstein equation.

It cannot be said that all problems in Einstein’s theory have been solved, and the initial provisions of the theory have been correctly understood and accepted by everyone without exception. The purpose of this chapter is to explain the insufficiently rigorous provisions of the theory and to warn the researcher against errors with the help of non-trivial results [7].

2. Local equivalence principle

In 1907, Einstein considered the equivalence principle [3] as a means to transfer the results obtained for accelerated reference systems to systems with a gravitational field. At the same time, Einstein limited himself to systems with a homogeneous field, which of course is not suitable for arbitrary reference systems. The modern approach to the equivalence principle is distinguished by a variety of interpretations and definitions, for example, [8, 9, 10, 11, 12].

We will talk about the local equivalence of the frames of reference of the Riemann space and the Minkowski space. In the cult book [11], Landau and Lifshitz try to prove the local equivalence of a system with a gravitational field to an inertial system:

“Formula (85.15) under condition (85.16)¹ allows us to prove the above statement about the possibility of such a choice coordinate system for which all ${\Gamma }_{\mathit{kl}}^i$ vanish at any predetermined point (such a system is called locally inertial or locally geodesic, see §87). Indeed, let a given point be chosen as origins and quantities ${\Gamma }_{\mathit{kl}}^i$ have in it initially (in $x^i$ coordinates) values ${\left({\Gamma }_{\mathit{kl}}^i\right)}_0$. We will produce near this point transformation

$$x^{\prime i}=x^i+\frac{1}{2}{\left({\Gamma }_{\mathit{kl}}^i\right)}_0{x}^k{x}^l$$

Then

$${\left(\frac{{\partial }^2{x}^{\prime m}}{\partial x^k\partial x^l}\frac{\partial x^i}{\partial x^{\prime m}}\right)}_0={\left({\Gamma }_{\mathit{kl}}^i\right)}_0$$

and according to (85.15)² all ${\Gamma }_{\mathit{kl}}^i$ vanish.”

Here we see that the authors consider the transition from the first expression to the second by incorrectly replacing small values of the $x^i$ coordinates with differentials, but.

$$x^i=\partial x^i+o\left(X\right).$$E2

The rejection of the finite value $o\left(X\right)$, contrary to popular belief, cannot be perceived as a coordinate transformation. Replacing coordinates with their differentials means replacing geodesics with their tangents at the point $P$.

A more transparent approach is used in Møller’s excellent monograph [13] (§ 9.6. Local pseudo-Cartesian coordinates and local inertial systems). Møller comes from the curvilinear coordinates $X$ to the local (near the origin) pseudo-Euclidean system $\stackrel{\mathrm{ˇ}}{X}$ using the transformation.

$${\check{x}}^i=\frac{\partial {\check{x}}^i}{\partial x^k}{x}^k;$$E3

$$x^i=\frac{\partial x^i}{\partial {\check{x}}^k}{\check{x}}^k,$$E4

here the derivatives are calculated at the origin. We note right away that the inverse transformation (4) is impossible, since transformation (4), as well as transformation (3), is a transformation into the gallium space. After such a transformation, all information about the parameters of the original Riemann space is lost, which cannot be restored.

Let us show that in this case, the equal sign in (3) is inappropriate. Indeed, the equalities must be written in terms of the total differential.

$${\stackrel{\mathrm{ˇ}}{x}}^i=d{\stackrel{\mathrm{ˇ}}{x}}^i+o\left(X\right)\approx d{\stackrel{\mathrm{ˇ}}{x}}^i;$$E5

Thus, we can only speak of an asymptotic transition (but not vice versa!) from curvilinear coordinates to pseudo-Euclidean ones.

$${\stackrel{\mathrm{ˇ}}{x}}^i\to d{\stackrel{\mathrm{ˇ}}{x}}^i.$$E6

As the radius δ of the neighborhood decreases (Figure 1), the geodesics begin to look more and more like straight lines. In other words, a small neighborhood of the pseudo-Riemannian space can be arbitrarily close to the inertial frame (Einstein’s freely falling elevator) but never coincide with it. In this case, as it should be in a finite region, neither the metric tensor nor, depending on it, the Riemann tensor with its convolutions and Christoffel symbols.

