We continue our journey to find an answer to the question whether the universe could be infinite in space. With some rather elementary considerations we have come up with an equation for the dynamics of the cosmos, namely the so called Friedman equation for a general scaling factor a(t), which appears to control cosmic distances.

In the 3rd post of this series we have discussed how we can integrate the Friedmann equation. We arrived at some well known conclusions for a universe filled with a kind of homogeneous gravitational dust. We saw that the Friedmann equations implied a dynamic evolution of the cosmos – with a starting point of infinite density in the past. We also found that the scaling factor a(t) stretches or in some cases also may shrink distances between elements in the dust during certain phases of three qualitatively different scenarios of the cosmological evolution.

We have formally related the scaling factor to a so called Hubble factor – which appears to be constant throughout space at a given cosmological point in time. However, I have not yet provided a direct interpretation of the Hubble factor which would link it to observations. This is one of the topics in this post.

Previous posts:

• Can the universe be infinite? – III – basic solutions of the Friedmann equation
• Can the universe be infinite? – II – Friedmann equation and scaling factor by Newtonian considerations
• Can the universe be infinite? – I – The cosmological principle

From simple assumptions to more solid physical principles

We have derived the Friedmann equation by putting the observer at the center of a coordinate system with spherical polar coordinates. One reason for this approach was a certain plausible variant of the cosmological principle (see the 1st post). This variant assumes isotropy for all observers at all times (for big enough scales) as a given pre-condition for the analysis of cosmological dynamics. To find a valid solution of the Friedmann equation, we pretty much “ad hoc” introduced a scaling factor which was time-dependent, only. So far, this approach has only been justified by the fact that it worked pretty well. At the same time we felt that a voluntarily chosen “center” of observation would be difficult to combine with the cosmological principle.

Additional physical principles …

What we have not done, yet, is to show that the assumptions of isotropy and of a position independent scaling factor are consistent with transformations between different observer systems. Regarding an isotropic velocity field varying linearly with radial distance, I have actually left it to the reader to show this.

While such an exercise appears to be a purely mathematical one, we actually have to combine it with more physics to gain safer ground in a Newtonian cosmology. More physics? What physics?

Well, in this post, we will apply homogeneity and isotropy conditions onto the continuity equation and the Euler equation of the cosmic fluid. And even more important: We will check under what conditions these equations hold in the coordinate systems of different observers at some distance from each other in the cosmic flow. Our guide line will be the functional invariance of the fluid equations.

I assume that both the continuity equation and the Euler equation are known to the spectrum of educated engineers, which is a target group of this post series.

As a nice side effect of our considerations, we will again find a time dependent expanding flow of the cosmic substrate which looks the same everywhere (!) at a given point in time. Leading to the conclusion that the Big Bang was an event in the past which happened simply everywhere in the space of a possibly infinite universe. We will, however, have to allow for “velocities” of observer systems faster than the speed of light.

Note 1: I will closely follow a derivation of basic cosmological equations in Newtonian physics by following invariance principles presented by E. Rebhan in his book on Relativity and Cosmology [1]. See chapt. 15 in particular. All credit belongs to Prof. Rebhan.

Note 2: Following [1], I call the moving substance of cosmic matter the cosmic substrate. This leaves room for characterizing it later by different components and related different equations of state.

Why is our endeavor not a totally trivial one?

There are two reasons while our new approach to describe the cosmic evolution by Newtonian physics is not trivial.

Reason 1: When applying transformations of physical equations between the coordinate systems of distant observers, we have to show that an equal representation of the cosmological principle is found by all observers (at any distance from us) – at least for a defined certain distribution of velocity and force components in space. What distributions? Well, the hope is that the requirement of isotropy and the demand for a functional invariance of physical equations when switching between observer systems would spit the dependencies on spacial coordinates out. I.e.: To get an answer, we combine an invariance principle regarding the functional form of differential equations in different coordinate systems with other symmetry requests.

