Short Communication - (2023) Volume 11, Issue 4

Department of Physics, Mahishadal Raj College, West Bengal, India

**Received: **03-Aug-2023, Manuscript No. jaat-23-110423;
**Editor assigned: **05-Aug-2023, Pre QC No. P-110423;
**Reviewed: **17-Aug-2023, QC No. Q-110423;
**Revised: **22-Aug-2023, Manuscript No. R-110423;
**Published:**
29-Aug-2023
, **DOI:** 10.37421/2329-6542.2023.11.266

**Citation:** Sarkar, Najmuj Sahadat. “Sarkar Singularity in Gravitational Cosmology.” J Astrophys Aerospace Technol 11 (2023): 266.

**Copyright:** © 2023 Sarkar NS. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

In this paper I have tried to derive an equation of my own using Professor Ray Chowdhury's equation which shows the constructing state of the universe just like the Ray Chowdhury equation. Hubble expansion rate and time act as special factors in my equation. In short it says we may get a singularity in the distant future not only in past.

Raychowdhury equation • Universe • Hubble construction rate • Time • Singularity • Penrose Diagram • Collapse

From Einstein's equation, we know that our universe is expanding. Indian physicist Amal Kumar Raychoudhury gave a solution to Einstein's equation, which is known as Raychoudhury equation. The equation is in the following form:

Here, = acceleration of the universe. R is system physical distance. G stands for the gravitational constant. ρ refers to the density of matter present in the system. p refers to the positive pressure of the substance present in the system. Now let's talk about what equation (1) actually means. The R/R term on the left represents acceleration. But the term on the right indicates something else. If both p and ρ are positive, i.e. greater than zero, then the value of (ρ+3p) will be much greater than zero. But according to the righthand side of equation (1) (ρ+3p) outside this term has negative sign. So [− (ρ+3p)] this total term will be less than zero. But R/R means Acceleration. R/R are not negative in general. So the entire term on the right-hand side represents attenuation. That is, if a system is considered, it cannot be stable. Due to increase in P and ρ, that means the system will start to compress or collapse [1]. The reason is also quite clear that increasing P and ρ means that the system will contract, which is what the Raychowdhury equation says. Take any system where matter is acting. Now positive gravitational pressure will act in the system in the presence of matter. After that the system starts to collapse due to presence of matter and squeezing into smaller and smaller volume. That is called Singularity we can easily analyze it from Pennors diagram. This is the key point of the Ray Chowdhury equation.

The equation says exactly the same thing. Although the equation I tried to explain is a different analogy, which analyses the singular state. Here I have used H and t. I got the idea from my paper "A Mathematical proof of the big crunch" (Journal of Physics and Astronomy, Vol 11/Issue 6). There was a term, which went something like this-

Ṙ= -3ηt^{2} × R Equation (2)

This means V=Ṙ=dR/dt. which is working as Hubble Velocity. As mentioned earlier, R is the physical distance of the system. t is the evolving time of the system. And η is a constant. Equation (2) looks a lot like V=HR or Hubble Law where

H= -3ηt^{2} Equation (3)

Derivative of equation (2) with respect to t;

R= dṘ/dt= -[6ηtR+3ηt^{2}Ṙ] Equation (4)

On the right side of equation (4) we get R, Common. dṘ/dt=-R[6ηt+3ηt^{2}Ṙ/R]

Or,

1/R(dṘ/dt)= -[6ηt+3ηt^{2}H] Equation (5)

Another thing, H= -3ηt^{2} Or, H^{2}=9η^{2}t^{4}

Or, H= ±3ηt²

But I am willing to work with positive values of H. Now substituting the
value of H instead of 3ηt^{2} in equation (5) we get-

(1/R) (dṘ/dt)= (6ηt+H^{2})

Equation (6) produces a singular situation. how? First, the term on the left side of equation (6) refers to acceleration [2]. That is, I am talking about the term R/R. Now come to the term on the right. Since; -

t>0 H>0

So (6ηt+H^{2})>0 will be.

