## The Gravitational Field in a Fluid Sphere of Uniform Invariant Density According to the Theory of Relativity. by Georges Lemaitre 1928 - Physics Thesis

This document contains the Physics Thesis "The Big Bang" by Georges Lemaitre written in 1925
To order to read the article select: https://dspace.mit.edu/bitstream/handle/1721.1/10753/36897534-MIT.pdf
• The text in italics is copied from the article.
• Immediate followed by some comments

### 1. Introduction

The gravitational field within a fluid sphere of uniform density has been the object of many investigations, specially by Schwarzschild, Nardström and De Donder and was considered as a solved problem until Eddington made some fundamental objections against the solution of these authors.
It is interesting that the universe is treated as a fluid, as sphere filled with uniform distribution of particles, which influence each other.
What is also interesting that the issue was considered solved.
The density which was supposed to be uniform in these works was the component T44 of the energy-tensor of the matter; Eddington contends that the true representation of the density is not T44 but the associated invariant T.
The most important part is the physical meaning of T44 and T.
If this conception is exact, the solution of Schwarzschild is but an approximation and a solution is required for which the invariant density T is uniform throughout the sphere.
The sphere meaning the universe.
Let us consider with Eddington a fluid formed of a great number of moving particles. The fluid will be incompressible if a given closed surface contains the same number of particles whatever may be the pressure. The velocities of the particles and the intensity of the electro-magnetic field which acts between them will be generally modified by a change of pressure.
The problem is, if this description is realistic, compared with the physical reality, which is not a closed surface. Better would be to mention the word volume.
The question is also to what extend an electro-magnetic field can be compared with a gravitational field.

### Page 2

Although Eddington insists chiefly on the objection we have spoken of, he indicates another objection which does not seem to have real foundations: The condition of fluidity is not expressed in natural measures and so would also require modification.
The condition of fluidity may be written
 ``` b b T = -g p (a,b, = 1,2,3) (1) a a ```
where the pressure p is an invariant. It is true that this relation is not tensorial for any change of the coordinates, but only for a change of the three spacial coordinates x1,x2,x3.
Schwarzschild's solution really refers to a perfect fluid but the density of the fluid (defined as Eddington has shown it must be defined) is not uniform.
The question is if the distribution of all the particles in the total universe is also uniform.
We have
 ``` 1 2 3 4 4 T = T + T + T + T = T -3p (2) 1 2 3 4 4 ```
and the natural units used in this formula are such that when the densities are expressed in gramms per c.c., the pressure p represents its value in C.G.S. units divided by the square of the velocity of light. Therefore, the pressure is small in any practical case.
The main question is: what has the velocity of light to do with the physical behaviour of this fluid i.e. distribution of particles?
The second question is about pressure: small in relation to what?

### 2.4. Uniform invariant density page 10

For an Uniform invariant density,
1. When the density is greater than that af an Einstein's universe of the same cosmological constant the radius of the sphere increases with the central pressure) passes through a maximum for a finite value of the central pressure and then diminishes until this pressure tends to infinity.
2. The density may be smaller than the density of an Einstein's universe of the same cosmological constant, but it cannot be smaller than about one half of this density.
Then the material sphere may fill up the whole space which has the same radius as an Einstein's universe of the same cosmological constant.
3. When the density approaches its minimum, the pressure curves have a minimum which corresponds to the boundary of a maximum sphere with free space outside of the sphere. The gradient of pressure vanishes at the boundary and the gravitation force is a repulsion outside of the sphere.
This describes the evolution of the universe in three different possibilities
It is not clear which one is the most realistic.

### Reflection 1 -

The article discussed is the physics thesis of Georges Lemaitre the degree of Doctor of Philosophy in the Department of Physics of the Massachusetts Institute of Technology
The article consists of three parts:
2. Equations
3. Numerical Computations. Specific these computations are a gargantuan effort and show an impressif result.

1. In part one the concept "invariant density" is discussed. The concept is, my understanding, a mathematical concept. The question is why should the universe be symmetrical? Of course considering the universe symmetrical, that means, my understanding, uniform in all directions, makes a mathematical description simpler. At the other side using observations, to demonstrate this, is impossible, because the present unviverse can not be observed. All what we can observe lies in the past. The further away the more in the past.

### Reflection 1.1 - Invariant density

The concept invariant density is often used in this document.
My interpretation that it can be used in two occasions: Mathematical and Physical
For a physical interpretation read this: https://en.wikipedia.org/wiki/Invariant_(physics)
The question is to what extend, from a physical point of view, discussions about transformations make sense. The same with the concept invariant. My understanding is that both are mathematical oriented.
A typical sentence in the mentioned document is this sentence: "Other quantities, like the speed of light, are always invariant."
It does not make sense to call the speed of light invariant when this is always the case
The point is that the speed of light is considered constant. The only experiment that supports this (?) is that when two lightsources meet each other and that when at the moment when they meet, both sources, each emit a lightflash (in all directions), that the sphere of both lightflashes is one and the same. That means that IMO, that only the center of this sphere can be considered at rest. This cannot be the same as the center of the sphere emitted in the coordination of both lightsources, which move relative to each other.

For a concept of density a similar issue exists, assuming density of mass throughout the whole universe. It makes sense to consider only one reference frame.

### Reflection 2 - Lemaitre - The shape of the universe

The following document shows the shape of the universe: https://www.bigbangroute.be/pages/nl_BE/lemaitre_premonstreit
The translation of the text:
 Het Universum is niet te groot voor de mensheid, het overstijgt noch de mogelijkheden van de wetenschap noch de capaciteiten van de menselijke geest Georges Lemaître (1894-1966) The Universe is not too big for humanity, it neither transcends nor exceeds possibilities of science nor the capabilities of the human mind Georges Lemaître (1894-1966)
The translation of the text below graph:
 Tijdsafhankelijke evolutie van de schaalfactor van het Heelal in functie van de kosmologische constante voor een ruimte met positieve kromming, door Georges Lemaître, 1927 Time-dependent evolution of the scale factor of the Universe as a function of the cosmological constant for a space with positive curvature, by Georges Lemaître, 1927

### Reflection 3 - The evolution of the universe - etc

Starting point of any phylosophical discussion that at present there exists an universe. A second point is that the universe is constantly changing as a result of events. Events can be classified as large or small.
The evolution of the universe from start, to present, to end, assuming there are three of such periods, are all physical processes.
The most import characteristic of each of these physical processes are that they are different chemical reactions, which will change in due course. The explanation of these reactions lie within the details of these same reactions.
To understand them, require a more detailed description of each of what is involved.
They have to do with objects composed of the physical elements and with stable configurations, which always involve multiple objects. These minimal stable configurations consists of at least 2 objects. The most important physical factor are forces, specific the physical interpretion of these forces.
One very important point is that the evolution of any process or reaction has nothing to do with any human involvement nor with our mind. Concepts like a coordinate system, time, a clock, which are largely human related are not important to understand the physics of the evolution of the Universe. The same with mathematics. Mathematics allows us to describe the evolution using numbers, based on observations and measurements. Mathematics does not tell us what is physical involved. The guiding factor of the evolution are chemical reactions. The overall strategy is that the measurements should not influence what is measured.

The current physical state of the universe is supposed to be expanding. That means that the outer shell is expanding.

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Created: 13 December 2023

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