Reason 2: The Euler equation includes accelerations due to force fields (per mass unit). This makes the whole thing particularly interesting. Why? Because in Newtonian physics, there appear so called “fictitious forces” (also called inertial forces) in systems accelerated or decelerated vs. an observer at rest or in constant motion (inertial system). As we already know from the solution of the Friedmann equations that the cosmic substrate is decelerated by the general gravitational pull of the expanding matter distribution, this is a non-trivial matter. Obviously, the interplay of gravitational forces and fictitious forces must adjust in a peculiar way between distant observers – when one observer O₀ regards himself force free, but regards the other one as being influenced by gravitational deceleration. The latter actually was exactly what we did when we started our derivation of the Friedmann equation.

Zero forces at any observer’s point of view (?)

Let us call the observer, which we used to derive the Friedmann equation, “O₀“. He claims that at his position, namely at the origin of her system of spherical coordinates, no force is exerted on him. Homogeneity combined with isotropy allows him to assume spherical shells of matter around him with a zero net gravitational force at their center. O₀ regards himself moving force free with the overall flow of a homogeneous cosmic substrate, which appears to be isotropic from his perspective.

To fulfill the cosmological principle, we have to show that the velocity field in combination with acting forces adjusts for another observer O_A at some distant position r_A from O₀, such that O_A does not feel any local forces either. r_A(t) is some specific vector at a given point in time t. Physically, a transformation of the Euler equation should give us a confirmation of this logical requirement as soon as we impose isotropy and request of functional form invariance for different observers.

At the same time a radially symmetric velocity field with a certain, but a priori unknown functional dependency of the velocity on radial distance should be found in the coordinate system of O₀ and be reproduced in the system of any distant observer O_A. We start our derivations with the latter point.

A spatially constant Hubble-factor for a given time in Newtonian Cosmology

We request the invariance of the equations describing the variation of matter density and momentum density with time when changing perspective from our original O₀-observer to another distant observer O_A. Let us first gather information about the variation of velocity of the cosmic substrate described by O₀ and O_A.

Isotropy from O₀‘s point of view is equivalent to the assumption that all vector fields for relative velocities v, forces per mass f and gravitational acceleration g are varying in a radially symmetric way, i.e. with radial distance r and time t, only. We mark our own position, i.e. of O₀, at the origin r=0 of a coordinate system [CS] “S₀” with spherical polar coordinates. When we write vectors in bold letters, we have in S₀:

Note that v refers to elements of matter moving relative to O₀ who sits at the origin of S₀. Eq. (1) guarantees that O₀ sees the same variation of v in all radial directions with distance r. We are interested in the functional form of v(r, t). Note that there cannot exist a homogeneous and at the same time isotropic vector field at all points in space. What variation with ||r|| is required to guarantee isotropy also for other observers?

Let us call another observer’s CS (at some changing position r_A(t)) S’_A. According to eq. (1), system S’_A obviously moves along a radial trajectory in S₀. This trajectory can be described by a specific radial vector r_A changing with a respective velocity

Let us mark all quantities in S’_A with a prime ‘. Time is global and the same everywhere in a Newtonian universe. So:

Because of the cosmological principle we expect the velocity of some matter element at r’ to fulfill the following relation in S’_A :

Eq. (7) demands that v'(r’,t) must have the same functional form as v(r,t). In Newtonian physics, relative velocities in two observer systems moving against each other just follow the law of vector addition. So:

and thus

r_A and r are independent vectors. So, all of the factors must become zero and therefore we must conclude:

The radial dependency of the isotropic velocity field is given by a constant with respect to the radial distance from the origin in both observer systems:

The expansive motion of the cosmic substrate is going on everywhere and for every observer with the same scaling factor H(t) of the velocities for distant matter elements of the cosmic substrate. We still have to identify a relation of H(t) with the radial scaling factor a(t) governing previously derived equations. But it is worth to note:

• The Hubble factor jumps out of invariance and symmetry demands – already in Newtonian physics.

