Relativistic Light Clock Experiments: Time Dilation or Time Contraction?

 

Relativistic Light Clock Experiments: Time Dilation or Time Contraction?

 

Pavle I. Premović, Laboratory for Geochemistry, Cosmochemistry&Astrochemistry, 

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

Abstract

Time dilation of Special relativity can be easily derived from the thought experiments using the traditional light clock. However, time contraction is also possible to derive but using the “novel” light clock described in this communication. The question is which of these two clocks is right? 

Keywords: Light clock, thought experiment, Special relativity, time dilation, time contraction.

Introduction

One of the concepts of Special relativity (SR) is time dilation which depends upon the second postulate of SR that the speed of light c (= 2.99792×10⁸ m sec^-1) is the same in all inertial frames of reference [1]. According to this theory if ΔT₀ is the (proper) time interval of an event that occurs at the same position in an inertial frame, then the (improper) time interval ΔT of the same event has a longer duration as measured by an observer in an inertial frame that is in uniform motion relative to the first frame. Of course, this initial choice of which of frames is stationary and which is moving is arbitrary and it could be vice versa. It appears that time dilation is successfully tested by - muon experiment [2] and the experiment of synchronizing two atomic clocks [3].

Many physics textbooks demonstrate time dilation using a device known as a light clock. In this note, we will consider a set of thought experiments for time intervals ΔT₀ and ΔT employing the two different light clocks. Of course, you may find some of the following derivations in many elementary physics texts.

The traditional light clock consists of two-plane parallel mirrors M₁ and M₂ facing each other at a distance d apart as in Fig. 1a. The lower mirror M₁ has a light source at the center that emits a photon (or light signal/pulse) at 90 degrees in the direction of mirror M₂. For the sake of simplicity, we will consider in this note only the time interval needed for the photon to travel from mirror M₁ to mirror M₂. For the light clock at rest this is the (proper) time interval ΔT₀ = d/c.

Fig. 1. The traditional light clock: (a) no relative motion and (b) the clock moving at speed υ.

Now allow the same clock to be moving with a relative speed υ horizontally in the direction of the positive x-axis, Fig. 1b. The photon will now travel a larger distance D, and thus will take a longer time: the (improper) time interval ΔT = D/c. Elementary SR shows that ΔT₀ and ΔT are related by the following formula: ΔT = ΔT₀/√(1-υ²/c²) where 1/√(1-υ²/c²) is the Lorentz factor or the time dilation factor. Thus, the stationary observer measures time dilation for the moving classical light clock.

Let us now perform the thought experiments using a somewhat different (“novel”) light clock. This clock is similar to the traditional light clock except that the two plane-parallel mirrors M₁ and M₂ not facing each other, and they are at a distance D away, as shown in Fig. 2a. The proper time interval required for the photon to reach then M₂ is now ΔT₀ = D/c.

Fig. 2. The “novel” light clock: (a) no relative motion and (b) the clock moving at speed υ.

In the next thought experiment, we assume that the “novel” light clock moves, as the previously classical light clock, in the direction of the positive x-axis with the same speed v. The stationary observer observes that the photon travels from the mirror M₁ to the mirror M₂ following the path shown in Fig. 2b. She/he now measures the improper time interval ΔT = d/c. ΔT₀ and ΔT are now related with the following expression: ΔT₀ = ΔT√(1-υ²/c²). In other words, the stationary observer measures time contraction with the “novel” light clock. 

Thus, the two light clocks give different results. The classical light clock shows time dilation but the “novel” light clock time contraction. The question is now which of these two clocks is relativistically right?

 

References

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

[2] J. Bailey, K. Borer K, Combley, et al., Measurements of relativistic time dilation for positive and negative muons in a circular orbit, Nature 268, 301–305 (1977).

[3] J. Hafele R. Keating, Around the world atomic clocks: observed relativistic time gains, Science,  177, 167–168 (1972).

