Variations of spectra of quasars with their velocities in the expanding Universe

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Variations of spectra of quasars with their velocities in the expanding Universe

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This preprint has been withdrawn

Alexander Agafonov^*

This version is not peer-reviewed

Submitted:

31 July 2024

Posted:

06 August 2024

Withdrawn:

18 December 2024

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Abstract

An atomic matter stores its motion information which is contained in velocity-induced shifts in the atomic energy levels. For redshifts less than or of the order of $\frac{1}{137}$, the velocity-dependent shifts of the energy levels of hydrogen atoms are calculated analytically. These velocity-dependent energy shifts lead to the changes of radiative transition wavelengths and doublet splittings of the quasar spectral lines. These spectral variations can be described by the effective value of the fine structure constant depended on the radiative source velocities. In the expanding Universe, these velocities are variable in time and space. It leads to the change of the effective value of the fine structure constant with the cosmological epochs, whereas the fundamental value of the fine structure constant does not.

Keywords:

Subject: Physical Sciences - Astronomy and Astrophysics

1. Introduction

Cosmological variation of the fundamental constants was hypothesized in (Dirac 11, Milne 14). Years after that, theories of this phenomenon in the expanding Universe have been proposed (see Chakrabarti 7, Chen 8, Ivanchik 12, Martins 13, Uzan 20,21, Wang 22 and references therein). Considering spectral features of absorption and emission lines of distant quasars and galaxies, the time and space changes in the fine structure constant,

α = \frac{e^{2}}{ℏ c}

, are of much interest.

In experimental astrophysics, the resonance doublet lines

λ_{1, 2}

of neutral H (Sargent 17, Scheuer 19) and ions C, N, O, Mg, Al, Si and other s(Bahcall 4, Webb 23, Wilczynska 24) are identified in emission or absorption spectra of quasars. These are doublet wavelengths are shifted due to the cosmological redshift z. The doublet splitting energy is proportional to

α^{4}

. Hence, the value

R (z) = (λ_{2} - λ_{1}) / (λ_{2} + λ_{1})

is proportional to

α^{2} (z)

. Comparing

R (z)

with the value

R (0)

obtained on Earth, upper and lower bounds of time-space variations of

α

were determined (Angstmann 3, Bahcall & Schmidt 5, Damour & Dyson 9, Ivanchik 12, Murphy 15).

The redshift can be explained by the relativistic Doppler effect. Then, the value z does not depend on the radiation wavelength, and is determined by the radiation source velocity. To improve measurement accuracy, the transition wavelengths from different multiples of several ions can be analyzed together (Webb 23).

Let us assume that there is a fundamental dependence of the fine-structure constant

α (z)

on the space-time points of the expanding Universe. Then the calculation results for all absorption and emission lines of distant quasars and galaxies should fit together or, taking into account errors, at least correlate with this fundamental law. Therefore, it does not matter which atom’s doublet is used for calculations. Further, keeping in mind the analytical calculations, we will consider the hydrogen atoms moving with a low velocity,

v < α c

The above value of

R (z)

does not depend on the Doppler effect. However, there is another reason for the dependence of

R (z)

on the velocity of the radiative source v. This reason is the velocity-induced shifts of the energy-levels of the moving atom (Agafonov 1).

First, let the hydrogen atom be initially at rest. let’s assume that we have calculated the rest energy of the atom in a

ψ

state. This energy is the sum of the proton and electron masses, the kinetic and potential energies of the particles, and the fine, hyperfine and recoil corrections. With account all these corrections, this rest energy is equal to the mass of the atom in the

ψ

state.

Now let the atom move with a low velocity. The energy of the moving atom will change. The kinetic energy of the particles will change, since the intraatomic motion of the proton and electron will be supplemented by their translational motion. All the above corrections are the mass corrections. Therefore, for moving atom energy in the state, each of these corrections should be multiplied by the Lorentz factor. The interaction of the particles will also change, since the magnetic interaction of the moving charges (or interaction through the vector potential) should be taken into account. All these changes will lead to the velocity-dependent shifts of the atomic energy levels.

