Special Relativity “Meets” the Uncertainty Principle

 

Special Relativity “Meets” the Uncertainty Principle

 

Pavle I. Premović 

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia

Heisenberg’s uncertainty principle is one of the fundamental concepts of quantum mechanics (QM). This principle defines a relationship between the uncertainties of energy (ΔE) and time (Δt) and it is given in expression form,

ΔEΔt ≥ ћ

where ћ (= h/2π = 1.05 × 10^-34 J sec) is the reduced Planck’s constant and h is the Planck’s constant (6.63 × 10^-34 J sec). This expression is valid for moving objects only.[1] Besides ћ, the energy-time uncertainty expression includes h (the absolute maximum), h/2, ћ and ћ/2 (the absolute minimum), depending on the case.

The above expression can be written as follows

ΔEΔt = (1J × 1sec)_QM  ≥ ћ

or

(1J × 1sec)_QM  ≥ ћ    … (1).

where the lower script denotes that (1J × 1sec) is a QM term.

If the uncertainty of the measured energy of 1J increases, the uncertainty for the measured time interval of 1 sec decreases. Conversely, if the uncertainty of the measured energy of 1J decreases, the uncertainty in the time interval of 1 sec increases. In other words, (1J × 1sec)_QM is always larger or equal to Planck’s constant ћ. Therefore, equation (1) represents a new mathematical form of the energy-time uncertainty.

Time dilation and length contraction are two important effects of Special relativity (SR). These two effects depend upon the second postulate of this theory that the speed of light c (= 2.99792 × 10⁸ m sec^-1) is the same in all inertial frames of reference. From this postulate, time dilation, as well as length contraction, inevitably results [1].

Simply speaking time dilation can be defined as a relativistic phenomenon in that a clock at rest shows the time interval T₀ which is shorter with respect to the time interval T_υ when it is moving with a relative speed υ. In equation

T_υ = T₀ /√(1 – υ²/c²)    … (2)

where 1/√1- υ²/c²) is the Lorentz-Einstein or the time dilation factor γ. 

As we noted above, another important relativistic phenomenon is length contraction. The formula for determining this contraction is

Lυ = L₀√(1 – υ²/c²)    … (3)

where L₀ is the length of the moving object in its rest frame of reference, Lυ is the length of this object measured by us in our rest frame of reference and υ is the relative velocity between these two frames of reference.

Multiplying the equations (2) and (3), we have

L_υ × T_υ = L₀ × T₀ = constant.

Hence

(1m)₀ × (1sec)₀ = (1m)_υ × (1sec)_υ (= constant)

where (1m)₀ and (1sec)₀ represent the length of a moving object of 1m measured in its frame of reference and a time interval of 1 sec measured also in this frame. On the right side of this expression, (1m)_υ and (1sec)_υ represent the length of this object and the time interval of its frame but measured by us in our rest frame. The relative velocity between these two frames of reference is υ.

This can be written as follows

(1m × 1sec)_SR = α    … (4).

where α is a constant and the lower script denotes that (1m × 1sec) is an SR term. Of course, this constant has the same value 1m sec for all observers, or in different frames of reference, in other words, it is invariant.

According to eqn. (4), if the length of the object of 1m increases with a speed υ, the time interval of 1 sec decreases and vice versa, so that (1m × 1sec)_SR is always the same or is equal to the constant α = (1m)₀ × (1sec)₀ = 1m sec. Therefore, this can be interpreted as the length-time uncertainty of the Special theory of relativity.

The relativistic formula for the variation of mass with velocity υ is as follows

m_υ = m₀/√(1 – υ²/c²)    … (5)

where m_υ is the mass of a moving object and m₀ is its rest mass. Multiplying this equation and eqn. (3) we obtain

m_υLυ = m₀L₀ = constant.

Hence,

(1m)₀ × (1kg)₀ = (1m)_υ × (1kg)_υ (= constant)

where (1m)₀ and (1kg)₀ represent a length of 1m and a mass of 1 kg of a moving object measured in its frame of reference. On the right side of this expression, (1m)_υ and (1kg)_υ represent the length and the mass of this object but measured by us in our rest frame. The relative velocity between these two frames of reference is υ.

In equation form,

(1kg × 1m)_SR = β    … (6)

where β is also an invariant constant.