 

 

 

A Simple Way to Show Space-Time Expansion

 

A Simple Way to Show Space-Time Expansion

 

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

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

Abstract

A time interval in the reference frame of nearby and distant galaxies is proportional to the distance between these galaxies and the Earth and is shorter if measured in Earth’s time frame. In other words, when space expands (or stretches), time also proportionally expands (or “stretches”) or, in the language of relativity, both dilate. Simply speaking, we are dealing with space-time expansion (or stretching).

Keywords: Earth, galaxy, age, distance, time, space-time expansion.

Introduction

The standard cosmology of the Universe is based on the General theory of relativity.  It states that the Universe was created about 14 Gy in an event called the Big Bang. From this event space and time started expanding and this expansion is still occurring creating galaxies, stars, Sun, Earth, planets and life. According to this cosmology, nearby and distant galaxies[1] recede from the Earth because the Universe is expanding at a constant rate. The light emitted by these galaxies is redshifted and is called a “cosmological redshift”. 

The best estimate of the age of the Earth (and the rest of the Solar system) is 4.55 ± 0.05 Gyr. This estimate for Earth's age is based on radiometric dating of iron meteorite fragments from the Canyon Diablo. Premović {2} concluded that the time of Earth’s formation is a more reliable zero-time standard for the Universe than the Hubble zero-time (see below).

All distant galaxies were formed before the Earth. Premović {2} used for this type of galaxies the term before the Earth and labeled it as BE. In rough analogy to the historical term: before Christ or BC. For galaxies created after the Earth, he applied AE in rough analogy to the historical term: after the death of Christ or AD. A galaxy can be created BE or AE. All distant galaxies were formed in BE. Let us first consider the BE galaxies.

Derivations and Discussion

Suppose a BE galaxy recedes emitting light. Denote with G its “present” position in respect to the Earth and with D_G the distance between them[1], Fig. 1a. Denote with G_E the position of this galaxy in respect to the Earth when the Earth was born. Of course, the light emitted by the BE galaxy would travel to the Earth as long as it has been since the birth of the Earth, i. e. during a time equal to A_E. In equation form

D_G = cA_E    … (1)

Fig. 1: a) position of the galaxies BE (1a) and AE (1b)

in respect to the Earth formation about 4.55 Gy.

where c (= 3 × 10⁵ km sec^-1) is the speed of light in a vacuum. (Obviously, the ratio D_G/A_E is equal to this speed and constant with increasing distance). This appears regardless of how far the BE galaxy is from the Earth. So, Earth’s observer concludes that the age of Earth A_E in the BE galaxy's reference frame is higher than the radiometrically measured age of the Earth, 4.55 Gy. In other words, a time interval in the reference frame of the BE galaxy is proportional to the distance between this galaxy and the Earth and is longer than measured in Earth’s time frame. Expressed in the language of cosmology, when space expands (or stretches), time also proportionally expands (or “stretches”). Simply speaking, we are dealing with space-time expansion or stretching. In the language of relativity, there is the relative time dilation between the identical clocks in the frames of reference of Earth and of a BE galaxy. In other words, the clock on a distant BE galaxy would run slower than an identical clock on Earth. According to the standard cosmology, the farther away the galaxy is, the older it is, the larger its redshift and the slower its time.

 

We know that the frequency of light emitted by the BE galaxy, ν, is inversely proportional to time and proportionally decreases with the distance between the Earth and this galaxy. Consequently, the energy of this light also decreases because it is proportional to its frequency. (In equation form: E = hν where E is light’s energy and h is Planck’s constant). Since the wavelength of light, λ = c/ν, increases as its frequency decreases, the light in the BE galaxy is redshifted. 

Previous discussion is based on zero time of the Universe is the birth of the Earth. However, if we assume that the time of birth of the galaxy G is zero time of the universe then D_G = cT where T is the time of the present day Earth relative to that zero time. However, as far as we know, the time of birth of distant galaxies is highly debatable.