The redshift z is a function of the radiative sources v. For the atom moving with a low velocity

v < α c

, the redshift

z = v / c

Hubble’s law predicts that galaxies are moving away from Earth at velocities proportional to their distances. It means that the farther they are, the faster they are moving away from Earth. Due to the velocity dependence of the atomic energy levels, the transition frequencies and doublet splittings should also be time- and space- variable in the expanding Universe. To clarify this, consider, for example, the doublet splitting of the

2 P_{3 / 2}

and

2 P_{1 / 2}

states. For the moving atom, the velocity-dependent part of the doublet splitting must include the factor

β v {(t)}^{2} / c^{2}

. Hence the effective value of the fine structure constant which corresponds to the complete splitting for the moving atom, is given by:

α_{e f f}^{4} (z) = α^{4} + β z^{2},

(1)

where

α

is the fundamental fine structure constant,

β

should to be determined.

According to (1), the effective value of

α

which is defines the parameters of optical transitions, varies with the epoch z. But the fundamental value of the fine structure constant does not change with z.

In this paper, using Hubble’s law, we examine the velocity dependencies of the wavelengths of the transitions

2 P_{3 / 2} \to 1 S_{1 / 2}

and

2 P_{1 / 2} \to 1 S_{1 / 2}

, and the doublet splitting of the

2 P_{1 / 2}

and

2 P_{3 / 2}

states. The article is organized as follows. In order to retain all the effects of the velocity-dependent doublet splitting, in Section 2, the relativistic-kinematic equation for bound states of the moving hydrogen atom is derived. Using this equation, the velocity-dependent perturbation operators are found in Section 3 for the case of

v < α c

. The hydrogen energy level shifts as a function of the atomic velocity are calculated in Section 4. In the previous sections, the states with the principal quantum numbers

n = 1, 2

were taken into account. Influence of the

3 S_{1 / 2}

state on the velocity-dependent shifts is analyzed in Section 5. In Section 6, using Hubble’s law and the results obtained for moving-atom energy-level shifts, we examine, according to the developed theory, the seeming cosmological variations of the fine structure constant. Conclusion and discussion are drawn in Section 7.

2. The Relativistic-Kinematic Equation for Bound States

As usually, the Hamiltonian of the particle system can be represented as:

H = E_{f r e e} + V_{i n t},

(2)

where

E_{f r e e}

is the energy of the free particles, and

V_{i n t}

is the interaction between them.

The energy at rest

E_{f r e e}^{'}

is equivalent the mass of the system,

E_{f r e e}^{'} = m_{f r e e} c^{2}

. In the relativity theory, the total energy and the total momentum of free particle system are additive quantities. The mass of an isolated system is not additive (Okun’ 16). In our case, we have:

m_{f r e e} = {[{(\frac{E_{e} + E_{p}}{c^{2}})}^{2} - {(\frac{P + Q}{c})}^{2}]}^{1 / 2} .

(3)

Here

E_{e}

and

P

are the energy and the momentum of the free electron, and

E_{p}

and

Q

are the energy and the momentum of the free proton.

Let the particles have the same velocities, for example equal to

v

. Then the electron momentum

P = γ m v

and the proton one

Q = γ M v

. Here

γ = {(1 - v^{2} / c^{2})}^{- 1 / 2}

is the Lorentz factor. From (3) we obtain that the mass of the system is equal to

m_{f r e e} = m + M

, and is the invariant that does not depend on the velocity of the charges. The energy of the moving system is equal to

E_{f r e e} = γ m_{f r e e}

Since the charges move with the velocity

v

, the electrical interaction should be supplemented by the magnetic interaction. Then, the interaction

V_{i n t} = V (r) (1 - v^{2} / c^{2})

, where

V (r) = - e^{2} / r

is the Coulomb interaction,

r = | r_{e} - r_{p} |

is the relative radius vector between the particles.

Now we take into account the specifics of the atomic system. In addition to the atom’s translational motion, the electron and proton are characterized by internal motion in the bound state. Let the electron momentum

p

and the proton momentum

q

are due to the relative motion in the composite. Therefore, the total electron momentum can be represented as

\hat{P} = \hat{p} + m v

(4)

the total proton momentum is

\hat{Q} = \hat{q} + M v .

(5)

Taking into account (4) and (5), we define the electron velocity operator,

{\hat{V}}_{e} = \frac{\hat{P}}{m}

(6)

and the proton one,

{\hat{V}}_{p} = \frac{\hat{Q}}{M}

(7)

As a result, we arrive at the following form of the Hamiltonian of the moving hydrogen atom:

\hat{H} = γ {[{(\sqrt{m^{2} + {\hat{P}}^{2}} + \sqrt{M^{2} + {\hat{Q}}^{2}})}^{2} - {(\hat{P} + \hat{Q})}^{2}]}^{1 / 2} - α \frac{(1 - {\hat{V}}_{e} {\hat{V}}_{p})}{r} .

