söndag 29 september 2019

Hubble's Law and Exponential Expansion vs Many-Minds Relativity

The Hubble law
• $v=Hx$
with $x$ distance to a galaxy with velocity $v=\frac{dx}{dt}$ vs the Earth with $t$ time and $H$ Hubble's constant, is the most celebrated law of cosmology. Solving the equation $\frac{dx}{dt}=Hx$ with $x=1$ for $t=0$, we have
• $x(t)=\exp(Ht)$  and $v(t)=H\exp(Ht)$ for $t\gt 0$.
The galaxy thus appears to be receding from the Earth with both distance and velocity increasing exponentially in time. The Universe thus appears to be expanding exponentially. What is driving such a violent expansion?

Newton's 2nd Law takes the following form in Many-Minds Relativity MMR:
• $\frac{dv}{dt} = (1+v)F$,
where again $v$ is velocity and $F$ force and mass is normalised. With $F$ a positive constant the solution with $v(0)=0$ for $t=0$ is given by
•  $v(t)=\exp(Ft)-1$.
We thus see an exponentially increasing velocity from a constant force in MMR. We compare with the standard form of Newtons 2nd law $\frac{dv}{dt}=F$, which gives $v(t)=Ft$  with much slower  linear increase in time in the case of a constant force.

In MMR a constant expansion force thus appears to be compatible with observations of exponential expansion. In a Big Bang scenario it is thinkable that such a constant expansion force was active over an initial period of time resulting in observations of exponential expansion later.

We thus find exponential expansion in both observations and MMR with constant expansion force.
Is this a coincidence?

Of course, exponential expansion appears to require massive dark energy, but nobody knows anything about this new form of energy.

In MMR, the exponential expansion appears as an optical effect from using (composite) Doppler shifts to determine velocities, and so the exponential expansion could be more illusion than reality. If so, less dark energy would seem to be needed which could easy the task of finding an explanation of its apparent presence.