Suppose that a distant galaxy is created when Earth is created and at a position relatively close[1] to the Earth in the then Universe. According to eqn. (1), now the distance of this previous galaxy’s position to the Earth is about (4.55c =) 4.55 Gly.  If the distance of the Earth to a galaxy is larger than 4.55 Gly then this galaxy is the BE-type. If this distance is smaller than 4.55 Gly, the galaxy is the AE-type.

Consider now an AE galaxy. Most of the previous consideration for BE-type galaxy is also applicable to this type of galaxy.

Let G_E to represent a position in respect to the Earth when this galaxy is born and G represents its position after 4.55 Gy, Fig 1b. Now, the light emitted by this galaxy would travel to Earth as long as it has been since the birth of this galaxy, i. e. A_G. In expression

D_G = cA_G    … (2).

The starting point of many papers related to D_G is the Hubble’s law equation which is valid only for nearby galaxies: cz_G = H₀D_G, where H₀ (= 7 × 10^-10 y^-1) is Hubble’s constant representing the constant rate of the Universe expansion caused by the expansion of space-time itself. After some rearrangement, we can write this equation as follows

D_G = z_Gc/H₀.

The terms 1/H₀ and c/H₀ in this equation are Hubble’s age (or Hubble’s zero-time) and Hubble’s length, respectively. For the above value of H₀ their values are about 14 Gy and 14 Gly.

Combining eqn. (2) and eqn. (3) we have

A_G = z_G/H₀    … (3).

Accordingly, if z_G ≤ 0.1 then A_G ≤ 1.4 Gy and if z_G > 0.001 then A_G > 0.0014 Gy.

The direct distance measurement method so-called the “megamaser” method has demonstrated its capability for precise distance measurement. But it appears this method is suitable for very few nearby galaxies -“megamaser” galaxies [1, and references therein]. For the present case, we select five of these galaxies whose redshift z_M and distance D_M from the Earth (determined by the megamaser method) is known, Table 1. The peculiar motion of these galaxies is negligible. Since their distance from the Earth is less than 4.55 Gly all of them are created in AE. This is consistent with our previous report {2}. The calculated age of the selected “megamaser” galaxies A_M is given in Table 1.

Table 1. Selected “megamaser” galaxies

Name of galaxy	zM*	DM [Gla]**	AM [Gy]***
NGC 1052	0.004930	0.065	0.07 (0.065)
UGC 3789	0.010679	0.162	0.15 (0.162)
NGC 6323	0.02592	0.349	0.37 (0.349)
NGC 5765B	0.02754	0.411	0.39 (0.411)
NGC 6264	0.03384	0.447	0.48 (0.447)

 *See {1} and the references therein. **Calculated using eqn. (2). ***Calculated using eqn. (3). In the round brackets calculated using eqn. (2). 

Finally, there is a possibility that the speed of light c is not constant and depends on the distance of a nearby or a distant galaxy to the Earth. One of the nearby galaxies NGC 1052 has a rather small distance D_G from the Earth: 65 Mly, Table 1. Introducing this value into eqn. (1) we estimate that in this

case, the speed of light c would be about 4300 km sec^-1. Of course, this result is complete nonsense that excludes this possibility.

Scientists estimate that in our galaxy alone (Milky Way), there are billions of planetary systems similar to our Solar system. On the other hand, in the Universe there are billions of galaxies. If the speed of light c is universal, we propose that the equations (1) and (2) are universal can be applied to all planetary systems of all galaxies in the Universe. Moreover, Hubble’s law is probably also universal so then eqn. (3) is also universal.

References

{1} P. I. Premović, Distant galaxies in the non-expanding (Euclidean) Universe: the light speed redshift. The General Science Journal, December 2021.

{2} P. I. Premović, The age of the “megamaser” galaxies in the Big Bang Universe. The General Science Journal, December 2021.