(8)

Considering the Hamiltonian (8), we obtain the Schrödinger-like equation which determines the dynamics of the quantum mechanical system:

i \frac{t i a l}{t i a l t} Ψ (r_{e}, r_{p}, t) = H Ψ .

For stationary states, this equation is reduced to the form:

H ψ (r_{e}, r_{p}) = E ψ

, where E is the energy of the two-particle system.

3. The Velocity Dependent Perturbation Operators

Equation (8) contains perturbation operators, some of which do not depend on the atom velocity, while others do. Considering the fine splitting of the energy levels, Equation (8) was expanded into a series up to

α^{4} m

. The subsequent terms of the series should not be taken into account, since they is less than the radiative corrections (the Lamb shift). Respectively, the

v -

dependent perturbation operators were determined by the expansion terms up to the order of

α^{4} v^{2} m

The first term in Equation (8) is the operator

E_{f r e e} (v)

. Considering

M > > m

and

v < α c

, we find:

{\hat{E}}_{f r e e} = γ (m + M) + \frac{1}{2 (m + M)} {(\sqrt{\frac{M}{m}} \hat{p} - \sqrt{\frac{m}{M}} \hat{q})}^{2}

- \frac{{\hat{p}}^{4}}{8 m^{3} c^{2}} (1 + \frac{v^{2}}{2 c^{2}}) - \frac{{(\hat{p} v)}^{2}}{2 m c^{2}} + \frac{{\hat{p}}^{2} v^{2}}{4 m c^{2}} - \frac{{\hat{p}}^{2} (\hat{p} v)}{2 m^{2} c^{2}} .

(9)

The second term on the right side of Equation (8) is the interaction function between the particles. Considering

M > > m

, we obtain:

V_{i n t} = V (r) - V (r) \frac{v \hat{p}}{m c^{2}} - V (r) \frac{v^{2}}{c^{2}},

(10)

where

V (r) = - \frac{α}{r}

Now we combine Equations (9) and (10). Combining the first and second terms on the right-hand side of Equation (9) and the first term on the right-hand side of Equation (10), we find the unperturbed Hamiltonian for the hydrogen atom:

{\hat{H}}_{r e s t} - γ (m + M) = \frac{1}{2 (m + M)} {(\sqrt{\frac{M}{m}} \hat{p} - \sqrt{\frac{m}{M}} \hat{q})}^{2} - \frac{α}{| r_{e} - r_{p} |},

(11)

The Hamiltonian (11) has an unusual form. In Appendix A we demonstrate that the Hamiltonian (11) represents the hydrogen atom at rest. Equation (11) predicts the resting center of mass, takes the ordinary form of the hydrogen Hamiltonian with the reduced mass, and, therefore, has well-known eigenfunctions and eigenvalues for the hydrogen atom.

The third term on the right-hand side of Equation (9) is the fine structure operator,

{\hat{W}}_{f} = - \frac{p^{4}}{8 m^{3}},

(12)

corrected for the moving atom. Since we are studying the fine splitting of spectral lines, the spin-orbit interaction operators in the atom should also be taken into account,

{\hat{W}}_{s l} = \frac{α}{2 m^{2} r^{3}} \hat{l} \hat{s} + \frac{π α}{2 m^{2}} δ (r)

(13)

The energy corrections (12) and (13) together determine the fine level splitting. These expressions can also be considered as corrections to the mass of the atom at rest. For a moving atom, the Lorentz factor will lead to a change in these mass corrections. Then, the total fine splitting operator is given by:

{\hat{W}}_{f}^{t} = ({\hat{W}}_{s l} + {\hat{W}}_{f}) (1 + \frac{v^{2}}{2 c^{2}}) .

(14)

Among the remaining operators, the operator, which is proportional to the atom velocity, is given by:

{\hat{W}}_{v}^{(1)} = - H_{0} \frac{v \hat{p}}{m c^{2}} .

(15)

where

H_{0}

is the unperturbed hydrogen Hamiltonian:

H_{0} = \frac{{\hat{p}}^{2}}{2 m} + V (r)

From Equations (9) and Equations (10), the operator

{\hat{W}}_{v}^{(2)}

, which is proportional the velocity squared, takes the form:

{\hat{W}}_{v}^{(2)} = - \frac{5}{4} V (r) \frac{v^{2}}{c^{2}} - \frac{{(v \hat{p})}^{2}}{2 m c^{2}} .

