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European Journal of Applied Sciences – Vol. 12, No. 4

Publication Date: August 25, 2024

DOI:10.14738/aivp.124.17327

Dialynas, T. E. (2024). Modified: Hubble’s Law. European Journal of Applied Sciences, Vol - 12(4). 431-436.

Services for Science and Education – United Kingdom

Modified: Hubble’s Law

Thanassis E. Dialynas

Department of Physics, University ofCrete,Heraklion, Crete, Greece

ABSTRACT

Based on natural grounds, we propose a (special - relativistic) “modification” of

Hubble’s Law [6, 7, 8, 9]. The advantage is that no galaxy can exceed the speed of

light, no matter how far it is. The disadvantage is that the equations appearing are

more difficult to handle and they are also non – invertible. The main tendencies of

the “theory” [8] remain the same.

Keywords: Hubble’s Law, "Modified" Hubble’s Law, Hubble’s constant, velocity of light,

“Maisie’s” galaxy.

INTRODUCTION

By Hubble’s Law [6, 7, 8], v = H * d and the necessary condition v < c, for H = 80 km/sec/Mpc,

a maximum radins of dmax = 12,2 Gly is expected. [ For H = 73,1 Km/sec/Mpc (dominant

value), dmax = 13,3 Gly].

The above “observation “[8] is purely phenomenological and does not come from a dynamical

theory (for instance General Relativity). Therefore, one can ask if finally, we have d > dmax (v

>c) from a dynamical theory what happens, i.e. can there be a velocity larger than light.

The usual answer given to this question is that the “observation” of velocities higher than c,

can happen and is due to the expansion of the space – time itself and not to the massive

objects themselves.

However, the light observed by spectroscopy comes from massive objects (galaxies) and their

velocity (if d >dmax exists) cannot exceed this of light.

We hope that the resolution of the “paradox” is given by the few next lines:

MAIN THEME

A) By Hubble’s law the velocity of a far galaxy is v = H * d where H is the Hubble constant

(dominant value):

H = 73,1 Km/sec/Mpc ≅ 2,37 * 10−18 sec−1

(1)

and d the distance of the far galaxy in appropriate units.

Therefore, if the mass of the galaxy is mG and is considered as one “compact” object its

(radial) momentum is

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Services for Science and Education – United Kingdom 432

European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024

PG = mG * H * d = mG * v (2)

However, the momentum of a mass (mG) in special relativity is

PG = mG * γ(v) v (3)

where γ = [1 − (

v

c

)

2

]

−

1

2 .

Subsequently equating expressions (3) and (2) we can have a transition from “Classical

Mechanics” to Special Relativity by demanding:

γ(v) v = H * d (4)

and of course, v = d = d

dt

(d).

Making the substitutions β = v

c

and χ = Hd

c

where

γ(β) * β = χ (5)

where γ(β) = = (1 − β

2

)

−

1

2.

Solving (5) for β we get

β = χ

√1+ χ2

< 1 (6)

for every χ greater than zero. (χ > 0)

Introducing the “dimensionless time” τ = H * t we have finally the o.d.e.

χ ’ = χ

√1+ χ2

(7)

with the initial condition χ (0) = χ0 – in general where χ0 ≥ 0. [Please note τ is the

“dimensionless time” t and χ the “dimensionless distance” d, of course χ’ = dx/dτ].

Equation (7) can be solved by the separation of variables (one indefinite integral) and after

four changes of variables one gets:

τ = f(χ) – f (χ0) (8)

where τ = H * t and χ = H∗d

c

and

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Dialynas, T. E. (2024). Modified: Hubble’s Law. European Journal of Applied Sciences, Vol - 12(4). 431-436.

URL: http://dx.doi.org/10.14738/aivp.124.17327

f(χ) = √1 + χ

2 +

1

2

ln |

√1+ χ2−1

√1+ χ2+1

| (9)

χ ∊ (0, +∞)

B) Let’s see some graphs:

(Fig I) [Ref 8]

(Fig II)

(Fig III) [Ref 8]