# Natural Boundaries of Exponential Grow

From: Christian Zielinski (zielinski@transhumanismus.org)
Date: Mon Jan 09 2006 - 08:36:36 MST

Hello.

Only an idea of the limits of exponential grow of a post-singularity
civilisation:
Let's assume an exponential grow of the technological progress of a type
II-III civilisation.
With growing technology the energy consumption of such a civilisation
should grow too.
Let's assume it for the sake as a linear relation, so that ("~" stands
for proportionality):

Available technology ~ Energy consumption

If the available technology grows exponential, then the same is valid
for the energy consumption:

Energy consumption ~ e^(a t) (with a as scaling factor)

The maximal available energy for the civilisation is all the included
usable energy in the area where the civilisation
has settled. Let's assume it further as a sphere. If the energy is
nearly (in averaged) homogenous
in space, than follows:

Available energy ~ Settled space

With the developed technology it should be possible to expand near the
speed of light.
So the volume of the sphere in space, which the civilisation has already
settled an
is using the energy, grows with the expansion in all three directions of
space:

Settled space ~ (radius of settled sphere)^3

The radius grows at the constant speed of light (r = c t). If we
substitute we get:

Settled space ~ t^3

Because the available energy is proportional to the settled space, it
follows that:

Available energy ~ t^3

So we have an civilisation which energy consumption grows exponential
with time while the
available energy follows only a cubic law:

Energy consumption ~ e^(a t)
Available energy ~ t^3

As the first of this curves is growing faster, we can find a point in
time where the
energy consumption would be greater than the maximal energy which can be
used.
Therefore the technological progress has to slow down. Am I right?

Regards,
Christian Zielinski

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