the Helland universe (as of 2 November 2023)
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Abstract.[/size]
The Helland universe analyzed herein is defined by the metric tensor field
Mike Helland proposed on
2 November 2023, which replaced
his previous best guess of 26 October. Using the same techniques I used to
analyze that 26 October version of the Helland universe, this note derives the following consequences of the 2 November version of the Helland metric:
- The Helland universe is static in these three senses:
- Its metric tensor field does not change over time.
- Its curvature tensor field does not change over time.
- Its distributions of mass-energy and pressure do not change over time.
The second and third of those are of course immediate consequences of the first.
- The Helland universe is geocentric in the following sense: Its spatial slices are spherically symmetric around a central point whose physical properties distinguish it from all other points of space.
- The Helland universe has a curvature singularity at that central point.
- If the cosmological constant is zero, then the mass-energy density is zero throughout the Helland universe.
- If the cosmological constant is positive, then the mass-energy density is negative throughout the Helland universe. That density is most negative at the center of the Helland universe, dropping to zero as Hr approaches c.
- If the cosmological constant is negative, then the mass-energy density is positive throughout the Helland universe. That density is greatest at the center of the Helland universe, dropping to zero as Hr approaches c.
- At the center of the Helland universe, the pressure is negative infinity, regardless of the cosmological constant. That means the center of the Helland universe, where Mike Helland imagines us to be, is uninhabitable.
- If the cosmological constant is zero or negative, then the pressure is negative throughout the Helland universe.
- If the cosmological constant is positive, then the pressure is negative throughout a region centered on the Helland universe's center, but becomes positive as Hr approaches c.
- The infinitely negative pressure at the center of the Helland universe means the center of the Helland universe is undergoing unimaginably rapid inflation/expansion, contradicting the static universe asserted by the Helland metric form.
- The static universe implied by the Helland metric is not consistent with Einstein's general theory of relativity, because a universe with the Helland universe's distribution of pressure would not be static.
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Outline.[/size]
- Abstract.
- Outline.
- The Helland universe in spherical coordinates.
- The Helland metric form
- Ricci curvature tensor
- Ricci scalar
- Mass-energy density
- Pressure
- The Helland universe in Cartesian coordinates.
- Summary of conclusions.
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The Helland universe in Cartesian coordinates.[/size]
The Helland metric form
On 2 November,
Mike Helland stated this new guess at a metric form for the Helland universe.
Working on this a little more with some other people, and it seems to be the right line element is:
http://latex.codecogs.com/gif.latex? ds^2 = -(1-\frac{rH}{c})^{2} c^2 dt^2 + dr^2 + r^2 d\theta^2 + r^2\sin^2 (\theta) d\psi^2
That makes sense intuitively that a tiny change in time multiplied 1/(1+z) equals the tiny change in space. The tiny change in space times (1+z) equals the tiny change in time.
Which is nice because that simplifies:
http://latex.codecogs.com/gif.latex?g_{tt} = -(1-\frac{rH}{c})^{2} c^2
http://latex.codecogs.com/gif.latex?g_{tt} = -\left(\frac{c- rH}{c}\right)^{2} c^2
http://latex.codecogs.com/gif.latex?g_{tt} = -(c- rH)^2
That metric form implicitly restricts the radial coordinate r to the range 0 ≤ r < c/H.
Note well that the H of
Mike Helland's metric form has absolutely nothing to do with the Hubble parameter or Hubble constant of mainstream cosmology, in which H expresses the rate at which the universe is expanding. The Helland metric describes a universe that is not expanding, which implies H=0 if the meaning of H is taken to be either the Hubble parameter or the Hubble constant.
In the Helland metric, H is an arbitrary positive constant. The author and sole proponent of Helland physics wants the value of H to be about 14 billion years, so he can pretend the Helland universe might have something to do with the Hubble radius, but the analysis and conclusions of this note are totally indifferent to the value of H.
