Does the Universe really expand faster than the speed of light: kinematic analysis based on~special relativity and Copernican principle Текст научной статьи по специальности «Физика»

Mathematical Structures and Modeling 2017. N. 4(44). PP. 66-72

UDC 530.12:531.1 DOI: 10.25513/2222-8772.2017.4.66-72

DOES THE UNIVERSE REALLY EXPAND FASTER THAN THE SPEED OF LIGHT: KINEMATIC ANALYSIS BASED ON SPECIAL RELATIVITY AND COPERNICAN PRINCIPLE

Ph.D. (Phys.-Math.), Professor, e-mail: vladik@utep.edu

University of Texas at El Paso, El Paso, Texas 79968, USA

Abstract. In the first approximation, the Universe's expansion is described by the Hubble's law v = H ■ R, according to which the relative speed v of two objects in the expanding Universe grows linearly with the distance R between them. This law can be derived from the Copernican principle, according to which, cosmology-wise, there is no special location in the Universe, and thus, the expanding Universe should look the same from every starting point. The problem with the Hubble's formula is that for large distance, it leads to non-physical larger-than-speed-of-light velocities. Since the Universe's expansion is a consequence of Einstein's General Relativity Theory (GRT), this problem is usually handled by taking into account GRT's curved character of spacetime. In this paper, we consider this problem from a purely kinematic viewpoint. We show that if we take into account special-relativistic effects when applying the Copernican principle, we get a modified version of the Hubble's law, in which all the velocities are physically meaningful - in the sense that they never exceed the speed of light.

Keywords: cosmological expansion, Copernican principle, special relativity, faster-than-speed-of-light.

1. Introduction

Universe's expansion and Hubble's law: reminder. Since the 1920s, it is known that distant galaxies are moving away, with a speed v which is proportional to the distance R: v = H ■ R. This empirical formula is known as the Hubble's law.

The empirical discovery of the Universe's expansion turned out to be in perfect accordance with Einstein's General Relativity theory, according to which the Universe cannot be stationary: it either expands or retracts. Moreover, the expansion predicted by General Relativity is in very good accordance with the Hubble's law; see, e.g., [1].

Hubble's law follows from the Copernican principle. Later, it turned out that the Hubble's law can be derived from the so-called Copernican principle,

Reynaldo Martinez

Student, e-mail: martinez76@miners.utep.edu Vladik Kreinovich

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according to which, from the cosmological viewpoint, there is no special location in the Universe, and thus, the expanding Universe should look the same from every starting point. This principle is named after Copernicus, which argued that, contrary to the then-prevalent opinion, there is nothing special about the location of Earth in space - and moreover, if we do not try to place Earth at the center of the Universe, our description of celestial mechanics becomes much clearer and simpler; see, e.g., [1].

The problem with the Hubble's law. From the physical viewpoint, the Hubble's law has a problem: for large distances R, the corresponding velocity v exceeds the speed of light c. This runs contrary to one of the main principles of special relativity, according to which physical velocities cannot exceed c (see, e.g., [1]).

How this problem is solved now. Since the Universe's expansion is a consequence of Einstein's General Relativity Theory (GRT), this problem is usually handled by taking into account GRT's curved character of space-time [1].

What we do in this paper. In this paper, we consider this problem from a purely kinematic viewpoint.

We show that if we take into account special-relativistic effects when applying the Copernican principle, we get a modified version of the Hubble's law, in which all the velocities are physically meaningful - in the sense that they never exceed the speed of light.

The structure of the paper. We start, in Section 2, by reminding the readers how, in the non-relativistic case, the Copernican principle leads to the Hubble's law. Then, in Section 3, we show that a special-relativistic modification of this derivation leads to a physically meaningful special-relativistic modification of the Hubble's law.

2. How the Hubble's Law Is Derived from the Copernican Principle: A Brief Reminder

What we want to analyze. We want to find out how the relative velocity v of two galaxies depends on the distance R between them.

We can safely assume that the dependence v(R) is continuous - even differen-tiable.

Copernican principle: reminder. With respect to the Universe's expansion, the Copernican principle states that the expansion should look the same from every starting point.

Consequences of this principle. The Copernican principle states that, for any real number R > 0, if we take an object A at a distance R from the Earth, then, from the viewpoint of this object, the Universe's expansion looks the same as from the Earth. In other words, an object B who is at a distance r from the object A along the line Earth A (and which is thus at the distance R + r from the Earth) moves with velocity v(r) relative to the object A.