Figure 1.

Transformation (5) is a transition from the original space with coordinates $x^i$ to the inertial tangent space with a common point $P$.

Let us illustrate this.

In curvilinear coordinates, the geodesic line will never coincide with the tangent in the vicinity of the tangent point, no matter how small this neighborhood is.

Let us formulate the principle of spatial equivalence of a Riemannian space of a neighborhood of a point P to a pseudo-Euclidean space.

If the point $P$ belongs to an open neighborhood $Q$ of a Riemannian space $X$ and has a derivative at the point $P$, then this space is locally equivalent to the Minkowski tangent space $\check{X}$ at the point P if, for any positive ε, there is a neighborhood radius δ such that for any $x^i\in Q$ will be carried out $\epsilon >\left|x^i-d{x}^i\right|$.

3. Coordinate system and reference system

It is necessary to distinguish between the concepts of reference system and coordinate system. The reference system can be specified in various coordinate systems.

When passing from the $X$ coordinates to the $X^{\prime }$ coordinates [11], the contravariant vector $A^{\mu }$ transforms as a differential:

$$A^i=\frac{\partial x^i}{\partial x^{\prime k}}{A^{\prime }}^k.$$E7

The transformation coefficient is called the Jacobi matrix, and the transformation is written

$$A^i={\mathbf{J}}_k^i{A^{\prime }}^k.$$E8

where the Jacobi matrix ${\mathbf{J}}_k^i=\left[\frac{\partial x^i}{\partial x^{\prime k}}\right]$.

The transformation of other contra-covariant tensors of arbitrary order is defined similarly to [14] as a sequence of linear transformations of the form (7). For example, the transformation of the covariant tensor of the second-rank

$$A_{\mathit{ik}}={\mathbf{J}}_i^l{\mathbf{J}}_k^m{A}_{\mathit{lm}}^{\prime }.$$E9

It is now clear that such transformations are not arbitrary. They must leave the frame of reference unchanged. General coordinate transformations transform a coordinate system into any other one with the same number of dimensions, including without preserving the reference system and its invariants.

$$x_i=x_i\left(x_0^{\prime },x_1^{\prime },x_2^{\prime },x_3^{\prime }\right).$$E10

With such a transformation, the reference system can change, therefore, if we want to explore a certain reference system, we must limit ourselves to linear coordinate transformations (7) or a sequence of such transformations.

However, transformations of the form (7), even if a transformation matrix exists, do not ensure the preservation of the reference system.

Example. A non-linear transformation that cannot be reduced to a linear one.

Converting a Cartesian coordinate system to a spherical one:

$$\left\{\begin{array}{c}r=\sqrt{x^2+y^2+z^2},\\ \theta =\mathit{arctan}\frac{\sqrt{x^2+y^2}}{z},\\ \phi =\mathit{arctan}\frac{y}{x}.\end{array}\right.$$E11

When trying to obtain an equivalent transformation according to (7), we obtain:

$$r=\frac{\partial r}{\partial x}x+\frac{\partial r}{\partial y}y+\frac{\partial r}{\partial z}z=\sqrt{x^2+y^2+z^2};$$

$$\theta =\frac{\partial \theta }{\partial x}x+\frac{\partial \theta }{\partial y}y+\frac{\partial \theta }{\partial z}z=\frac{x^2+y^2}{z^2}\left(\frac{x^2+y^2}{z\sqrt{x^2+y^2}}-\frac{\sqrt{x^2+y^2}}{z^2}z\right)=0;$$E12

$$\phi =\frac{\partial \phi }{\partial x}x+\frac{\partial \phi }{\partial y}y+\frac{\partial \phi }{\partial z}z=\frac{x^2}{x^2+y^2}\left(-\frac{y}{x^2}x+\frac{1}{x}y\right)=0.$$

Such a transformation is, of course, not equivalent to transformation (11). In this example, instead of the expected spherical coordinates, we got a one-dimensional space.