(16)

4. Calculations of Energy Level Shifts

The relativistic correction due to the operator (14) is equal to:

Δ E_{n j} = - \frac{α^{4} m}{2 n^{3}} (\frac{1}{j + 1 / 2} - \frac{3}{4 n}) (1 + \frac{v^{2}}{2 c^{2}}) .

(17)

In our consideration, the principal quantum number is

n = 1, 2

and the quantum number of the total momentum is

j = \frac{1}{2}, \frac{3}{2}

The energy correction due to the operator

{\hat{W}}_{v}^{(2)}

(16) depends only on the principal quantum number:

< n l j m_{j} | W_{v}^{(2)} | n l j m_{j} > = - \frac{13}{12} < | V (r) | > \frac{v^{2}}{c^{2}} = + \frac{13}{12 n^{2}} α^{2} m \frac{v^{2}}{c^{2}} .

(18)

This correction leads to the same shift of all levels with the same n. It changes the transition frequencies.

The last perturbation operator (15) gives the off-diagonal matrix elements which are of the order of

α^{3} m v / c

. In the second order of the perturbation theory, this operator leads to the energy level shifts of the order of

α^{4} m v^{2} / c^{2}

. The fine structure (17) does not completely remove the degeneracy, since the

2 S_{1 / 2}

and

2 P_{1 / 2}

states remain degenerate. The operator (15) removes this degeneracy, and the energies of the states

2 S_{1 / 2}

and

2 P_{1 / 2}

become different.

For calculations of these shifts, we use the spin-angle wave functions

Φ_{l, \frac{1}{2}, j = l \pm \frac{1}{2}, m_{j}}

, which are expressed in terms of the spherical functions

Y_{l, m \pm \frac{1}{2}}

and the

\frac{1}{2}

-spin functions

χ_{σ = \pm \frac{1}{2}}

([10]). Taking into account non-zero off-diagonal matrix elements, we obtain:

δ E (2 P_{1 / 2}) = \frac{8}{3 α^{2} m} \times \frac{1}{2 j + 1} \sum_{m_{j} = \pm \frac{1}{2}, m_{j 1} = \pm \frac{1}{2}} < R_{21} Φ_{1, \frac{1}{2}, \frac{1}{2}, m_{j}} | W_{v}^{(1)} | R_{10} Φ_{0, \frac{1}{2}, \frac{1}{2}, m_{j 1}} >

< R_{10} Φ_{0, \frac{1}{2}, \frac{1}{2}, m_{j 1}} | W_{v}^{(1)} | R_{21} Φ_{1, \frac{1}{2}, \frac{1}{2}, m_{j}} >

(19)

Here

R_{n l}

is the hydrogen radial wave function, the spin-angle wave functions are given by:

Φ_{0, \frac{1}{2}, \frac{1}{2}, \pm \frac{1}{2}} = Y_{00} χ_{\pm \frac{1}{2}},

(20)

Φ_{1, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}} = - \sqrt{\frac{1}{3}} Y_{10} χ_{\frac{1}{2}} + \sqrt{\frac{2}{3}} Y_{11} χ_{- \frac{1}{2}},

(21)

Ψ_{1, \frac{1}{2}, \frac{1}{2}, - \frac{1}{2}} = - \sqrt{\frac{2}{3}} Y_{1, - 1} χ_{\frac{1}{2}} + \sqrt{\frac{1}{3}} Y_{10} χ_{- \frac{1}{2}},

(22)

Substituting (20)–(22) into Equation (19), we obtain the shift of the

2 P_{1 / 2}

level:

δ E (2 P_{1 / 2}) = \frac{2^{8}}{3^{10}} α^{4} \frac{v^{2}}{c^{2}} m .

(23)

For the

2 P_{3 / 2}

state we have:

δ E (2 P_{3 / 2}) = \frac{8}{3 α^{2} m} \times \frac{1}{2 j + 1} \sum_{m_{j} = \pm \frac{1}{2}, \pm \frac{3}{2}, m_{j 1} = \pm \frac{1}{2}} < R_{21} Φ_{1, \frac{1}{2}, \frac{3}{2}, m_{j}} | W_{v}^{(1)} | R_{10} Φ_{0, \frac{1}{2}, \frac{1}{2}, m_{j 1}} >

< R_{10} Φ_{0, \frac{1}{2}, \frac{1}{2}, m_{j 1}} | W_{v}^{(1)} | R_{21} Φ_{1, \frac{1}{2}, \frac{3}{2}, m_{j}} >] .

