# [ExI] Dark mass = FTL baryons?

John Clark johnkclark at gmail.com
Wed Aug 16 16:01:47 UTC 2017

```On Wed, Aug 16, 2017 at 7:25 AM, Stuart LaForge <avant at sollegro.com> wrote:

> *​> ​What fraction of reality, by 4-D volume, lies inside​ ​of our past
> lightcone?*

​
We know the Big Bang happened 13.8 billion years ago but it seems to me
that to answer your question I'd have to know how big our future lightcone
is, and I don't know what that is
​, for all I know it's infinite.​

> ​>* ​*
>
>
>
> *I modelled reality as a 4-D hypersphere or 4-sphere​ ​in Planck units
> given by ​ ​S4. t^2+x^2+y^2+z^2 <= R^2*

*​ ​ where t is time and R is the radius of the 4-sphere. *

​
The
​ ​
Pythagoras Theorem
​ ​
works for 4-D Euclidian space but that's not what Einstein used, he used 4-D
​ ​
Minkowski
​ ​
space (sometimes called
​ ​
Hyperbolic
​ ​
space)
​ ​
with a different distance formula
​ ​
than the one Pythagoras gave:

R^2= X^2 +Y^2 +Z^2 - (c^2)T^2

Where
​
R is the distance between events in spacetime
​ and​
X,Y, and Z are the distance in space in those dimensions and (c^2)T^2 is
the distance in the time dimension. The c is the speed of light and the c^2
is a conversion factor
​ to get the units right;​
if you didn't have that you'd be adding apples and oranges, or rather
meters and seconds
​,​
which would make no sense.
​

​
​
It can be proven that
​ ​
in
​ ​
flat space mathematically there are only 2 possible definitions of distance
such that distance remains
​ ​
the same for all observers
​ ​
in any frame of reference, the one in
​ ​
the
​ ​
Pythagoras Theorem
​ ​
and the one above. Both are equally valid mathematically but
​ ​
Minkowski
​ ​
works better for physical reasons. Minkowski treats time differently than
space and that's why the minus sign is in there, if it were a plus as in
regular old 4-D Euclidian space then causality would not be
​ ​
preserved
​ ​
and a event could happen before the thing that caused it, and that would be
unphysical. And the above formula is only an approximation, when Spacetime
becomes highly curved, as it is around massive stars,
​  ​
4-D
​ ​
Tensor Calculus must be used
​ ​
to find the distance between two events in spacetime.

John K Clark
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