Components of the metric tensor are coordinate-dependent. With the spherical spatial coordinates of the Helland metric form, the off-diagonal components are zero and the diagonal components are
gtt = − (c − Hr)2grr = 1
gθθ = r2gφφ = r2 sin2 θ
Ricci curvature tensor
Components of the Ricci curvature tensor are coordinate-dependent.
Mike Helland has used the
EinsteinPy software to calculate those components for the spherical coordinate system he used in his metric form. In that coordinate system, all off-diagonal components of the Ricci tensor are zero.
The diagonal components are
Rtt = − 2 H (c − Hr) / r
Rrr = 0
Rθθ = H r / (c − Hr)
Rφφ = (H r sin2 θ) / (c − Hr)
Note that R
tt converges to negative infinity as r approaches zero, while all other components of the Ricci tensor remain finite.
That implies a curvature singularity at r=0.
That singularity is not just a coordinate singularity, as can be confirmed by converting the Helland metric form to use Cartesian coordinates.
Ricci scalar
The Ricci curvature scalar does not depend on the coordinate system. For the Helland universe, the Ricci scalar is
R = giiRii = 4 H / ((c − Hr) r)
The Ricci scalar is positive throughout the Helland universe, and converges to positive infinity (i.e. increases without bound) as r approaches zero.
In other words, the curvature blows up at r=0. That is a curvature singularity.
That curvature singularity at r=0 says an observer situated at the center of the Helland universe is experiencing unimaginably extreme physical forces. The next two subsections analyze the nature of those forces.
Already, however, we can see that the center of the Helland universe is uninhabitable.
Mass-energy density
Einstein's field equations say
Rμν − ½ R gμν + Λ gμν = κ Tμν
where κ = 8πG/c
4 is Einstein's gravitational constant, with a numerical value of about κ = 2.077 × 10
−43 s
2/(kg m).
Taking both μ and ν to be t, we get the equation for the mass-energy density T
tt:
Rtt − ½ R gtt + Λ gtt = κ Ttt
Using the values of R
tt and R computed from the g
μν components by the EinsteinPy software, we find that
so the first two terms cancel. The mass-energy density of the Helland universe is therefore proportional to
If the cosmological constant Λ is zero, then the Helland universe contains no matter or energy.
That surprised me.
Yesterday I claimed the Helland "metric form says he and we reside at the singularity of a black hole." I was mistaken.
As seen in the next subsection, the center of the Helland universe is even more bizarre than a black hole.
Pressure
Einstein's field equations give us this equation for the pressure T
rr:
Rrr − ½ R grr + Λ grr = κ Trr
Plugging in the values of R
rr, R, and g
rr tells us the pressure at a distance r from the center of the Helland universe is proportional to
− 2 H / ((c − Hr) r) + Λ
The center of the Helland universe is experiencing infinitely negative pressure, regardless of the value of the cosmological constant.
Gravitationally, positive pressure acts a lot like mass-energy and would work against expansion of the universe. Negative pressure acts like negative gravity, and therefore contributes to expansion of the universe.
The infinitely negative pressure at the center of the Helland universe acts like a repulsive gravitational field of infinite strength.
The center of the Helland universe is therefore uninhabitable.
Because of that unimaginably negative pressure, the central portions of the Helland universe should be expanding at an unimaginably rapid rate.
But the Helland metric form says the Helland universe is static. That means the Helland metric form is not consistent with general relativity.
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The Helland universe in Cartesian coordinates.[/size]
As I explained 8 days ago, all spherical coordinate systems have coordinate singularities. One of those coordinate singularities occurs at r=0.
