[AntonioLao]

Both hole arguments were used to justify the desired linearity of all applicable equations of their corresponding theories.

Einstein used the hole argument to arrive at the general covariance (analogous to a modern view of canonical transformations in classical dynamics) of his nonlinear field equations of general relativity, while Dirac used the hole argument to justify the existence of negative energy states from a nonlinear relativistic wave equation to a linear relativistic wave equation.

Their “linearized” equations do not explicitly contain any force terms. For Dirac’s, it is dominated by the positive and negative solutions for kinetic energy, which led to the concept of intrinsic angular momentum or spin and the existence of elementary anti-particles; while for Einstein’s, it’s the energy-momentum tensor, which led to the concept of BB theory and the expanding universe. Note that the tensor product of energy and momentum is not an ideal nonlinear form of , but where the left and right terms are identically zero and only the middle term is positve definite and can be zero when p=0 or mv=0, which is the same as v=0 for the existence of the BB singularity.
In TQS, nonlinear equations of square of energies containing forces can satisfy a generalized equation for spacetime quantization, which contains both Dirac’s and Einstein’s as linear and almost linear approximations.



Expanding by using Lagrange’s identity gives



and



where

and are the two distinct nonequivalent topologies of quantized spacetime.
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[michellemfry]

Is it true that anything in the universe can be mathematically squared? Is there anything that cannot be mathematically squared? I once had some strange thought about (everything)^2. It looks silly now. But is there anything that cannot be mathematically squared?

[baudrunner]

Watch those Margaritas, not to mention those roll-yer-owns.

Here's a really, really, really deep thought : (R E A L I T Y)³

That should keep you for awhile...

And oh, yes, go ahead and square whatever you like.

[AntonioLao]

Originally Posted by michellemfry

But is there anything that cannot be mathematically squared?

Anything in reality should be capable of being squared. By squaring, we are really removing the directional property of the thing squared. When we square distances as by the Pythagorean theorem we are able to add them without worrying about the signs of pluses or minuses. When we square energy we don't need to explain the concept of negative energy, which was a serious problem for Dirac but he got out of this quagmire by postulating the existence of antiparticles. When we square force we change it into zero, which is neither repulsive or attractive. Therefore, the existence of force implies that it must not be squared.

[michellemfry]

You break my heart. All that talent, and you're intolerant of ignorance as well. I know my limitations. As a student, I had as many A's as I did F's. I was never consistent. You have PHD's up the wahzoo don't you.

I can tell you right now there is no end of making fun of my math skills. It is my weak spot. From an A in Trig to a D in Calc in one year. Linear Algebra was no better and I gave up from there.

What's the difference between Nirvana^2 and Everything^2? Or are you just a mean drunk coming home to beat the kids. Hey teacher, leave them kids alone.
You break my heart^2. Now I can have a Happy New year's Eve. The hard part is I really like your writing, so I'm beyond confused. Your move, Baudrunner.

Peace