According to eqn. (6), if the mass of an object of 1kg increases with a speed υ, its length of 1m decreases and vice versa, so that (1kg × 1m)_SR is always the same or is equal to the constant β = (1m)₀ × (1kg)₀ = 1 kg m. Therefore, eqn. (6) can be interpreted as the mass-length uncertainty of the Special theory of relativity.

If we multiply eqn. (5) with c² we get the following equation for the total relativistic energy of a moving object E_υ

E_υ = m_υc² = m₀c²/√(1 – υ²/c²).

We know that the rest energy of this object E₀ = m₀c², so we write the above equation as follows

E_υ = E₀/√(1 – υ²/c²).

Using a similar approach as above we find that

(1J × 1m)_SR = ί    … (7)

where ί (= βc²) is an invariant constant, of course.

According to eqn. (7), if the energy of an object of 1J increases with a speed υ, its length of 1m decreases and vice versa, so that (1J × 1m)_SR is always the same or is equal to the constant ί = 1J m. Therefore, eqn. (7) can be interpreted as the energy-length uncertainty of the Special theory of relativity.

This is how we see that Einstein’s Special theory of relativity “meets” Heisenberg’s uncertainty principle. 

Reference

{1} Taylor E. F. and Wheeler J. A. Spacetime Physics: Introduction to Special Relativity, 2nd ed.  Freeman & Company, 1992.

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[1] The uncertainty principle applies to all objects but is only significant for the atomic or subatomic particles.

 

The Redshift and Age of Nearby and Distant Galaxies

 

The Redshift and Age of Nearby and Distant Galaxies

Pavle I. Premović,

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia 

Abstract

An equation is derived that correlates the natural logarithm of the age of distant galaxies in the Big Bang Universe with the redshift of the light they emit. This equation is based on published data for their age. A simple analysis of the equation implies that that the Big Bang Universe may be old as low as 4.7 Gy. That age is comparable to the age of the Sun and its planetary system and it is about three times younger than the age (about 13.8 Gy) of the Big Bang Universe adopted by the vast majority of cosmologists. This and other issues are briefly discussed.

Keywords: cosmology, galaxy, redshift, age, Earth

Introduction

According to the Big Bang theory, the age of the Universe is about 13.8 billion years. This theory states that cosmological redshift (or more commonly just redshift), denoted here as zG, is the result of the Universe expansion during the flight of the light from nearby/distant galaxies¹ to the Earth.

In contrast, Lerner [3, and references therein] reported that the ultraviolet surface brightness data of nearby/distant galaxies, over a very wide redshift range, support the non-expanding (Euclidean) Universe (NEEU). Moreover, Sanejouand {4} performed a detailed analysis of the gamma-ray burst sources. This analysis indicates that the observable Universe has been Euclidean and static over the last 12 Gy. The Big Bang Universe is finite in space and time. In contrast, NEEU is unlimited in both space and time.

The age-dating of nearby/distant galaxies is an important topic in the study of galaxy evolution. To determine their accurate age is exceedingly difficult.

For our purpose, we select sixteen distant galaxies (with zG > 1) and one nearby galaxy (with z < 0.1), Table 1. This selection is based on an emerging consensus among cosmologists (or a current consensus among cosmologists but open for further evaluation, of course).

Table 1. Selected galaxies.

Name of galaxy	Redshift zG	Age (AG)*	lnAG*	References
GN-z11	11.09	0.4[0.3]	− 0.9[−1.2]	{5}
MACS0647-JD	10.7	0.5[0.5]	− 0.8[− 0.7]	{6}
GRB 0904231	8.26	0.6[0.6]	− 0.5[− 0.5]	{7}
EGS-zs8-1	7.73	0.7[0.7]	− 0.4[− 0.4]	{8}
z8 GND 5296	7.51	0.7[0.7]	− 0.4[− 0.4]	{9}
ULAS2 1120+0641	7.085	0.8[0.8]	− 0.2[− 0.2]	{10}
IOK-1	6.96	0.8[0.8]	− 0.2[− 0.2]	{11}
Cosmos Redshift 7	6.60	0.7[0.9]	− 0.4[− 0.1]	{12}
Abel 383**	6.027	0.8[1.0]	− 0.2[0]	{13}
APM 08279+52552	3.91	2.1[1.8]	0.7[0.6]	{14}, {15}
zf-COSMOS- 20115	3.72	1.7[1.9]	0.5[0.6]	{16}
QSO B1422- 231	3.62	1.4[1.9]	0.4[0.6]	{17}
A1689B11	2.54	2.6[2.5]	1[0.9]	{18}
53W091	1.55	3.5[3.2]	1.3[1.2]	{19}
53W069	1.43	4.0[3.3]	1.4[1.2]	{20}
3C 65	1.175	4[3.5]	1.4[1.3]	{21}
NGC 6872	0.01594	5[4.7]	1.6[1.55]	{22}