This unexpected result is a good illustration of the danger that lies in waiting for the researcher in case of ill-conceived application of arbitrary transformations of the form (10), which can lead to erroneous results. The phrase “arbitrary transformation of coordinates” should also not be abused, if it does not follow from the context that this is a transformation of coordinates of the reference system (7).

At the same time, one should not forget about the restrictions on relativistic frames of reference. For example, the speed limit should not exceed the speed of light ([13] § 8.7. General accelerated reference systems. The most general allowable coordinate transformations).

4. Some solutions of the Einstein equation

Einstein’s equation has proven itself well for solving problems in small and moderate gravitational fields. Let us consider some non-trivial solutions of this equation.

4.1 Uniform fields

The equations of motion in the general theory of relativity [11] allow one to calculate the covariant or contravariant acceleration vector.

$$\frac{d^2{x}_i}{d{s}^2}-{\Gamma }_{i,\mathit{kl}}\frac{d{x}^k}{\mathit{ds}}\frac{d{x}^l}{\mathit{ds}}=0;\frac{d^2{x}^i}{d{s}^2}+{\Gamma }_{\mathit{kl}}^i\frac{d{x}^k}{\mathit{ds}}\frac{d{x}^l}{\mathit{ds}}=0.$$E13

For a fixed point, the acceleration vectors ${\alpha }_i$ or ${\alpha }^i$ are calculated by the formulas.

$${\alpha }_i=\frac{d^2{x}_i}{d{t}^2}=c^2{\Gamma }_{i,00};{\alpha }^i=\frac{d^2{x}^i}{d{t}^2}=-c^2{\Gamma }_{00}^i.$$E14

In this work, we will use the scalar acceleration value.

$$\alpha =\sqrt{{\alpha }_i{\alpha }^i}.$$E15

The simplest metric.

$$d{s}^2=\left(1+\frac{\mathit{\alpha x}}{c^2}\right)d{t}^2-\left(d{x}^2+d{y}^2+d{z}^2\right)$$E16

has a constant acceleration $\alpha ={\mathrm{\alpha }}_{\mathrm{i}}={\alpha }^i$ in the region $\frac{\mathit{\alpha x}}{c^2}>-1$. The rest of the space does not have any physical meaning. Note that the scalar curvature of the metric is negative

$$R=-\frac{2a^2}{{\left(2\mathit{ax}+c^2\right)}^2}.$$E17

We use a general approach to search for homogeneous accelerated frames of reference. Consider the gravitational field in the metric of the form.

$$d{s}^2=s\left(x\right)d{t}^2-p\left(x\right)\left(d{x}^2+d{y}^2+d{z}^2\right)$$E18

and one-dimensional metric.

$$d{s}^2=s\left(x\right)d{t}^2-p\left(x\right)d{x}^2-d{y}^2-d{z}^2.$$E19

Calculations of parameters of metrics are carried out. Metric (18) has a curvature tensor and its convolutions are much more complicated than metric (19). We did not consider it necessary to present these data in full. Calculations of vectors (14) give the same result for both metrics, and their scalar acceleration is related to the components of the fundamental tensor:

$${\alpha }^2=\frac{s^{\prime }{\left(x\right)}^2}{4p\left(x\right)}$$E20

or

$$p\left(x\right)=\frac{s^{\prime }{\left(x\right)}^2}{4{\alpha }^2}.$$E21

This makes it possible to write homogeneous metrics.

$$d{s}^2=s\left(x\right)d{t}^2-\frac{s^{\prime }{\left(x\right)}^2}{4a^2}\left(d{x}^2+d{y}^2+d{z}^2\right)$$E22

and

$$d{s}^2=s\left(x\right)d{t}^2-\frac{s^{\prime }{\left(x\right)}^2}{4a^2}d{x}^2-d{y}^2-d{z}^2$$E23

where $s\left(x\right)>0$ is an arbitrary function with a continuous derivative.

