As the speed of light is related to Permittivity and Permeability, and the speed of light slows down near a black hole due to the curvature of space, then what is the relationship of Permittivity and Permeability to the curvature of space near a black hole? Are both Permittivity and Permeability or only one effected by the curvature of space. And what is the formulae relating them to the curvature in a simple Schwarzchild black hole. Similarly for vacuum impedance.
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1Related: Does gravity slow the speed that light travels? – Michael Seifert Nov 30 '21 at 21:57
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Below is a section of a speculative paper I wrote that looks at the permittivity of the space of loop quantum gravity in free space versus in a gravitational field. I compare free space to Earth, the Sun and the largest known star:
It occurred to me that in a different universe, if the size of the particulate space of loop quantum gravity was larger or smaller, then the speed of light would itself be slower or faster. This works directly in conjunction with Maxwell’s equations with c=√(1/µ0ɛ0). That is, c is inversely proportional to the permeability and permittivity of free space. In this formula, ɛ0 (8.8542 x 10-12) is the permittivity of free space. This can be thought of as the resistance of free space to the formation of fields, or the viscosity of space.
Paraphrasing from Arvin Ash; Why are µ0 and ɛ0 these exact values? These are the constants of nature. These are properties of free space that tell us how fast magnetic fields and electric fields can interact with each other. This sets a limit on how fast these fields can propagate through space. In a different substance, or in a different universe, these constants could be different. Thus if ɛ0, the permittivity of space was lower, c would increase. Likewise if ɛ0 was larger, as with the dilation of particulate space, c would decrease as we see with time dilation in gravity. Paraphrasing from Review of the Universe, “Just as space is defined by a network's discrete geometry, time is defined by the sequence of distinct moves that rearrange the network. Time flows not like a river but like the ticking of a clock, with "ticks" that are about as long as the Planck time: 10-43 second. Or, more precisely, time in the universe flows by the ticking of innumerable clocks - in a sense, at every location in the network where a quantum "move" takes place, a clock at that location has ticked once.”
To show how this works in a gravitational field, we want to compare the permittivity of free space versus the permittivity of space on the surface of the gravitational body:
• In Free Space: c=1/√µ0ɛ0, where µ0=1.25663706 x 10-6 and ɛ0=8.85418782 x 10-12
• And using the Schwarzschild metric to determine time dilation relative to free space: t!=t/√(1-2Gm/rC2)
On the Earth: Mass=5.9722x1024kg Radius=6.371x106m We find a time dilation factor of tE=1.000,000,000,699,68 Thus we find a permittivity factor of ɛE = ɛ0 x tE=8.85418783x10-12 or 21 centimeters per second in spacetime dilation compared to free space
On the Sun: Mass=1.989x1030kg Radius=6.9634x108m We find a time dilation factor of tS =1.000,002,121,041,69 Thus we find a permittivity factor of ɛS = ɛ0 x tS=8.85420660x10-12 or 635 metres per second in spacetime dilation compared to free space
On star R136a1: Mass=6.263x1032kg Radius=2.089x1010m (The largest known star)
We find a time dilation factor of tR =1.000,002,226,755,19
Thus we find a permittivity factor of ɛR = ɛ0 x tR=8.85438498x10-12
or 6,675 metres per second in spacetime dilation compared to free space
Thus we can see the extremely small distortion of spacetime surrounding Earth when compared to the larger distortion around the Sun and much large distortion surrounding star R136a1.
foolishmuse
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