Wrong mathematical formulas in
"Cosmology and the Zero Point Energy"
Key reference is the monograph by Barry
Setterfield: Cosmology
and the Zero Point Energy
(Natural Philosophy Alliance Monograph series, No.1, 2013, ISBN 978-1-304-19508-1).
The following is not a criticism of the physics in the book.
It concerns only the pure mathematics, quite apart from the physics.
It should be empasized in general that wrong mathematical formulas do no good to the argument.
Instead of wrong mathematics, it's better to have no mathematics at all!
At page 190 we find equation (65) and it is wrong, unless $K=1$ but it isn't because $K = 4.745 \times 10^9$ :
$$
\int \left[(1+T)/\sqrt(1-T^2)\right]dt = K\left[\arcsin T -\sqrt(1-T^2)+1\right] \qquad wrong
$$
That must be, with $C$ an arbitrary constant, for example $C=1$:
$$
\int \left[(1+T)/\sqrt(1-T^2)\right]dT = \arcsin T - \sqrt(1-T^2) + C \qquad correct
$$
As can be checked eventually by invoking a computer algebra system (MAPLE):
> int((1+T)/sqrt(1-T^2),T);
2 1/2
arcsin(T) - (1 - T )
It's a pity, because I have the impression that equation (65) is supposed to have some
significant consequences and because of this error some of the conclusions cannot be
properly drawn. The wrong formula is repeated as (99) on page 202 and elsewhere in the book.
But the main criticism concerns Appendix A: The ZPE and the redshift equation derivation.
A suitable Wikipedia reference is:
Redshift
Where we have that:
$$
1+z = \frac{\lambda_{\mbox{observed}}}{\lambda_{\mbox{emitted}}}
$$
A basic formula from the Wikipedia reference is the one for relativistic Doppler shift:
$$
1+z = \sqrt{\frac{1+v/c}{1-v/c}} = \frac{1+v/c}{\sqrt{1-(v/c)^2}} =
\left(1+\frac{v}{c}\right)\gamma \quad \mbox{with} \quad \gamma = \frac{1}{\sqrt{1-(v/c)^2}}
$$
This formula has a counterpart in equation (A) on page 410, where it reads:
$$
(1+z) = (1+T)/\left[\sqrt(1-T^2)\right]
$$
The two formulas would be identical for $v/c = T$. But instead, $T$ is interpreted as
an "orbital time" running backward ( hence $t = 1-T = $ time running forward ), scaled
in such a way that $T=0$ is at here and now and $T=1$ is at the origin of the universe
- hope I get that correct.
As has been said before, I shall not criticize the physics, only the mathematics.
Let us start with equation (5) on page 410:
$$
dN/dt = q-rN^2 = q-N^2/(Nt) = q-N/t \qquad (5)
$$
On the same page, we find that $0 \le t \le 1$ . So far so good (i.e. no mathematical
errors). Followed however by equation (6) on page 411:
$$
\int dN/dt = qt-N \qquad (6)
$$
For a less educated person, this might look like a mathematical formula, but it isn't.
It's an ill formed, completely meaningless formula. Equation (6) is no mathematics.
Equation (5) may be correct and can be integrated - it's actually an ordinary differential
equation (ODE). We need a few additional data, such as an initial condition for $N(t)$;
indeed that is available in the text as "the number that were originally
present" $N_1$ : $N(0)=N_1$. Furthermore we need to know what the term $q$ looks like.
This is clarified on page 411, where it can be inferred by combining formula (7) with
formula (9) that: $q(t) = 1/\sqrt{t}$ . Now we have all the ingredients necessary and
sufficient, mathematically, to solve the abovementioned ODE. And once we have done so,
there is no other way around. With a well known formula from Operator Calculus, we have:
$$
\frac{dN}{dt} + \frac{N}{t} = \sqrt{t} \quad \Longleftrightarrow \quad
\left[\frac{d}{dt} + \frac{1}{t}\right] N = \sqrt{t} \quad \Longleftrightarrow \\
e^{-\int dt/t}\frac{d}{dt}e^{+\int dt/t} N = e^{-\ln(t)}\frac{d}{dt}e^{+\ln(t)} N
= 1/t\,\frac{d}{dt}\,t N = \sqrt{t} \quad \Longrightarrow \\
\frac{d(t N)}{dt} = t^{3/2} \quad \Longrightarrow \quad tN = \frac{2}{5}t^{5/2} + C
\quad \Longrightarrow \quad N(t) = \frac{2}{5}t^{3/2} + C/t
$$
with C an arbitrary constant.
If you don't believe this, let a computer algebra system (MAPLE) confirm:
> dsolve(diff(N(t),t) = sqrt(t)-N(t)/t,N(t));
5/2
2 t
------ + _C1
5
N(t) = ------------
t
Here we have the boundary condition $N(0) = N_1$. But $t=0$ would give an infinite value
for $C/t$ with $C\ne0$ ; therefore we must conclude that $C=0$ . Then we have:
$$
N(t) = \frac{2}{5}t^{3/2} \quad \Longrightarrow \quad N(0) = 0 = N_1
$$
With other words: the proposed ODE has zero as its only solution.
So there is no Planck Particle Production at all, with this model:
$$
N(t) \equiv 0
$$
But we are not finished yet. On page 411 we have the mathematics-look-alike:
$$
dU/dt = \left[(\sqrt(t) + (N_1-N)\right] dt \qquad (11)
$$
It's again an ill formed, completely meaningless formula: equation (11) on page 411 is
no mathematics. But it goes on: equations (14) and (15) on page 412 are no mathematics.
And no, for $0 \le T \le 1$ we certainly cannot write:
$$
\sqrt(1-T) \cong \sqrt(1-T^2) \qquad (19)
$$
I think it's safe to conclude that these errors are enough to render the rest of the argument
in appendix A completely invalid. Effectively, there is NO derivation of the redshift
equation there. Though I really would have been delighted to see such an alternative.