---

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

[2] Just to remind the reader that at that moment Earth’s observer “Adam” and a bit later “Hubble” came into a “cosmic contact” with nearby or distant galaxy. Emission of light by the BE galaxy occurred in the past and this light has reached the Earth when “Hubble” was able to measure its redshift.

[3] Its distance to the Earth should be large enough, which would prevent its recession motion from being affected by the gravitational forces of the Milky Way. For example, the nearby Andromeda galaxy is about 2.2 Mly away from Earth and yet it is exposed to the gravitational influence of the Milky Way.

The Age of the “Megamaser” Galaxies in the Big Bang Universe

 

The Age of the “Megamaser” Galaxies in the Big Bang Universe

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

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

The Big Bang theory is widely regarded as the leading explanation for the origin of the Universe. According to this theory, all the matter and energy in the current Universe emerged from a singularity. The Universe then expanded rapidly to the present version and this expansion of space itself (or better of space-time itself) is still occurring.

 

Hubble’s law states that the recession speed vG of distant galaxies is proportional to their distance DG from the Earth. This law can be expressed as follows

vG = czG = H0DG … (1)

where c (= 3×10⁸ m s^-1) is the speed of light, zG is the galaxy’s redshift and H0 (= 72 km sec^-1 (Mpc)^-1 = 7×10^-11 y^-1) is the Hubble constant. According to cosmology, H0 represents the constant rate of cosmic expansion. The above formula is valid for distant galaxies with zG ≤

0.1. For zG > 0.1 this law is no longer relevant and depends on the particular cosmological model.

Most cosmologists consider that when the Big Bang occurred, time began. The time scale for the age of the Universe and its galaxies is set by the Hubble time 1/H0. For a Hubble constant H0 of 72 km sec^-1 (Mpc)^-1, as given by current measurements, the Hubble time 1/H0 is around 13.65 Gy. However, the true value of the Hubble constant is still debatable and its estimate varies between 50 km sec^-1 (Mpc)^-1 to 100 km sec^-1 (Mpc)^-1. Thus, the Hubble time varies between 10-20 Gy and hence the beginning or zero time of the Universe.

The best estimate of the age of the Earth (and the rest of the Solar system) is 4.55 ± 0.05 Gyr. (This value is derived from several different lines of evidence). Therefore, it appears that Earth’s formation is a more reliable zero-time standard for the expanding Universe and the non-expanding Universe than the Hubble zero-time. In other words, the birth of the Earth can be considered as “coordinated astronomical time”[1] (CAT) for each of the universes. This would imply that each of them is “geocentric in time”.

Most of the distant galaxies were formed before Earth's creation. For this type of galaxies, we will use the term before the Earth and label it: BE. This is a rough analogy to the historical term before Christ or BC. For example, the (active-type) galaxy APM 08279+5255 was formed at about 7 Gy BE. The age of this galaxy (~ 2.1 Gy) was determined by measuring the Fe/O ratio [1 and references therein].

A much smaller number of distant galaxies were formed after the birth of the Earth. For this type of galaxy, we will use the term after the Earth and label it: AE. This is a rough analogy to the historical term Anno Domini (in the year of the Lord) or AD.

In what follows, we will only consider the galaxies whose peculiar motion is insignificant.

 

In the further text, we will consider only galaxies that are formed during BE. In Fig. 1, E and G represent the relative position of the Earth and galaxy G when the Earth is born. Let us denote with D the distance between these two astronomical objects at that time. After Earth’s formation and to the present, galaxy G moves away from it at an average recession speed vG for an additional distance of 4.55vG.[2]

Fig. 1. Relative positions of the Earth and galaxy G and their distances.

At present, we measure the total distance DG between the Earth and the galaxy G which can be expressed with a simple formula

DG = D + 4.55vG

where DG and D are expressed in Gly, the unit of 4.55 is Gy. For convenience, this equation can be written as

D = DG – 4.55vG     … (2).

