SHRINKING THE UNIVERSE

[George Hammond]

I have a simple Relativity question:

Suppose spacetime is being observed using a lattice of
clocks and rulers in the usual fashion. Then suppose
unknown to the observer all the clocks speeded up and the
rulers got longer such that the APPARENT METRIC of the
spacetime became multiplied by a SCALE FACTOR, a(t), such
that:

ds^2 is replaced by a(t) ds^2

Note that this would LOOK LIKE the familiar Hubble expansion
of the universe with scale factor a(t) even though of course
it physically would not be.

Let me interject here that this problem has a very real
practical application; namely this:
As a person ages from age 0 to 18 years old his physical
size and mental speed (intelligence) both increase
approximately linearly. From the SUBJECTIVE standpoint your
personal ruler (your foot size say) and your personal clock
(your mental speed in bits/sec) both increase dramatically
during the first 18 years of life. This means,
subjectively, that the universe gets dramatically smaller
and slower as you grow up.

Now, my question is this:

1. Strictly mathematically speaking is this describable as
a time varying "curvature" of subjective spacetime ?

2. Is either the Riemannian Curvature or the scalar
curvature expressable as a function of a(t)
as it is in the familiar Hubble expansion ?

3. Or are all the curvature invarients equal to zero
because the metric is conformally Euclidean, even
though the metric is (apparently) expanding ?

Note: Einstein's field equations are not involved here
because the expansion is only APPARENT and not physically
real.

I just want to know if such an APPARENT EXPANSION is
correctly mathematically described as a "curvature" of
"subjective spacetime"?

Thanks in advance for your expertise

[Phillip Helbig---undress to reply]

In article <jjssq79o0st5jhbdr...@4ax.com>, George Hammond

<george_...@verizon.net> writes:

> Suppose spacetime is being observed using a lattice of
> clocks and rulers in the usual fashion. Then suppose
> unknown to the observer all the clocks speeded up and the
> rulers got longer such that the APPARENT METRIC of the
> spacetime became multiplied by a SCALE FACTOR, a(t), such
> that:
>
> ds^2 is replaced by a(t) ds^2
>
> Note that this would LOOK LIKE the familiar Hubble expansion
> of the universe with scale factor a(t) even though of course
> it physically would not be.

What you describe would correspond to an apparent CONTRACTION.

> As a person ages from age 0 to 18 years old his physical
> size and mental speed (intelligence) both increase
> approximately linearly. From the SUBJECTIVE standpoint your
> personal ruler (your foot size say) and your personal clock
> (your mental speed in bits/sec) both increase dramatically
> during the first 18 years of life. This means,
> subjectively, that the universe gets dramatically smaller
> and slower as you grow up.

SUBJECTIVELY, you can have one or the other, but not both. Either one
feels that one's personal scales (foot, intelligence) are the same and
the universe changes or vice versa, but not both.

> Now, my question is this:
>
> 1. Strictly mathematically speaking is this describable as
> a time varying "curvature" of subjective spacetime ?

SPACETIME curvature (as opposed to space curvature) is proportional to
the ACCELERATION of the universe (the constant of proportionality is
negative). So, without acceleration, there is no SPACETIME curvature.
(SPACE curvature is a different matter.)

> 2. Is either the Riemannian Curvature or the scalar
> curvature expressable as a function of a(t)
> as it is in the familiar Hubble expansion ?

Probably not. In "normal" cosmology, the curvature of space depends on
the density and the cosmological constant (which also determine the
expansion history); these are not included in your model.

> 3. Or are all the curvature invarients equal to zero
> because the metric is conformally Euclidean, even
> though the metric is (apparently) expanding ?

Note that in "normal" cosmology there can be flat (i.e. Euclidean) space
which can expand.

> I just want to know if such an APPARENT EXPANSION is
> correctly mathematically described as a "curvature" of
> "subjective spacetime"?

No; see above.

I don't think there is much point in trying to pursue this analogy
further. In any case, you need to understand the difference between the
curvature of space and the curvature of spacetime.

(In his 1993 Saas-Fee lecture, Alan Sandage recounts the following story
(footnote 4 on page 9; ISBN 3-540-58913-9) which I have shortened a bit:
The young Sandage meets the old Lema?tre. The latter asks the former if
he can really understand the curvature of space. After thinking a bit,
Sandage says "no" and Lema?tre suggests that he should change fields.)