Note that eq. (12) has the direct consequence of velocities bigger than the velocity of light beyond a certain distance r_H(t)

r_H is called the radius of the Hubble sphere. Obviously, this radius is time dependent. In what way will be a question to be answered in future posts.

Freely floating observers and equations to describe the cosmic flow

Let us consider our movement (as O₀) with the cosmic fluid. We claim that no gravitational force is acting on us. Due to the assumption that we have a homogeneous matter/energy distribution around us, we can safely assume that any pressure gradient due to gaseous contributions will disappear, too.

As all force fields go to zero at r=0 in S₀, we are freely moving with the general cosmic flow. This is consistent with the fact that v(r=0) = 0. The latter tells us that at this point there is no relative motion of the cosmic fluid vs. the observer.

What physical rules govern the cosmic flow?

A basic assumption would be that no new physical matter is generated out of the blue. We refrain from assuming any uncontrolled sources of matter on average cosmic scales. (Note, however, that certain stationary cosmological models actually introduce such a source of matter.)

We also assume that only force fields change the momentum density in a given piece of volume. Let us call the force per unit of mass (i.e. an acceleration) “f“. f typically would include gravitational acceleration g and potentially also other forces. Even fictitious forces appearing when we transform observations into accelerated systems. If we disregard further forces we could be tempted to include fictitious forces even in the local g of freely falling systems.

Continuity equation and Euler equation

All in all: We assume that the basic continuity equations for mass-density ρ_m and momentum-density are fulfilled. With P_m giving the pressure, we have the standard continuity equation for matter (m) and the Euler equation, which results from a continuity equation for momentum density (ρ_m*v) in the non-relativistic case.

The Nabla operator applied in a scalar product (indicated by the bullet “•”) in (15) is equivalent to the divergence. The gradient applied to v works on every component of the vector. Due to homogeneity we set the gradient of the pressure to zero. Note that the gradient ∇’ fulfills

because of equ. (8).

Transformation of the continuity equation

Let us first transform the continuity equation. Using (6), defining the specific velocity vector given at r_A as

and using the homogeneity of ρ_m, we get:

We have used the fact that the divergence of a constant vector is zero. As expected, there is no problem with the continuity equation. Together with (12) we get a first result:

It expresses mass conservation in an expanding or contraction universe for H > 0 or H < 0, respectively. We have not seen this relation in our previous derivations! We do not know the time dependency of H(t), yet, but even if it were a constant, we see the ingredients for an exponential behavior of ρ.

Transformation of the Euler equation and radial variation of accelerations

The transformation of the Euler equation requires some preparation. For convenience reasons let us write

and indicate what we keep constant during differentiation by an additional hint behind a vertical line. Now, let us write down the left side of the Euler equation in S’_A, replace terms with unprimed ones and see what we must require to get functional invariance for the full Euler equation

. Because of (9) we have

where we used

The complicated terms include fictitious forces due to the transformation.

The next step is a bit complicated as we have to think about the velocity changes with time at constant r‘. The point is that we need to go to tie derivatives at fixed coordinates r to be able to use the relations conditions in S₀. So, how do we change to fixed r?

The point is that as we keep r‘ fixed changes during an infinitesimal time difference dt at a position
r = r_A + r‘
from the perspective of O₀ (in S₀) not only come about due to the time variation of v at r. The also come about, because S’_A is moving its position in the spatial vector field. For constant r‘ the distance passed during dt is given by v_A dt both at the origin of S’_A and at (constant) r’ in S’_A. For infinitesimal distances the change in velocity due to the velocity field variation in space is given by the projection of the gradient of v onto v_A. Therefore, we can write

and eq. (20) becomes

What about the last term? Well, with a consideration similar to the one above we can change to time derivatives for constant r, this time at r=r_A and get

Why I have written the expression in this form will become clear in a minute. However,

So, the Euler equation in S’_A gives us

We have used equ. (16) in the last steps. This has a message for us: The forces must transform like the velocities !