We see that the resulting metric has a non-zero scalar curvature.

$$R=-\frac{2a^2}{s{\left(x\right)}^2}.$$E24

System (22) also has a negative scalar curvature. However, we did not present it because of the cumbersome expression. This suggests that in a flat space, there is no uniformly accelerated metric at all.

A good example of a uniformly accelerated reference system is one with the metric:

$$d{s}^2=e^{2\mathit{ax}}d{t}^2-e^{4\mathit{ax}}d{x}^2-d{y}^2-d{z}^2.$$E25

With scalar curvature $$R=-2a^2{e}^{4\mathit{ax}}$$ .

4.2 Universe with constant curvature

At present, the cosmological hypothesis of the expanding Universe is being successfully used. The first attempts to describe a homogeneous stationary Universe on the basis of the general theory of relativity were made by A. Einstein and W. de Sitter. However, soon A.A. Friedman published an article [15], in which he proposed a relativistic model of a homogeneous non-stationary Universe. Somewhat later, E. Hubble discovered that the redshift of the optical lines of galaxies increases with increasing distance to them. This made it possible to interpret the redshift as a Doppler one, related to the speed of the galaxy’s receding. Thus, the hypothesis of a homogeneous expanding universe has now become practically indisputable.

We are primarily interested in homogeneous spaces. Not only spaces with a uniform distribution of matter and constant curvature, but also spaces with a constant speed of light.

A metric with a constant speed of light belongs to exceptional metrics. Metrics with a constant speed of light were first considered by Einstein and Fokker [16]. The authors of the article [16] used a similar metric to discover its connection with the Nordström theory, which, as the authors showed, cannot describe the gravitational field, since the speed of light in this theory is constant, which, as the authors of the work showed, is incompatible with the general theory of relativity.

Metrics with a constant speed of light have the form:

$$d{s}^2=f^2\left(c^2{t}^2-d{x}^2-d{y}^2-d{z}^2\right),$$E26

where $f$ is a function of coordinates. For each metric of this kind, one can construct the Einstein equation by calculating the Einstein tensor for the given function $f$. Consequently, all metrics of this kind are related to solutions of the Einstein equation.

It is easy to see that in the space given by the metric (26), the speed of light is isotropic and constant. Indeed, the wavefront equation $\mathit{ds}=0$ for the metric (26) coincides with the wavefront equation in the Minkowski space.

Let us require that the metric of the required space be homogeneous not only in the speed of light but also be homogeneous in curvature, i.e. had a constant scalar curvature $R$.

The Einstein equation for such metrics has the form.

$$G_b^a=-\frac{1}{4}R{\delta }_b^a.$$E27

Einstein’s equation is a system of differential equations of ten unknown functions of four space-time coordinates.

It is essential that time is orthogonal to spatial coordinates. This allows you to separate the variables into spatial and temporal components. Then, the complete solution can be represented as two independent solutions:

$$d{s}_X^2=A\left(X\right)c^2{t}^2-B\left(X\right)d{x}^2-C\left(X\right)d{y}^2-D\left(X\right)d{z}^2,$$E28

and

$$d{s}_t^2=A\left(t\right)c^2{t}^2-B\left(t\right)d{x}^2-C\left(t\right)d{y}^2-D\left(t\right)d{z}^2.$$E29

The solution of the form (28), which depends on the spatial coordinates, is the well-known metric of the de Sitter space [17]. The metric tensor of this space, expressed in terms of scalar curvature [18], looks like this in spherical coordinates:

$$d{\left(s_X\right)}^2=\left(1+\frac{R{r}^2}{12}\right)c^{2}d{t}^2-\frac{d{r}^2}{1+\frac{R{r}^2}{12}}-r^2\left(d{\vartheta }^2+{\mathit{sin}}^2\mathit{\vartheta d}{\phi }^2\right).$$E30

