[ExI] Dark mass = FTL baryons?

[Stuart LaForge]

In response to my question:

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

John Clark wrote:

>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.​

Actually that's beauty of the method I used. By taking the ratio of the
4-D contents of the lightcone to the contents of the 4-D ball, the
distances and times cancel out leaving a constant- no matter how big those
distances and times get. Check the limit at infinity.

In a way, you can think of 1/(6*pi) as the *maximum* fraction of
space-time that can be in your past when your lightcone catches up to the
expansion of the universe. You can never see it all, just a maximum of
5.30 percent of it.
​ ​
>[. . .]with a different distance formula
​ ​
>than the one Pythagoras gave:

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

The spacetime interval is irrelevant to the calculation I made because we
are dealing with the size of the lightcone. The spacetime interval is zero
because the lightcone is a null geodesic. We are only interested in X^2
+Y^2 +Z^2 - (c^2)T^2 = 0.

> if you didn't have that you'd be adding apples and oranges, or rather
>meters and seconds
​,​
>which would make no sense.
​
That's why I use Planck units. By setting c=1, it simplifies the math. I
am certain that even if you leave c in the equation, it will cancel during
the division of the lightcone by the hypersphere. Surely you have heard of
the utility of natural units?
​
​
>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.

Yes. And I have divided the 4D volume of Minkowski space by the 4D volume
of the same-sized Euclidean space. This works even both are infinite in
size and gives an answer that is a constant: 1/(6*pi)

>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
​
Yes. This is precisely why my result makes sense. Minkowski treats time
differently than spatial dimensions. There is only 1 time axis and 3 space
axes. So the vast majority (89.4%) of all possible spacetime intervals to
all possible events will be spacelike.

>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.

Yes, John, my calculation is only valid in flat spacetime. But the
universe seems relatively flat at the largest scales. Because of cosmic
expansion, the curvature of spacetime has an inflection point at a
distance proportional to the third root of the mass involved. You can
think of it as the point where the expansion of spacetime exactly balances
the gravitational attraction.

In any case, the universe's density kind of smooths out at a scale of
several billion lightyears. But spacetime not being perfectly flat at that
scale might be why my number is off from the measurements by +0.4 percent.

Stuart LaForge