Relative to the Earth, the object A moves with the velocity v(R). When B moves with velocity v(r) relative to the object A, and the object A moves relative to the Earth with the velocity v(R), we can conclude, in the non-relativistic case, that B moves with the velocity

v(R) + v(r)

relative to the Earth.

On the other hand, since the object B is located at the distance R + r from the Earth, it moves with the velocity

v(R + r)

relative to the Earth. By comparing the above two expressions for the B-relative-to-Earth velocity, we conclude that

v(R + r) = v(R) + v(r) (1)

for all R > 0 and r > 0.

This formula implies the Hubble's law. Indeed, by applying the formula (1) multiple times, we conclude that

v(ri + ... + r,) = v(ri) + ... + v(r,)

for all possible values r1,... , rn > 0. In particular, for every natural number n, for

ri = ... = r„ = 1, we have ri + ... + m = 1 and thus,

n

v(1) = v(1 ) + ... + v( -nn

v

n times

So, v(1) = n ■ v ( — ), hence v ( — ) = — ■ v(1).

\nj \njn

Similarly, for any natural number m, for r1 = ... = rm = —, we get

n

v ( = v ( 1 ) + ... + v ( 1 n n n

m times

thus

(m \ m m

v [ — = m ■ v \ — = — ■ v(1).

V n/ \nj n

m

So, for rational numbers R = —, we have v(R) = H ■ R, where we denoted

n

H d=f v(1).

Since we assumed that the dependence v(R) is continuous, and every real number can be approximated, with arbitrary accuracy, by rational numbers, we conclude that v(R) = H ■ R for all real values R > 0. This is exactly the Hubble's law.

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3. What If We Take Special Relativity into Account

Let us recall the above situation. Let us consider the same situation: we have the Earth, we have an object A at distance R from the Earth, and we have an object B at the distance R + r from the Earth along the same line as the object A. Relative to the Earth:

• the object A moves with velocity -(R), and

• the object B moves with the velocity -(R + r).

The expansion should look the same from the viewpoint of the object A as it looks from the viewpoint of the Earth.

Let us take relativistic effects into account. In the non-relativistic case, from the viewpoint of the object A, the object B was at the distance r. However, in the relativistic case, since the object A is moving with velocity -(R) relative to Earth,

the distance AB shrinks to r = r

1-

; see, e.g., [1]. Therefore, from

the viewpoint of the object A, B moves with velocity -(F) relative to A.

We need to combine the A-relative-to-Earth and B-relative-to-A velocities into the B-relative-to-Earth velocity. In the non-relativistic case, we simply added the given velocities. In the relativistic case, we need to use the special-relativity

-1 + -- ; see, e.g., [1]. In particular, for

1 + -V2

formula for such a combination: v

-1 = -(R) and -2 = -(?), we conclude that

v(R + r)

v(R) + v(r)

1 +

v(R) ■ v(r)

v(R) + v I r ■ W1 -

v(R)'

1 +

v(R)■v I r■\ 1 -

v(R)'

This formula can be simplified if we consider an auxiliary function u(R)

def

v(R)

instead of the desired function -(R). For this auxiliary function, the above formula takes the following simplified form:

u(R + r) =

u(R) + u - (u(Ä))2)

1 + u(R) ■ u ir^J 1 - (u(R))2

(2)

2

c

c

c

c

2

c

What can we derive from this equation? Since we assumed that the dependence -(R) is differentiable, we can differentiate both sides of the equality (2) by r and take r = 0.

In the left-hand side, we get the derivative u'(R). In the right-hand side, we can use the usual formula for the derivative of the ratio:

thus

(//g)'(r) =

(//g)'(0) =

//(r) ■ g(r) - / (r) ■ g'(r) (g(r))2 ;

/'(0) ■ g(0) - /(0) ■ g'(0)

(g(0))2 :

For /(r) = u(R) + u (V Vi - (u(R))2) , we have /(0) = v(R) and /'(r) = u' (rVi - (u(R))2) ■ Vi - u(R)2.

So, for r = 0, we have

/'(0) = u'(0) Vi - (u(R))2

Similarly, for g(r) = i + u(R) ■ u ^r^i - (u(R))2) , we have g(0) = 1 and g'(r) = u(R) ■ u' (rVi - (u(R))2) ■ Vi - (u(R))2.