(24)

Here we use the spin-angle functions:

Φ_{1, \frac{1}{2}, \frac{3}{2}, \frac{1}{2}} = \sqrt{\frac{2}{3}} Y_{10} χ_{\frac{1}{2}} + \sqrt{\frac{1}{3}} Y_{11} χ_{- \frac{1}{2}},

(25)

Φ_{1, \frac{1}{2}, \frac{3}{2}, - \frac{1}{2}} = \sqrt{\frac{1}{3}} Y_{1, - 1} χ_{\frac{1}{2}} + \sqrt{\frac{2}{3}} Y_{10} χ_{- \frac{1}{2}},

(26)

Φ_{1, \frac{1}{2}, \frac{3}{2}, \pm \frac{3}{2}} = Y_{1, \pm 1} χ_{\pm \frac{1}{2}} .

(27)

After calculations, for the

2 P_{3 / 2}

level we obtain the same shift as for the

2 P_{1 / 2}

level:

δ E (2 P_{3 / 2}) = \frac{2^{8}}{3^{10}} α^{4} \frac{v^{2}}{c^{2}} m .

(28)

Taking into account (17), (18) and (23), the energies of the considered states become different to each other . With the accuracy of radiation corrections (the Lamb shift), the energy of the

2 P_{1 / 2}

state is:

E (2 P_{1 / 2}) / m = - \frac{1}{8} α^{2} - \frac{5}{2^{7}} α^{4} + \frac{13}{48} α^{2} \frac{v^{2}}{c^{2}} - (\frac{5}{2^{8}} - \frac{2^{8}}{3^{10}}) α^{4} \frac{v^{2}}{c^{2}},

(29)

The energy level of the

2 P_{3 / 2}

state is given by:

E (2 P_{3 / 2}) / m = - \frac{1}{8} α^{2} - \frac{1}{2^{7}} α^{4} + \frac{13}{48} α^{2} \frac{v^{2}}{c^{2}} - (\frac{1}{2^{8}} - \frac{2^{8}}{3^{10}}) α^{4} \frac{v^{2}}{c^{2}} .

(30)

Also, in the second order of perturbation theory, the operator (15) leads to the energy shift of the

1 S_{1 / 2}

level. It is easy to obtain that

δ E (1 S_{1 / 2}) = - δ E (2 P_{1 / 2}) - δ E (2 P_{3 / 2})

. Then the energy of the

1 S_{1 / 2}

state is given by:

E (1 S_{1 / 2}) / m = - \frac{1}{2} α^{2} - \frac{1}{2^{3}} α^{4} + \frac{13}{12} α^{2} \frac{v^{2}}{c^{2}} - (\frac{1}{2^{4}} + \frac{2^{9}}{3^{10}}) α^{4} \frac{v^{2}}{c^{2}} .

(31)

The velocity-dependent energy shifts in Equation (18) does not affect the splitting of the doublet

2 P_{1 / 2}

and

2 P_{3 / 2}

, but changes the frequencies of optical transitions

2 P_{3 / 2} \to 1 S_{1 / 2}

and

2 P_{1 / 2} \to 1 S_{1 / 2}

Considering all the states with

n = 1, 2

only, the velocity-dependent operator (15) has no correction to the energy level of the

2 S_{1 / 2}

state. Then the energy of the state is written as:

E (2 S_{1 / 2}) / m = - \frac{1}{8} α^{2} - \frac{5}{2^{7}} α^{4} + \frac{13}{48} α^{2} \frac{v^{2}}{c^{2}} - \frac{5}{2^{8}} α^{4} \frac{v^{2}}{c^{2}} .

(32)

As is known, the energy level with the principal quantum number n is split into n fine structure components. The operator (15) removes the remaining degeneracy of the levels

2 S_{1 / 2}

and

2 P_{1 / 2}

. As a result, all the states with

n = 2

2 S_{1 / 2}

2 P_{1 / 2}

and

2 P_{3 / 2}

, have different energy levels.

The known expressions for the radiation corrections

O (α^{5})

do not depend on the hydrogen atom velocity. For the quantum numbers

n = 2

l = 1

and

j = 3 / 2

, the radiative correction is equal to:

δ E_{21 \frac{3}{2}} = \frac{α^{5}}{6 π} (L_{21} + \frac{1}{16}) m,

(33)

where

L_{21} = 0.03

. Formally, for the moving atom with

v < < c

, the Lorentz factor could be introduced into Equation (39). Then the velocity-dependent correction could be appeared:

δ E_{21 \frac{3}{2}}^{v} = \frac{α^{5}}{12 π} (L_{21} + \frac{1}{16}) \frac{v^{2}}{c^{2}} m,

(34)

It is easy to see that (34) is negligible compared to (28). Therefore, corrections

O (α^{5})

can be omitted.