*In the square brackets are AG (Gy) and lnAG calculated using eqn. (1); **multiply imaged by the cluster Abell 383;

                  ¹gamma ray burst (GRB) host; and ²quasar (a type of active galaxy).

Age of distant galaxies

Table 1 contains the name of selected galaxies, their redshift zG and published age AG  (hereinafter AG = 0 corresponds to the currently accepted age of Big Bang Universe ≈ 13.8 Gy). It appears that the ages of these galaxies were calculated assuming the flat model of the Universe², with the matter density parameter ΩM = 0.3 and the "dark energy" density parameter ΩΛ = 0.7, and H0 = 70 km sec⁻¹ Mpc⁻¹. These ages can be calculated using the calculator of Wright {24}.

To find a relationship between the age of these galaxies, AG, and their redshift zG, we plot their age vs. their redshift, Fig 1. This graph implies that the age of these galaxies can be approximately expressed as a (natural) exponential function [f(x) = e^x] of their redshift.³ For this reason, we plot the natural logarithm (ln)AG vs. zG, Fig. 2. The graph shows an approximate linear dependence of lnAG on zG, confirming the approximate (natural) exponential dependence of AG on zG. Using the regression method, we obtained the following linear function for the above graph

lnAG = − 0.25zG + 1.55 … (1).

In Table 1 are also given the age of selected galaxies AG and lnAG calculated using eqn. (1).

Fig. 1. Galaxy’s age AG vs. their redshift zG for selected galaxies.

After the conversion of this equation into the (natural) exponential form we have

AG = e0.25zG - 1.55.

The dimension of the left side of this expression is T but its right side is dimensionless.⁴ Hence, eqn. (2) is dimensionally incorrect. The simplest way to make this equation dimensionally correct is to multiply its right side by the constant α equals or close to 1 and expressed in time unit of AG (e.g. Gy)

AG = e − 0.25zG + 1.55 × α.

Since lnα is equal or close to 0 the natural logarithmic form of this equation is identical to the initial eqn. (1).

To estimate possible uncertainty in determining AG we employed eqn. (1) to the galaxy APM 08279+5255 with redshift zAPM = 3.91 and age AAPM = 2.1 Gy, Table 1. A simple calculation based on this equation shows that the age of this galaxy should be about 1.8 Gy. The difference between the experimental (2.1 Gy) and the calculated one (1.8 Gy) is (about) 0.3 Gy or the percentage difference is about 15 %.

a

Fig. 2. Natural logarithm of galaxy’s age lnAG vs. their redshift zG for selected galaxies.

Let us denote with AG(X) the age of unknown distant galaxy X and with AG(S) the age of the galaxy’s age standard with known corresponding redshifts zG(X) and zG(S). Introducing these two

ages into eqn. (1) we get two equations lnAG(X) = − 0.25zG(X) + 1.55 and lnAG(S) = − 0.25zG(S) +

1.55. Subtracting these two equations yields

                                              lnAG(X) – lnAG(S) = − 0.25[zG(X) – zG(S)].

For convenience, we write

                                             lnAG(X) = lnAG(S) − 0.25[zG(X) – zG(S)]    … (2).

This equation can be used to determine the age of distant galaxies of unknown age but with a known redshift. For this purpose, we need a reliable age standard for galaxies. The author prefers the galaxy APM 08279+5255.