Substituting the metric (29), in spherical coordinates, into Eq. (27), after simple transformations, we obtain the second solution of the equation.

$$d{\left(s_t\right)}^2=\frac{-12}{R{c}^2{t}^2}\left[c^{2}d{t}^2-d{r}^2-r^2\left(d{\vartheta }^2+{\mathit{sin}}^2\mathit{\vartheta d}{\phi }^2\right)\right].$$E31

Let us immediately note that from one time value to another, a space with a given metric transforms conformally, i.e. all scales of space-time change in the same way. Then, if we use light standards to locally measure linear dimensions and time, we will not be able to detect any space changes. This means the real homogeneity of space-time.

The choice of the multiplier f(t) in the metric (26) sets the proper time scale [14, 19] as a function of time at the origin. In the case of the metric.

$$d{\left(s_t\right)}^2=\frac{1}{{\left(\mathit{Ht}\right)}^2}\left[c^{2}d{t}^2-d{r}^2-r^2\left(d{\vartheta }^2+{\mathit{sin}}^2\mathit{\vartheta d}{\phi }^2\right)\right].$$E32

is the dependence of the time τ at the origin of coordinates on the time of the spatial region located at a finite distance from the origin.

$$\mathit{d\tau }=\frac{1}{\mathit{Ht}}\mathit{dt},$$E33

here $H$ is a constant, $t$ is the time it takes the light signal to reach the origin. If we have a set of identical generators with a unit period ${\mathrm{\lambda }}_0$, then the frequency signal of the generator located at a distance $r=\mathit{ct}$ is a signal from the past and the point of view of the observer at the origin, the period of this generator [20].

$$\lambda -1=\mathit{Ht}{\lambda }_0.$$E34

Thus, we have a space with a redshift. The offset is given by Eq. (34). However, this law does not coincide with the Hubble law, which states that the dimensionless speed of the removal of galaxies is proportional to the distance to them³.

$$v=\mathit{Ht}.$$E35

Usually, deviations from Hubble’s law are shown using a graphical comparison with this law. The signal period of the receding signal generator is related to the dimensionless velocity.

$$\mathrm{\lambda }=\frac{1+v}{\sqrt{1-v^2}}{\mathrm{\lambda }}_0.$$E36

This makes it possible to compare the redshift laws (34) and (35). If we substitute the value λ into (34), we get.

$$\frac{1+v}{\sqrt{1-v^2}}-1=\mathit{Ht}.$$E37

Of course, the galaxies in the new space remain motionless, and v is a parameter of this space (Figure 2).

Figure 2.

Redshift in the universe with metric (32). In a stationary universe, stationary galaxies are assigned a speed v according to their redshift for comparison with Hubble’s law. The horizontal axis shows the redshift in units of velocity. Vertical distance to the galaxy. The thin green line is Hubble’s law. The thick red line is the distance to a stationary galaxy in a stationary universe.

The figure shows the dependence of the redshift on the “velocity” of the recession of galaxies as if the redshift were Doppler. Of course, the galaxies in the universe (34) are stationary, and this picture was needed for comparison with the Hubble law.

Thus, in the stationary Universe, a redshift is observed, equivalent to the observed redshift with the effect of “accelerated expansion”.

It remains to add that the density of such a universe⁴.

$$\epsilon =\frac{3H^2{c}^2}{8\mathit{\pi G}}$$E38

coincides with the Friedmann critical energy density. The motion of test bodies in this Universe is determined by the equation of geodesics with Christoffel symbols.

$${\Gamma }_{10}^1={\Gamma }_{01}^1={\Gamma }_{20}^2={\Gamma }_{02}^2={\Gamma }_{03}^3={\Gamma }_{30}^3={\Gamma }_{00}^0=-\frac{1}{T};$$

$${\Gamma }_{11}^0=-\frac{1}{c^{2}T};{\Gamma }_{22}^0=-\frac{r^2}{c^{2}T};{\Gamma }_{33}^0=-\frac{{\mathit{sin}}^2\left(\theta \right)r^2}{c^{2}T}.$$E39

These 10 quantities describe the behavior of a test particle at a distance $\mathit{cT}$ from the observer. Who knows, perhaps Christoffel’s symbols will make it possible to identify the new Universe with ours.