5. Influence of Higher Energy States

Above we considered the group of the hydrogen states with the principal quantum number

n = 1

and

n = 2

, and obtained the velocity-dependent shifts of these energy levels. Besides, other states of the hydrogen atom can contribute to the shifts. To clarify this influence, we examine the effect of the

3 S_{1 / 2}

state. Using (20)–(22), the corresponding energy shift of the

2 P_{1 / 2}

level is given by:

δ^{'} E (2 P_{1 / 2}) = - \frac{2^{3} 3^{2}}{5 α^{2} m} \frac{1}{2 j + 1} \sum_{m_{j} = \pm \frac{1}{2}, m_{j 1} = \pm \frac{1}{2}} < R_{21} Φ_{1, \frac{1}{2}, \frac{1}{2}, m_{j}} | W_{v}^{(1)} | R_{30} Φ_{0, \frac{1}{2}, \frac{1}{2}, m_{j 1}} >

< R_{30} Φ_{0, \frac{1}{2}, \frac{1}{2},_{j 1}} | W_{v}^{(1)} | R_{21} Φ_{1, \frac{1}{2}, \frac{1}{2}, m_{j}} >

(35)

The result of our calculations of (35) is:

δ^{'} E (2 P_{1 / 2}) / m = - \frac{2^{7} 3^{2}}{5^{11}} α^{4} \frac{v^{2}}{c^{2}}

(36)

Comparing (36) with (23), we conclude that the

3 S_{1 / 2}

state does not influence on the velocity-dependent shift of the

2 P_{1 / 2}

state.

6. Interpretation of Equations (29)–(32) as Time- and Space- Variations of the Fine Structure Constant

According to (29)–(32), the energy levels depend not only on the fine structure constant, but also on the velocity of the radiative sources. It allow us to introduce the effective value of the fine structure constant. It is this latter quantity, and not the fundamental constant

α

, that is the time- and space- dependent in the expanding Universe. Hence, the doublet splitting of the quasar spectral lines is determined by the velocity-dependent of the effective value of the fine structure constant.

From (29)–(30), we obtain the doublet energy splitting:

Δ E_{d} / m = \frac{1}{2^{5}} α^{4} + \frac{1}{2^{6}} α^{4} \frac{v^{2}}{c^{2}} .

(37)

Here v is less than or of the order of

α c

. The radiative source velocity v and the redshift z are related to each other. For

v < < c

v = z c

We introduce the effective value of the fine structure constant,

Δ E_{d} / m = \frac{1}{2^{5}} α_{e f f}^{4} (z) .

(38)

Then, using (38), we have:

α_{e f f} (z) = α (1 + \frac{1}{8} z^{2}) .

(39)

The result (39) can be compared with the experimental values of

α^{2} (z)

for small z. In the work [18], data on the time dependence of

α

using the fine-structure splitting of emission lines of N II and Ne III in the spectrum of the nearby radio galaxy Cygnus A were reported:

α^{2} (z = 0.057) / α^{2} = 1.0036 \pm 0.003

. Substituting in (39), we obtain:

\frac{α_{e f f}^{2} (z = 0.057)}{α^{2}} = 1.000812,

(40)

which correlates with the experimental data [18].

The difference between the effective value

α_{e f f} (z)

and the fundamental value

α

Δ α = α_{e f f} (z) - α

. Then Equation (39)can be written in the form:

\frac{Δ α}{α} = \frac{1}{8} z^{2}

(41)

Substituting the same redshift

z = 0.057

in Equation (41), for the same radiation source ([18]) we obtain:

\frac{Δ α}{α} = 4.06 \times 10^{- 4} .

(42)

Note that the value (42) is slightly higher than the upper limits that have been obtained in the most precise of these previous absorption line studies ([4]). These limits are generally

| Δ α / α | < 2 \times 10^{- 4}

. These studies where performed, as a rule, for relatively large redshifts, z is about 1 (see [4] and references therein), while the presented theory is applicable only for

z < < 1

. Also, in known works, the change of

α

is often given in the range of z. In our approach, each value of z corresponds to the certain

α

Using the Hubble law, the redshift z of the emission or absorptilon lines is given by:

z = H_{0} \frac{r}{c},

(43)

where

H_{0}

is the Hubble constant, r is the distance to the galaxy.

