Math of General Relativity - Part 19
If you want to look at the equation relating the matrix G with the mass, energy and motion (or Stress-Energy Tensor) of the gravitating matter, see the below link.
So now we know how to assign a Matrix of 16 numbers to each point P in space inside MSG at each point in time. On Earth, these numbers won't change with time, to a huge degree of accuracy. Now we want to translate this point P inside MSG to a point P4 on a Riemann surface in Riemann Space.
Relating a space-time point P (x ,y, z, t) in real space-time to the surface, or Riemann Space R4, can be done using the simple "mapping" equations x = u1, y = u2, z = u3 and u4 = ct, where c is the speed of light , and the U are the coordinate axes on the surface of R4.
This gives us a point P4 = (u1, u2, u3, u4) in R4, or on the surface R4, whichever way you want to look at it.
Theory: This point P4 was mapped from MSG to R4, and the value of the gravitational field G at the point P4 is the value of the Metric Tensor at P4.
In other words, the theory is that real space-time and gravity in real space-time can be translated into a purely GEOMETRIC mathematical scenario. Gravity in 3D space-time produces a 4D surface in an alternate 4D mathematical Universe. How this is not the "real" Universe will be shown next.
A Short Story
Inspired by the song HOTEL CALIFORNIA
by the Eagles
The desert stretched endlessly before me, a blackened sea of sand under a moonless sky. My old pickup rattled along the desolate highway, the cool wind whipping through my hair, carrying a strange, sweet scent, like burning herbs, sharp and intoxicating.
Colitas, maybe, though I didn’t know the word then. It curled into my lungs, making my thoughts hazy. Up ahead, a faint light flickered, a beacon in the void. My eyelids drooped, my vision blurred, and the weight of exhaustion pressed me down. I had to stop. I didn’t have a choice.
The building materialized like a mirage, a sprawling, dilapidated structure, its neon sign buzzing faintly: Hotel. The light shimmered, unnatural, pulling me closer. I parked and stumbled out, my legs heavy as lead. At the doorway stood a woman, her silhouette framed by the dim glow of the entrance.
Her eyes glinted, sharp and unblinking, like a predator’s. A distant bell tolled, low and mournful, vibrating...
I appreciate your perspective and your emphasis on the metric tensor as the central factor in spacetime dilations, and I acknowledge your understanding of the distinction between kinematic and gravitational effects. Your interpretation that all space and time dilations are caused by the metric tensor is indeed consistent with the mathematics of General Relativity (GR), as the metric tensor ( g_{\mu\nu} ) fully describes the geometry of spacetime, which governs all relativistic effects, including time dilation. Let me align with your viewpoint, clarify the role of the metric tensor in the scenario, and address the time dilation between the two clocks at the same spatial location, ensuring we stay consistent with the mathematics.
Your Scenario and the Metric Tensor
You’ve specified two clocks at the same spatial location in a given coordinate system, with Clock 1 at rest and Clock 2 in motion relative to that system. The metric tensor ( g_{\mu\nu} ) defines the spacetime geometry at that point, and all time dilation effects are indeed encoded in ...
Oh, Peg, you’re standing there in the spotlight’s glare, aren’t you? The camera loves you, they say, and who am I to argue?
Your face, all sharp cheekbones and that practiced pout, is plastered across the call sheets, the casting director’s desk, the daydreams of every nobody who ever wanted to be a somebody.
You’ve got that role, Peg, the one you clawed your way through auditions for, the one you cried over in that dingy Hollywood motel when you thought the callback wasn’t coming.
It’s a big part, they tell you, big enough to make people whisper your name in line at Schwab’s, big enough to get you that photoshoot with Vanity Fair.
You’re on the cusp, Peg, teetering on that razor’s edge where dreams either bloom or bleed out. But you know how this town works, don’t you? You’ve seen the ghosts of starlets past, their faces fading from billboards, their names scratched off the marquee.
I see you now, Peg, in that rented gown, posing for the magazine spread. The photographer’s ...
