Hyperspace?

 It Ain't Necessarily So

It ain't necessarily so

It ain't necessarily so

They tell all your children

The devil he's a villain

It ain't necessarily so

Believing only what you can see locally is wrong

Pythagoras’ theorem is that any number can be expressed by two smaller numbers according to the relationship c²=a²+b². This was originally solved in Euclidean geometry, space with a curvature of zero, as the hypotenuse of a triangle. The relationship is true for a triangle in any geometry, including non Euclidean. It is just that under other geometries, the formula for the hypotenuse of a triangle is different because the sum of the angles in a triangle will be different. If the sum of the angles in a triangle is greater than 180 degrees, the curvature is positive and the non-Euclidean geometry is spherical. If the sum of the angles in a triangle is equal to 180 degrees, the curvature is 0, flat, and the geometry is Euclidean. If the sum of the angles in a triangle is less than 180 degrees, the curvature is negative and the non-Euclidean geometry is hyperbolic.

In Euclidean space, with a curvature of zero, the formula for a hypotenuse is

c=sqrt(a²+b²)

But this is only the formula in Euclidean space. The Flat Earth fallacy is because while locally space is Euclidean, the Earth is a large sphere, with a radius R, has positive curvature, and the formula for a hypotenuse which is its Great Circle Distance is

cos (c/R)=cos(a/R)*cos (b/R)

When the radius of the sphere is exceptionally large compared to c, a, and b, this is virtually indistinguishable from the formula in Euclidean space, i.e. locally space is Euclidean but over the entire surface of the Earth the geometry is spherical.

If the curvature is negative, hyperbolic, the formula is

c=ln(cosh(a)cosh(b)±sinh(a)sinh(b))

Mabkhout[1] proposed that the universe is globally hyperbolic and locally flat. If Einstein’s equations are solved for globally flat space, then dark matter and dark energy must be invoked in order  to be consistent with cosmic inflation immediately after the Big Bang;  to solve the paradox of rotating galaxies; to solve for the apparent difference in the size and the age of the observable universe; and to result in an expanding universe. If the universe is globally hyperbolic, then all of these issues can be resolved without invoking unseen dark energy or matter. Additionally gravity may be only an apparent force and the time dilation, length contraction, relativistic mass transform would not be the Lorentz transform, but would be

γ=1+ln(cosh(v/c)±sinh(v/c))

Just because something is true locally, it is not necessarily so that it is true globally.

 

 

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[1] Mabkhout, Salah A. "The infinite distance horizon and the hyperbolic inflation in the hyperbolic universe." Phys. Essays 25.1 (2012): 112.