Substituting (42) into (41), we have:

\frac{Δ α}{α} = \frac{1}{8} \frac{r^{2}}{{(c t_{H})}^{2}},

(44)

where

t_{H}

is the Hubble time. Using

t_{H} = 14.4

billion years, Equation (43) gives us:

\frac{Δ α}{α} = {(0.7992 \times 10^{- 4} r)}^{2}

(45)

where the distance r in units of Mpc.

Equation (44) predicts that

α_{e f f}

increases with distance as

r^{2}

Note that in our consideration z is less than or of the order of

α

. Hence, according to Equation (42), the distance to the galaxy is less than or of the order of 32 Mpc.

Now we find the time derivative of the equation (39), represented as:

α_{e f f} (v (t)) = α (1 + \frac{1}{8} \frac{v^{2}}{c^{2}}) .

(46)

In Equation (45),

α_{e f f} (v (t))

is the effective value of the fine structure constant at much earlier time t with respect to the present time

t = 0

. Let the time

t_{1}

be less than the time

t_{2}

t_{2} > t_{1}

. This means that the epoch corresponded to time

t_{2}

is an earlier compared to the epoch of

t_{1}

. Taking into account the Hubble law for the expanding Universe, the distance

r_{2}

of the radiation source from the Earth at the moment

t_{2}

is less than the distance

r_{1}

at the moment

t_{1}

. Similarly, the velocity

v (t_{2})

of the radiating source at the earlier time

t_{2}

is less than the velocity

v (t_{1})

at time

t_{1}

. This means that in the equation

\frac{1}{α} \frac{d α_{e f f}}{d t} = \frac{1}{4} \frac{v}{c^{2}} \frac{d v}{d t}

(47)

the derivative

\frac{d v}{d t} < 0 .

(48)

Using Hubble’s law, we obtain:

d v / d t = H_{0} d r / d t = - H_{0} v = - \frac{v}{t_{H}} .

(49)

Using (49) and substituting the Hubble time in Equation (47), we have:

\frac{1}{α} \frac{d α_{e f f}}{d t} = - 1.736 \times 10^{- 11} z^{2} y r^{- 1}

(50)

For the redshift

z = 0.057

that corresponds to the source velocity

v = 17100

km/s, from (50) we obtain:

\frac{1}{α} \frac{d α_{e f f}}{d t} = - 0.564 \times 10^{- 13} y r^{- 1}

(51)

To set a robust upper limit on the time dependence of the fine structure constant, the strong nebular emission lines of O III, 5007

\overset{˚}{A}

and 4959

\overset{˚}{A}

, are used [4]. It was reported:

| α^{- 1} d α (t) / d t | < 2 \times 10^{- 13}

{yr}^{- 1}

. The result (51) is in agreement with this constraint.

7. Discussion and Conclusions

In the present theory the fine structure constant is the fundamental value which does not change with time. However, the quasar absorption and emission lines depend on the quasar velocities. This dependence is due to velocity-dependent shifts of the energy levels of atom or ions which are the radiative sources of the quasars. Then the transition wavelengths depend not only on the fine structure constant, but also on the velocity of the radiative sources. It allow us to introduce the effective value of the fine structure constant. Just this quantity, but not the fundamental constant

α

, is the time- and space- dependent in the expanding Universe.

The reason for the shifts in energy levels of moving atoms or ions can be understood from the following. The hydrogen atom’s mass is different in its different bound states

ψ_{n l j m_{j}}

m_{H} (ψ_{n l j m_{j}}) = m + M + < ψ_{n l j m_{j}} | T + V | ψ_{n l j m_{j}} > + Δ E_{f i n e} (ψ_{n l j m_{j}}) + O (α^{5}) (ψ_{n l j m_{j}}) + . . .

(52)

Here

O

includes the hyperfine and recoil corrections and so on. From (52) one can see that the atom mass is a function of its bound state. Hence the atom energy at rest is given by

E_{n l j m_{j}} = m_{H} (ψ_{n l j m_{j}}) c^{2}

What will change for a moving atom? We consider low velocity motion, v is less than or of the order of

α c

. The kinetic energy of the particles T will change, since the intraatomic motion of the proton and electron will be supplemented by their translational motion. All the above corrections in Equation (52) are the mass corrections. Therefore, for moving atom energy in the state, each of these terms should be multiplied by the Lorentz factor. The interaction of the particles V will also change, since the magnetic interaction of the moving charges (or interaction through the vector potential) should be taken into account. All these changes will lead to the velocity-dependent shifts of the atomic energy levels. At high velocities of atoms

v > α c

, the account of the Lorentz factor will not be enough because all terms, except for

m + M

in Equation (52)will change.