Denote with z the redshift of a galaxy at the distance D from the Earth. Elementary physics and Hubble’s law [eqn. (1)] imply that this galaxy travels a distance 4.55vG with an average recession speed vG = (czG + cz)/2. Of course, this formula is valid only for zG and z lower than or equal to 0.1 (or ≤ 0.1). A bit of algebra applied to eqn. (1) yields z = zG(D/DG). After simple mathematical manipulation, we can write eqn. (2) in the following form

D = DG(2DG – 4.55czG)/ (2DG + 4.55czG)    … (3).

Of course, if galaxies are formed during BE then

DG – D = 4.55c(zG + z)/2.

Substituting z with zGD/DG and plugging the value of c, and after a bit of      algebra, we get

D = DG – 2.275zG(1 + D/DG)    … (4).

Here, the unit of 2.275 is Gly.

As one of the direct distance measurement methods, the “megamaser” method has demonstrated its capability for precise distance measurement of distant galaxies, i. e. galaxies beyond our Local Group. But it appears this method is suitable for very few of distant galaxies -“megamaser” galaxies. For the present case, we select five of these galaxies whose redshift zG and distance DG from the Earth (determined by the megamaser method) are known, Table 1. The peculiar motion of these galaxies is negligible.

Table 1. “Megamaser” galaxies.

Name of galaxy	RedshiftS zG	DG [Mly] Megamaser	D [Mly]1
NGC 1052	0.004930	65 {2}	50
UGC 3789	0.010679	162 {3}	120
NGC 6323	0.02592	349 {4}	250
NGC 5765B	0.02754	411{5}	300
NGC 6264	0.03384	447 {6}	320

 aSFrom Simbad (Astronomical database, Centre de données astronomiques de Strasbourg,

Université de Strasbourg) and 1calculated using eqn. (3).

Plugging into eqn. (3) DG and zG, given in Table 1, we calculated the distance D of these galaxies to the just-born Earth, Table 1.

A simple calculation shows that the average value of D/DG of selected galaxies is about 0.72. Since zG of these galaxies is much smaller than 0.72 (Table 1) eqn. (4) can be approximate to

                             D = DG – 2.275zG(1 + D/DG) = DG – 1.72 × 2.275zG = 1.7 Gly – 3.9zG.

Since zG of the selected galaxies is much smaller than 0.1 (Table 1) then the minimum D should be ~ 1.3 Gly. In contrast, the calculated D for the “megamaser“ galaxies is < 0.350 Gly therefore they are formed after the Earth or during AE.

Of course, the above approach can be applied to any galaxy with zG ≤ 0.1 if we know its distance DG from the Earth.

References

{1}   Y.-H. Sanejouand, A simple Hubble-like law in lieu of dark energy.arXiv:1401.2919[astro- ph.CO]. (2015).

{2}   P. van Dokkum , S. Danieli1, Y. Cohen , et al., The Distance of the dark matter deficient galaxy NGC 1052–DF2. Astrophys. J. Lett. 864: L18 (2018).

{3}  M. J. Reid, J. A. Braatz, J. J. Condon, et al., The megamaser cosmology project. IV. A direct measurement of the Hubble constant from UGC 3789. Astrophys. J., 767, 154-165 (2013). Updated by Braatz.

{4}  C. Y. Kuo, J. A. Braatz, K. Y. Lo, et al. The megamaser cosmology project. VI. Observations of NGC 6323. Astrophys. J. 800, 26-35 (2015).

{5}  F. Gao, J. A. Braatz, M. J. Reid, et al., The megamaser cosmology project .VIII. A geometric distance to NGC 5765b. Astrophys. J., 817, 128-145 (2016).

{6}  C. Y. Kuo, J. A. Braatz, M. J. Reid, et al., The megamaser cosmology project. V. An angular-diameter distance to NGC 6264 at 140 Mpc. Astrophys. J. 767, 155-168 (2013).

[1] In rough analogy with the world standard "zero" time: coordinated universal time (UTC).

[2] Fig. 1 would be drawn by the observer “Adam” created when Earth was created.