Let us attribute lnAG(S) to the APM 08279+5255 with zAPM = 3.91 and lnAG(X) to the galaxy GN-z11 with zGN = 11.09 (Table 1). Inserting these two redshift values into eqn. (2) we get that the age of GN-z11 is about 0.11 Gy. As expected, this value is in a rather reasonable agreement with the value (0.4 Gy) given in Table 1. A similar calculation for 3C 65 indicates that its age is about 4 Gy which is equal to the value in Table 1.

ArArp’s peculiar galaxies

In 1966 Harald Arp, a famous American astronomer published his Atlas of Peculiar Galaxies in which he presented 338 images of rather peculiar galaxies. He later published many papers, articles and books. His work is updated in his last book, “Catalogue of Discordant Redshift Associations”. In all of his publications, Arp argued that many quasars with high-redshift are somehow linked to nearby low-redshift active galaxies. He hypothesized that these quasars are ejected from these galaxies and exhibit intrinsic redshifts. This is a controversial view which does not accord with the current cosmological thinking. Most cosmologists reject Arp's interpretation arguing that his observations could be explained by perspective effects. In this work, we will only consider briefly one of Arp’s classic cases.

Arp discovered that five quasars around the distant central Seyfert galaxy, NGC 3516 with redshift z = 0.009. Chu et al. [23] reported redshift measurements of these quasars and found that they are distributed along the minor axis of the galaxy, Fig. 3. Since zG = 0.009 ≈ 0 this implies that the age of NGC 3516 within the Big Bang Universe (or NEEU) is probably ≥ 4.7 Gy. The redshifts of quasars are: 0.33, 0.69, 0.93, 1.40 and 2.10. Using this equation we estimated the quasar’s ages: 4.5 Gy, 4.1 Gy, 3.7 Gy, 3.3 Gy and 2.7 Gy. Therefore, it appears that the central galaxy NGC 3516 is formed after these quasars. In other words, NGC 3516 did not “deliver the five baby quasars”. Of course, this is true if eqn. (1) is valid.

Fig. 3. The five quasars related to the central galaxy NGC 3516. Redshifts are

written to the upper right of each quasar. The redshift of NGC 3516 is z = 0.009, see also {25}.

Age of Universe and Earth

A simple analysis of eqn. (1) shows that if zG = 0. AG = 4.7 Gy (lnAG = 1.55). This implies that the age of the observable Universe is about 4.7 Gy which is approximately the same as the age of the Sun and its planetary system. This value is about three times less than the generally accepted value (about 13.8 Gy) by astronomers for the Big Bang Universe. If this is true then no galaxy can be older than about 4.7 Gy. Indeed, most distant galaxies are between 0.2 Gy and 3.8 Gy old. This corresponds to their AG between 0.2 Gy and 3.8 Gy.⁵ However, this hypothesis is odd given that, as noted above, a vast majority of cosmologists believe that the age of the Big Bang Universe is about 13.8 Gy. We tentatively propose eqn. (1) is not valid for nearby galaxies of this universe. The dependence of AG on zG has probably another (defined or undefined) mathematical form so that for zG = 0, AG ≈ 13.8 Gy. However, it appears that the observable Universe is looking younger every day {26, 27}. Therefore, it is necessary to consider carefully the possibility that the observable Universe is much younger. 

The age of the Sun and its planets in NEEU would be also about 4.7 Gy. In addition, a recent study implies the galactic crashes influenced the Milky Way, and found that one of them coincided with the birth of our Sun around 4.7 billion years ago {28}. This kind of crash could occur in the Big Bang Universe and in NEEU as well. Therefore, it is conceivable that Sun and its planetary system in NEEU could begin to form about 4.7 Gy.

References

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 [1] We define nearby galaxies as those whose redshift zG is from 0.001 to 0.1 (or 0.001 ≤ zG ≤ 0.1) and with distant galaxies having zG > 0.1 {1, 2}. Of course, there is no sharp boundary between nearby and distant galaxies.

[2] It is worth noting here, that an analysis of major cosmological data by Valentino, Melchiorri, Silk {23} favors a closed universe although other evidence suggests the Universe is flat.

[3] The mathematical reason for this exponential dependence for the calculated ages of distant galaxies on their redshift is beyond the scope of this paper.

[4] Just to remind the reader that T is a symbol for basic dimension-time.

[5] We name the period after it the “galaxy-barren epoch”. The question now is why no or few galaxies were created in this epoch which has been going on for 10 Gy?

 

 

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