α_{e f f}

. This latter quantity, but not the fundamental constant

α

, is the time- and space- dependent in the expanding Universe. At the same time, the doublet splitting of the quasar spectral lines is determined by

α_{e f f}

. In the expanding Universe, the quasar velocities vary in time and in space. As a result, the velocity-dependent variations of the quasar spectra could be interpreted as cosmological space-time variations of the fine structure constant.

Above we shown that matter stores information about its movement. This information is contained in velocity-induced shifts in the energy levels of atomic systems. However, what is the velocity in this case? First of all, a component of this velocity is the speed of the galaxy which caused by the expansion of the Universe. In turn, within galaxies, stars and gaseous formations, which can provide a bright spectrum of radiation at optical frequencies, can have radial and orbital velocities. Further, emitting objects in the gas formations can be characterized by their own velocity distribution. We denote this velocity with all these components as

v_{a b}

. The direction of this velocity is not important at all.

From the equations (29), (30) and (31) we find the frequencies of the transitions

^{2} P_{1 / 2} \to {^{1} S}_{1 / 2}

and

^{2} P_{3 / 2} \to {^{1} S}_{1 / 2}

registered on the Earth:

ω_{1 / 2, 1 / 2} = [\frac{3}{8} α^{2} + \frac{11}{2^{7}} α^{4} - \frac{13}{2^{4}} α^{2} \frac{v_{a b}^{2}}{c^{2}} + (\frac{11}{2^{8}} + \frac{2^{8}}{3^{9}}) α^{4} \frac{v_{a b}^{2}}{c^{2}}] (1 - \frac{v_{d}}{c}),

(53)

and

ω_{3 / 2, 1 / 2} = [\frac{3}{8} α^{2} + \frac{15}{2^{7}} α^{4} - \frac{13}{2^{4}} α^{2} \frac{v_{a b}^{2}}{c^{2}} + (\frac{15}{2^{8}} + \frac{2^{8}}{3^{9}}) α^{4} \frac{v_{a b}^{2}}{c^{2}}] (1 - \frac{v_{d}}{c}) .

(54)

Here

v_{d}

is the relative velocity of the radiative source with respect to the Earth. The velocity

v_{r}

has any sign.

Equations (53) and (54) are the system of two equations in two unknown,

v_{a b}

and

v_{r}

. In principle, these two velocities should be different. One of them,

v_{r}

, is the radial velocity for the Doppler-shift measured from the spectra on the Earth, while the other,

v_{a b}

, represents the velocity of the light-emitting object in the Universe.

Appendix A. The center of mass frame

Using the unperturbed Hamiltonian (11), we have the equation for eigenfunctions:

[\frac{1}{2 (m + M)} {(\sqrt{\frac{M}{m}} \hat{p} - \sqrt{\frac{m}{M}} \hat{q})}^{2} - \frac{α}{| r_{e} - r_{p} |}] ψ (r_{e}, r_{p}) = E ψ (r_{e}, r_{p}),

(A1)

where the binding energy

E = E - m - M

Equation (A1) can be transformed in terms of new independent variables:

R = \frac{m r_{e} + M r_{p}}{m + M}, r = r_{e} - r_{p},

(A2)

Then, it is easy to obtain that the transformed equation is the Schrödinger equation:

(- \frac{ℏ^{2}}{2 μ} Δ_{r} - \frac{α}{r}) ψ (r) = E ψ (r) .

(A3)

Here

μ = \frac{m M}{m + M}

is the reduced mass.

Note that the variable

R

vanishes in the equation (A3) after the transformation (A2). It means that the center-of-mass of the atom is at rest. Also, the total momentum of the system is zero.

Let us note that the original equation (A1) is the equation of two independent variables

r_{e}

and

r_{p}

. after the replacement (A2), the equation (A3) is the equation only for the relative radius of the vector

r

. In this case, the radius vector of the center of mass essentially turned out to be undefined.

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Variations of spectra of quasars with their velocities in the expanding Universe

Abstract

1. Introduction

2. The Relativistic-Kinematic Equation for Bound States

3. The Velocity Dependent Perturbation Operators

4. Calculations of Energy Level Shifts

5. Influence of Higher Energy States

6. Interpretation of Equations (29)–(32) as Time- and Space- Variations of the Fine Structure Constant

7. Discussion and Conclusions

Appendix A. The center of mass frame

References

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