However, the dependence of the Christoffel symbols on time does not allow us to consider such a Universe as ideally homogeneous.

4.3 Homogeneous universe with cosmological redshift

Meanwhile, the solutions of Einstein’s equation as applied to the gravitational field show the presence of a redshift, which is possibly a gravitational redshift. However, solutions of the Einstein equation without a gravitational field were demonstrated above, which have a redshift within the parameters according to the Hubble law, but at the same time do not serve to describe the expanding space:

$$d{s}^2=\frac{1}{{\left(\mathit{Ht}\right)}^2}\left[d{t}^2-d{r}^2-r^2\left(d{\vartheta }^2+{\mathit{sin}}^2\mathit{\vartheta d}{\phi }^2\right)\right].$$E40

A natural question arises. Are there truly stationary non-trivial homogeneous universes with Christoffel symbols that are independent of time?

Consider a metric of the form:

$$d{s}^2=a{\left(t\right)}^2\left[d{x}_0^2-d{x}_1^2-r^2\left(d{x}_2^2+{\mathit{sin}}^2\mathit{\vartheta d}{x}_3^2\right)\right].$$E41

The time-dependent Christoffel symbols of this metric are:

$${\Gamma }_{10}^1={\Gamma }_{20}^2={\Gamma }_{30}^3={\Gamma }_{01}^1={\Gamma }_{02}^2={\Gamma }_{03}^3={\Gamma }_{11}^0={\Gamma }_{00}^0=\frac{\dot{a}\left(t\right)}{a\left(t\right)};$$

$${\Gamma }_{22}^0=\frac{r^2\dot{a}\left(t\right)}{a\left(t\right)};{\Gamma }_{33}^0=\frac{\dot{r^2{\mathit{sin}}^2\vartheta }\dot{a}\left(t\right)}{a\left(t\right)}.$$E42

The only solution to the equation.

$$\frac{\dot{a}\left(t\right)}{a\left(t\right)}=-H=\mathit{\text{const}}$$E43

is an.

$$a\left(t\right)=e^{-\mathit{Ht}}.$$E44

In contrast, the metric with Christoffel symbols independent of time is:

$$d{s}^2=e^{-2\mathit{Ht}}\left[d{x}_0^2-d{x}_1^2-r^2\left(d{x}_2^2+{\mathit{sin}}^2\mathit{\vartheta d}{x}_3^2\right)\right].$$E45

The equations of motion in the universe do not change with time and do not differ from the equations of the motion of flat space for motionless bodies. Therefore, we can conclude that there is no gravitational field in such a universe, even the Riemann tensor and its convolutions are nonzero.

Metric (2) is described by the Einstein equation:

$$G_{\mathit{\mu \nu }}=\mathit{diag}\left(3H^2,-H^2,-H^2{r}^2,-H^2{r}^2{\mathit{sin}}^2\vartheta \right)$$E46

We associate the constant $H$ with the Hubble constant. Then the right side of the equation can be expressed in terms of the (critical) density of the Friedmann universe ${\rho }_0=\frac{3H^2}{8\mathit{\pi G}}$:

$$G_{\mathit{\mu \nu }}=8\mathit{\pi G}\mathit{diag}\left({\rho }_0,-\frac{1}{3}{\rho }_0,-\frac{1}{3}{\rho }_0{r}^2,-\frac{1}{3}{\rho }_0{r}^2{\mathit{sin}}^2\vartheta \right).$$E47

The equations of motion in the universe do not change with time and do not differ from the equations of the motion of flat space for motionless bodies. Therefore, we can conclude that there is no gravitational field in such a universe, even though the Riemann tensor and its convolutions are nonzero.