Because of that coordinate singularity at the central and most interesting point of the Helland universe, I did not bother to use the spherical coordinates of the Helland metric form to calculate components of the Ricci scalar. My first move was to transform the Helland metric to the Cartesian coordinate system (τ, x, y, z) via
τ = c t
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
In that coordinate system, the Helland metric is
ds2 = − (1 − f(x, y, z))2 dτ2 + dx2 + dy2 + dz2
where f(x, y, z) is the function of spatial coordinates defined by
0 ≤ f(x, y, z) = (H/c) sqrt(x2 + y2 + z2) < 1
Because my primary interest in Helland physics is as a source of exercises in first-year calculus, I calculated the 64 coordinate-dependent Christoffel symbols by hand, finding that only 9 of the Christoffel symbols are non-zero: Γ
ttx = Γ
txt, Γ
tty = Γ
tyt, Γ
ttz = Γ
tzt, and
Γxtt = (H/c)2 ((1 − f(x, y, z)) x) / (f(x, y, z))
Γytt = (H/c)2 ((1 − f(x, y, z)) y) / (f(x, y, z))
Γztt = (H/c)2 ((1 − f(x, y, z)) z) / (f(x, y, z))
To calculate components of the Ricci tensor, it is necessary to calculate the partial derivatives of those nonzero Christoffel symbols. Upon doing so, I saw that three of those partial derivatives blow up at the center of the Helland universe, signalling a probable curvature singularity.
Instead of boring you (and me as well) by typesetting those partial derivatives, I'm going to offer the intuitive reason for that curvature singularity.
In Cartesian coordinates,
gττ = − (1 − f(x, y, z))2
To focus our attention on what happens in the vicinity of x=0, we can set y=z=0, for which the equation above becomes
gττ = − (1 − (H/c) sqrt(x2))2
It would be easy to make the mistake of thinking sqrt(x
2) = x, but that isn't true when x is negative. The correct relationship is sqrt(x
2) = |x|, where |x| is the absolute value of x. Thus
gττ = − (1 − (H/c) |x|)2
when y and z are zero.
Although −(1 − (H/c) |x|)
2 is continuous at x=0, it is not differentiable at x=0. The graph of −(1 − (H/c) |x|)
2 is thorn-shaped, with a sharp point at x=0. As x approaches x=0 from below (i.e. x < 0), the slope converges toward negative infinity. As x approaches x=0 from above, the slope converges toward positive infinity. The slope at x=0 is therefore undefined, which is to say the derivative with respect to x is undefined at x=0.
The mathematical definition of a differentiable manifold requires the manifold to be sufficiently smooth, where "sufficiently smooth" means all of the derivatives you might need to calculate are well-defined at all points of the manifold.
The Helland universe is not sufficiently smooth at x=y=z=0, which is to say it is not sufficiently smooth at r=0.
Mathematically, that means events with r=0 must be excluded from the manifold.
So the center of the Helland universe, at which
Mike Helland believes himself to reside, is not a part of the mathematical model he defined on 2 November by stating his most recent guess at a Helland metric form.
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Summary of conclusions.[/size]
- According to the Helland metric form, the Helland universe is static in these three senses:
- Its metric tensor field does not change over time.
- Its curvature tensor field does not change over time.
- Its distributions of mass-energy and pressure do not change over time.
- Those consequences of the Helland metric form should not be trusted, because the Helland metric form implies other consequences that are not consistent with the general theory of relativity.
- In particular, a universe with the extreme negative pressure implied by the Helland metric form would not be static.
- The Helland universe is geocentric in the following sense: Its spatial slices are spherically symmetric around a central point whose physical properties distinguish it from all other points of space.
- The Helland universe has a curvature singularity at that central point.
- If the cosmological constant is zero, then the mass-energy density is zero throughout the Helland universe.
- If the cosmological constant is positive, then the mass-energy density is negative throughout the Helland universe. That density is most negative at the center of the Helland universe, dropping to zero as Hr approaches c.
- If the cosmological constant is negative, then the mass-energy density is positive throughout the Helland universe. That density is greatest at the center of the Helland universe, dropping to zero as Hr approaches c.
- At the center of the Helland universe, the pressure is negative infinity, regardless of the cosmological constant. That means the center of the Helland universe, where Mike Helland imagines us to reside, is uninhabitable.
- The infinitely negative pressure at the center of the Helland universe means the center of the Helland universe is undergoing unimaginably rapid inflation/expansion, contradicting the static universe asserted by the Helland metric form.
- The static universe implied by the Helland metric is not consistent with Einstein's general theory of relativity, because a universe with the Helland universe's distribution of pressure would not be static.