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User:Sesquihypercerebral/sandbox
Lange's Theorem
The fourth dimension has a residual spatial component. The size of the spatial component is to the size of the temporal component as the planck length is to the age of the universe. The spatial component is assumed to be orthogonal to the temporal component and to all the other spatial ordinates. This leads to the following adjustment of Einstein’s field equations. xº rather than being interpreted as t (i.e. pure time) should now be interpreted thus.
(dxº)²= (dt)² – (lp²/t²)(dt)²
Therefore the calculation of the line element in any metric would necessitate the following correction.
(ds)²= (ds0)² + (lp²/t²)(dt)²
This may explain cosmic acceleration. If it does then it obviates the need for the cosmological constant to have a non-zero value.
Glossary
xº The fourth dimension. Formally interpreted as pure time.
lp The planck length.
ds The line element.
ds0 The line element before any adjustment is made.
t Time.
Where a bare t occurs it should be interpreted as cosmological time, i.e. the age of the universe.
Preamble.
In "The Grand Design" page 172 it is posited that the fourth dimension began as a dimension of space. This being a consequence of the no-boundary condition, which allows the universe to avoid having a singularity at the beginning of time. It being implied that the fourth dimension went on to become the dimension of time that we are all familiar with today.
I therefore posit the following. If the fourth dimension began as a dimension of space then it should still to some extent be a dimension of space. On the simplest of assumptions the size of its spacial component would have remained the same while the size of its temporal component would have increased linearly with time. Again on the simplest of assumptions the size of its spacial component at the begining of time would be the planck length. The obvious consequence of this is that over any period of time there would be an additional expansion of space, and futher, the amount predicted is in good agreement with the amount found for cosmic acceleration.