“The Moon Is New” (a book by John Dobson – Berbeo Publishing, 2008) has the potential to completely change our understanding of the universe. On page 14, it’s stated that “Einstein’s equation (E=mc squared) says that mass and energy are the same thing …” and “The c squared is just how many ergs are equal to one gram” (making the equation E=m). In pages 38-40, the book asks “… how many centimeters (are) equal to a second. That ratio, what is known in the trade as the speed of light, is about 30 billion centimeters to a second.” This question, and these pages, could lead to us saying “space and time are the same thing.” But as the book tells us on p. 38, “… time is the opposite of space in the geometry of this world …” and “… the space and time separations between (any) two events are equal and the total space-time separation is, therefore, zero.”

Suppose a star we are viewing is at a distance of 100 light years (this can be represented as +100). Since we see nothing as it presently is but as it was when the light left it, we are seeing the star as it was 100 years ago (represented as the opposite of space i.e. as -100). Repeated experimental verification of Einstein’s Relativity theory confirms its statement that space and time can never exist separately but form what is known as space-time. The space-time distance between us and the star is therefore 100 + (-100) i.e. 100-100 i.e. 0 and there is actually zero separation between us and the star’s gravity, heat etc.

So saying space and time are equivalent (“equal” or “the same thing”) is incomplete and, to be accurate, we need to say space-time separation is equal (and zero). This possibly explains cosmic unification and because the inverse-square law of famous English scientist Isaac Newton (1642-1727) says the force between two particles is infinite if the distance of separation goes to zero; also possibly explains the existence of an all-powerful, and super-intelligent (since those particles could be brain particles), God.

Is it also incomplete to say mass and energy are the same thing? Sure, we can add c squared to E=m. But we can think differently and think of E=m as 10=10 exponent 1. To make the equation totally complete, we must add something without altering the meaning e.g. by writing 10=10 exponent 1+0. Now we have E=m exponent 1+0. Where do we find 1’s and 0’s? In the binary language used by computers. Does this mean the Underlying Existence spoken of in the book is energy as the book suggests – but to be more specific, the energy of a computer (perhaps a quantum supercomputer) processing?

Maybe this quantum supercomputer resides in the same place as the purported Big Bang. Carl Sagan said there is no centre to the universe where the Big Bang could have taken place and initiated expansion. Therefore, the Big Bang (and for our purposes, the quantum supercomputer) would exist outside space and time in what we might call hyperspace. Page 34 suggests “… the rest mass of the proton (is) just the energy represented by its separation … from all the rest of the matter in the … universe.” Since that separation is zero, the universe must be unified with each of its constituent subatomic particles and those particles must follow the rules of fractal geometry being similarly composed of space and time and hyperspace.

I think the correct spin is that the classical formula for kinetic energy is the approximation, and that E=mc² is to be thought of as the correct formula for the energy of a body of mass m; then for low velocities, the increase in energy approximately agrees with the classical formula for kinetic energy.

But, as I said, this business of approximation makes me uneasy!

The question is: do both your and Terry Tao’s derivation involve approximation?