Metric (45) is described by the Einstein equation:

$$G_{\mathit{\mu \nu }}=\mathit{diag}\left(3H^2,-H^2,-H^2{r}^2,-H^2{r}^2{\mathit{sin}}^2\vartheta \right)$$E48

We associate the constant $H$ with the Hubble constant. Then the right side of the equation can be expressed in terms of the (critical) density of the Friedmann universe ${\rho }_0=\frac{3H^2}{8\mathit{\pi G}}$:

$$G_{\mathit{\mu \nu }}=8\mathit{\pi G}\mathit{diag}\left({\rho }_0,-\frac{1}{3}{\rho }_0,-\frac{1}{3}{\rho }_0{r}^2,-\frac{1}{3}{\rho }_0{r}^2{\mathit{sin}}^2\vartheta \right).$$E49

Eq. (48) does not depend on the choice of origin of the spatial coordinates. The proper time at the origin is related to the time $t$, which is counted from the initial time $t=0$:

$$\mathit{d\tau }=e^{-\mathit{Ht}}\mathit{dt}\approx \frac{1}{\mathit{Ht}}\mathit{dt}.$$E50

here $t$ is the time for which the light signal reaches the origin. If we have a set of identical generators with a unit period ${\lambda }_0$, and the frequency signal of the generator located at the distance $r=\mathit{ct}$ is a signal from the past and the point of view of the observer at the origin, it is the period of this generator:

$$\frac{\lambda }{{\lambda }_0}=e^{\mathit{Ht}}$$E51

Thus, the space under consideration is redshifted. The offset is given by Eq. (51). However, this law does not coincide with Hubble’s law, which states that the speed of the recession of galaxies is proportional to the distance to them.

$$\mathit{cz}=v=\mathit{Hr}.$$E52

In our case, the galaxies are motionless. However, we can introduce a fictitious speed and, accordingly, a redshift. For this, we use the relativistic Doppler shift formula:

$$\frac{\lambda }{{\lambda }_0}=e^{\mathit{Ht}}=\sqrt{\frac{1+v}{1-v}},$$E53

or

$$r=\mathit{ct}=\frac{c}{2H}\mathit{ln}\frac{1+v}{1-v}.$$E54

This allows for the comparison of the redshift laws.

Figure 3 shows the dependence of the redshift on the recession “velocity” of the galaxies as if it is a Doppler redshift. The fact is, of course, that the galaxies in the universe are stationary and this picture (54) was needed to show a comparison with Hubble’s law.

Figure 3.

A redshift in the universe shown through a metric (45). In a stationary universe, stationary galaxies are assigned a speed “v” according to their redshift properties for comparison with Hubble’s law. The horizontal axis shows the redshift in units of velocity. The vertical is the distance to the galaxy. The thin green line is Hubble’s law. The thick red line is the distance to a stationary galaxy in a stationary universe.

5. Reducing the entropy of gas flows in a gravitational field

A while ago, an amazing result was published [21]. The authors demonstrated a magnet-controlled superconducting diode in a multi-layer structure [Nb/V/Ta]n having no inversion center. During the diode transition from the superconductivity to the normal state, the curve of the current-dependent non-reciprocal resistance was observed when measuring the current direction. Notably, it is thought that the difference in the critical current is due to the magnetic field-induced anisotropy caused by the inversion of space & time. Theoretically, such a superconducting diode rectify some thermal current noises; in so doing it can play a role of the Maxwell’s demon, when converting the thermal energy into the current.

Another well–known example of non-reciprocal devices is found in electromagnetic fields. The reciprocity theorem was proved for the electromagnetic radiation in the absence of a magnetic field [22]. In the practice, the non-reciprocal devices are implemented in the form of microwave or optical devices with one-way bandwidth [23, 24]. Those devices are based on the Faraday Effect, i.e. on rotation of a polarization plane of electromagnetic waves in a medium located in a magnetic field.