But (6ηt+H^{2}) has negative sign before this term. That is, − (6ηt+H^{2}), so
the entire term will have a large negative value. As H and t increase, the term
on the right-hand side will have a large negative value. But the acceleration
term is not negative. This means that the system we consider is decelerating.
That is, the system is generally not stable. Due to deceleration, the system
will eventually form a singularity before contracting. In a way, H will act as the
Construction Rate in the far future. I think H and t their direction is always one
way [3]. But currently no singular situation is emerging. There is still time for
that to happen. My equation is not a special equation. This is a variant of the
Raychowdhury equation. Only I am giving more importance to H and T here. In
short, matter can indirectly affect the rate of expansion of a system. Since the
Hubble Expansion Rate will act as the Contraction Rate in the distant future,
we can expect a Possible Big Crunch. As said earlier H<0 cannot be. A long
time ago we got a singularity, just like that i think we will get it in the future
and this cycle will continue forever. Let's see if my equation is Dimensionally
Consistent. First we apply [η] to get the units of η. Here H=3ηt^{2} is a number of
3, which is not so important to us. Then we can say;-

[H]=[η][t]^{2}

Or, [η] = [H]/[t]^{2}

Or, [η]= km/s^{3}/Mpc

The units of the terms on the right-hand side of equation (6) will be,

[η][t]+[H]^{2}= km/s^{2}/Mpc+km^{2}/s^{2}/Mpc^{2}

= (km/s^{2}/Mpc)(1+km/Mpc)

= (km/s^{2}/Mpc)(Mpc+km)/Mpc

Now we can apply a convention. 1Mpc>>1km

From here it can be said, (km/s²/Mpc)(Mpc/Mpc) = km/s^{2}/Mpc

Now let's consider the term on the left side i.e. R/R.

[R]= [dv/dt]= km/s^{2} Again, [R]= Mpc So,

[R]/[R]= km/s^{2}/Mpc

That is, the unit on the right side and the unit on the left side are equal
to each other. My equation is completely Dimensionally Consistent [4]. Now I
want to find the value of η at H=3ηt^{2}. First, the current value of H= 70 km/s/
Mpc and the current value of t, t= 5 × 10^{17}s. So;-

H=3ηt^{2}

Or, η= H/3t^{2}

Or, η= 70/(75 × 10^{34})

Or, η= 9 × 10^{⁻35} Km/s^{3}/Mpc

In this case η is a constant, whose value is very close to zero (**Figure 1**).

My equation is also a 2^{nd} order differential equation like Prof.
Raychowdhury's. And just as his equation had a negative sign on the righthand
side, my equation has a negative sign on the right-hand side, which
is very useful for contracting systems. But the equation implies something
quite unstable. If there is enough matter in the system, the combined gravity
of the matter will compress everything back into a singular state. And to
explain all those conditions, Sarkar singularities are needed. Because the
hubble expansion rate will act as the constructing rate in the far future. This
construction is only for the presence of matter. This does not mean that H<0.
This is not possible, because H is always positive. But the presence of matter
is somehow generating this construction. Sarkar Singularity indirectly reveals
the truth of Raychowdhury equation. Just as we got the Singularity in the past,
we will surely get the Singularity in the distant future.

None.

None.

- Raychaudhuri, Amalkumar. "Arbitrary concentrations of matter and the Schwarzschild singularity."
*Phys Rev*89 (1953): 417. - Raychaudhuri, Amalkumar. "Condensations in expanding cosmologic models."
*Phys Rev*86 (1952): 90. - Raychaudhuri, Amalkumar. "Relativistic cosmology. I."
*Phys Rev*98 (1955): 1123. - Raychaudhuri, AmalKumar "Relativistic and Newtonian cosmology."
*Z Astrophys*43 (1957): 161.

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