So, for r = 0, we have

g'(0) = u(R) ■ u'(0) ■ Vi - (u(R))2.

Let us denote u'(0) by h. Then, by equating the derivatives of both sides of the formula (2), we conclude that

u'(R)

h ■ Vi - (u(R))2l ■ i - u(R) ■ |u(R) ■ h ■ Vi - (u(R))2

i2

hence

h Vi - (u(R))2l - i(u(R))2 ■ h Vi - (u(R))2

dR = u'(R) = h ■ Vi - (u(R))2 ■ (i - (u(R))2

By moving all the terms related to u to the left-hand side and all the terms related to R to the right-hand side, we get

du

VT-u2 ■ (i - u2)

h • dR.

By integrating both sides, we get

f du

Vi-u2 ■ (i - u2)

h • dR = h • R + C,

for some integration constant C.

To find the expression for the integral in the left-hand side, we can substitute u = sin(0), then du = cos(0) ■ d0, and the integral takes the form

f cos(0) d0 f cos(0) d0 f d0

J y/1 - sin2(0) ■ (1 - sin2(0)) J vW(0) ■ cos2(0) J cos2(0)'

This integral is known - it is equal to tan(0), hence tan(0) = h ■ R + C. For R = 0, we have v(0) = sin(0), hence 0 = 0, tan(0) = 0, and thus, C = 0 and tan(0) = h■ R. Here,

sin(0) sin(0) u

tan(0) =

oos(0) yi sin2 (0) V1 - u2 ;

so u

, U = h ■ R.

J1—u2

By squaring both sides and multiplying both sides by the resulting denominator, we get

u2 = (1 - u2) ■ h2 ■ R2 = h2 ■ R2 - u2 ■ h2 ■ R2. By moving the terms containing u2 to the left-hand side, we get

u2- (1 + h2- R2) = h2- R2,

hence

u2

therefore

u(R)

So, for v(R) = c ■ u(R), we get

v(R)

h2 R2

1 + h2 ■ R2 '

h ■ r V1 + h2 ■ R2 '

c ■ h ■ r V1 + h2 ■ R2 '

If we denote H = c ■ h, so that h = H, we get the following formula. Resulting formula.

v(R)= H'R

H ■ RN 2

1+

For this formula, as one can easily see, the velocity never exceeds the speed of light.

Acknowledgments.

This work was supported in part by the National Science Foundation grant HRD-1242122 (Cyber-ShARE Center of Excellence).

References

1. Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics // Addison Wesley, Boston, Massachusetts, 2005.

ДЕЙСТВИТЕЛЬНО ЛИ ВСЕЛЕННАЯ РАСШИРЯЕТСЯ БЫСТРЕЕ, ЧЕМ СКОРОСТЬ СВЕТА: КИНЕМАТИЧЕСКИЙ АНАЛИЗ НА ОСНОВЕ СПЕЦИАЛЬНОЙ ТЕОРИИ ОТНОСИТЕЛЬНОСТИ И ПРИНЦИПА

КОПЕРНИКА

Р. Мартинес

студент, e-mail: martinez76@miners.utep.edu В. Крейнович

к.ф.-м.н., профессор, e-mail: vladik@utep.edu

Техасский университет в Эль Пасо, США

Аннотация. В первом приближении расширение Вселенной описывается законом Хаббла V = Н ■Д, согласно которому относительная скорость V двух объектов в расширяющейся Вселенной растёт линейно с расстоянием R между ними. Этот закон может быть получен из принципа Коперника, согласно которому космологически нет особого местоположения во Вселенной, и, следовательно, расширяющаяся Вселенная должна выглядеть одинаково с каждой отправной точки. Проблема с формулой Хаббла заключается в том, что для больших расстояний это приводит к нефизическим скоростям, превышающим скорость света. Поскольку расширение Вселенной является следствием общей теории относительности Эйнштейна (ОТО), эту проблему обычно решают учитывая искривлённость пространства-времени в ОТО. В этой статье мы рассматриваем эту проблему с чисто кинематической точки зрения. Мы показываем, что если учесть эффекты специальной теории относительности при применении принципа Коперника, мы получим модифицированную версию закона Хаббла, в которой все скорости физически значимы — в том смысле, что они никогда не превышают скорость света.

Ключевые слова: космологическое расширение, принцип Коперника, специальная теория относительности, сверхсветовая скорость.

Дата поступления в редакцию: 30.08.2017