The Russian physicist Nikolai Nikolayevich Pirogov published a paper back in 1887 in which he considered gas in a gravitational field based on the methods of statistical physics [25]. Here he proves that the temperature of an ideal gas column in a gravitational field and in an equilibrium state does not depend on altitude and derives the now well-known exponential barometric dependence of altitude pressure.

Pirogov did not limit himself to the stationary case and considered gas flows in a potential field. At the same time, he showed that.

“the well-known theory of the Carnot cycle is correct only under the assumption that the working substance is not affected by external or long-range forces, or, which is all the same if the action of these forces can be neglected.”

Further, Pirogov gives an example of a heat engine operating due to the expansion of gas in a gravitational field. The operation of the machine was ensured by the supply of heat from the surrounding space. At the same time, the author did not believe that such a machine violates the second law of thermodynamics because this law is applicable only to finite regions of space, while the gravitational field is infinite.

First of all, we are interested in the law of expansion/compression of gas in pipes of constant cross-section. Unlike the well-known (in hydrodynamics) adiabatic gas flow in pipes of variable cross section (for example, in a Laval nozzle), gas expansion under the action of some external forces cannot be described using the usual adiabat (Poisson’s adiabat). Fortunately, the problem is solved simply [26] based on the laws of conservation of mass, energy, and momentum. Let us choose two sections in the vertical gas column; now the ratio of pressure and density is described by the relation:

$$\frac{V_2}{V_1}=\frac{\left(\gamma +1\right)p_2+\left(\gamma -1\right)p_1}{\left(\gamma -1\right)p_1+\left(\gamma +1\right)p_2},$$E55

where adiabatic index is $\gamma =C_p/C_v$; $C_p$ is heat capacity at constant pressure; and $C_v$ is constant heat capacity.

A gas flow with a shock wave or in a flow with compression along a potential field is accompanied by an increase in entropy, which can be estimated using the difference in entropy in the final and initial stationary flow [26]:

$$s_2-s_1=\frac{1}{12T_1}{\left(\frac{{\partial }^{2}V}{\partial p_1^2}\right)}_s{\left(p_2-p_1\right)}^3.$$E56

It is usually stated that the difference between the entropies of the adiabatic compression of a gas in a plane-parallel flow cannot be negative. This should follow from the second law of thermodynamics. However, N.N. Pirogov drew attention [25] to the fact that the gas flow in a potential field can, depending on the direction of the flow, both compress and expand the gas. But then, according to Eq. (56), we can conclude that the entropy of the expanding gas in the potential field decreases.

The main results of the theory of shock waves [26] allow us to compare two adiabatic processes, the shock adiabat and the isentropic Poisson adiabat.

So the dependence of temperature on pressure in the Poisson adiabat in a polytropic gas.

$$\frac{T_P}{T_0}={\left(\frac{p}{p_0}\right)}^{\frac{\gamma -1}{\gamma }}.$$E57

where γ is the adiabatic exponent. Flow temperature in the shock adiabat.

$$\frac{T_s}{T_0}=\frac{p}{p_0}\left[\frac{\left(\gamma +1\right)p_0+\left(\gamma -1\right)p}{\left(\gamma -1\right)p_0+\left(\gamma +1\right)p}\right].$$E58

In Figure 4, we see that when expanding in the shock adiabat, the gas cools faster, and the adiabatic efficiency is always greater than unity. Similar phenomena are observed more often than many people think. Tornadoes, atmospheric cyclones, and almost certainly cosmic jets are stable, thanks to the drop in entropy and the release of the internal energy of the gas/plasma. In this case, the potential force of the vortex motion to the opposing gravitational forces (Figure 5).

Figure 4.

Expansion of gas with γ = 1.4 in a potential field (red line) compared to the Poisson adiabat (blue line): Dependence of the normalized temperature on the normalized pressure and their ratio (adiabatic efficiency) (green line).

Figure 5.

Hypothetical anti-Carnot cycle.