Variation of isotropic acceleration vectors with radial distance from the observer

We have shown that the Euler equation is invariant regarding a transformation from S₀ to S’_A, if all forces f per mass unit (i.e. accelerations, including the gravitational acceleration g) follow

Now, we can again apply the CP and follow the discussion for the velocity above, but for the accelerations to find

At the origin of S’_A given by

we therefore find

This means that also the other observer with his reference frame S’_A floats freely with the cosmic substrate! As would any other element of matter. Fictitious forces of the deceleration of the cosmic flow just compensate the gravitational force there. For force fields (per mass unit) linearly varying with radial distance with a common and only time-dependent proportionality factor from the origins in all coordinate systems !

Logically, the only component of gravitational force on matter elements remaining for an observer in S’_A is the gravitational acceleration imposed by local spherical mass shells around O_A and the origin of S’_A. You can see this by applying our results between three observers.

There are two points to keep in mind:

1. Working with a gravitational acceleration field and a velocity field, both varying linearly with radial distance from an observer and and an only time-dependent proportionality factor, has been shown to be consistent with a form of the cosmic principle that explicitly includes isotropy.
2. We can add any other isotropic acceleration varying linearly and only time dependent with radial distance to the gravitational acceleration without becoming inconsistent with Euler equation for a dynamic universe in a Newtonian cosmology.

This gives our whole approach, which we followed in previous posts in a much more intuitive way, a lot more substantial strength due to an overall consistency between the descriptions of the cosmic flow by different distant observers within this flow.

Gravitational acceleration

Indeed, when we apply eqs. (22), (23) for the gravitational acceleration, we find (with equ. (17)) :

This makes also the equation for the gravitational force per mass unit invariant between our observer systems.

Due to our special settings for velocity and force fields, the systems S₀ and S’_A appear to work similar to inertial systems – at least locally. Still, a strange feeling remains. And we will indeed find later on that General Relativity tells us that the Friedmann equations describe the universe in a non-inertial system.

Nevertheless, for the time being, we have to take the Friedmann equations and its solutions seriously in Newtonian cosmology.

Simplified control equations for the Newtonian cosmic development

Let us turn back to the equations (15), (16), (26) governing the development of our cosmic matter flow. With our results and

we get

with f(t) including isotropic accelerations varying linearly with radial distance or just equal to g(t).

Questions to the reader

• What is the definition of an inertial system?
• Do force free bodies moving with constant velocity relative to an observer O_A also move with constant relative velocity to observer O₀?

Conclusion

In this post we have shown that the cosmological principle together with symmetry assumptions (homogeneity, isotropy) and the demand of functional invariance of the continuity equation and the Euler equation between observers in the cosmic substrate impose strong conditions on the distribution of scalar quantities as well on the velocity and acceleration fields. A linear and only time dependent variation of the velocity and acceleration vectors with radial distance and an isotropic orientation along the radius vectors is required and then given in all observer systems.

The requirements following our invariance principles are fortunately consistent with an isotropic gravitational acceleration (with linear dependency on distance), which we have used in previous posts to derive the Friedmann equations. We can even add other other isotropic accelerations with linear dependency on radial distance without breaking transformation consistency.

We have also found that the structure of the vector fields is consistent with observers just co-moving with the expanding or contracting cosmic substrate – without any forces accelerating them versus the general cosmic flow.

Remarkably, a Hubble factor (constant over space at a given point in time) jumped out of the resulting equations for the vector field.

In the next post we will use the degree of freedom regarding force fields to derive extended Friedmann-Lemaitre equations including a cosmological constant from our intermediate results above.

Stay tuned …

Literature

[1] E. Rebhan, 2012, “Theoretische Physik: Relativitätstheorie und Kosmologie”, Springer Verlag Berlin, Heidelberg