Main Question or Discussion Point

I am playing with thought experiments to try to improve my comprehension of relativity. Hopefully I can get some feedback to further enlighten my understanding.

I begin with the assumption that orbital trajectories are inertial straight-lines in space-time curved by gravity. Objects in motion tend to stay in (circular) motion around a gravity well except to the extent that they are forced further into or out of the well, right?

Also, any given altitude of a given gravity well should have a specific velocity of free fall, correct? In other words, at any given distance from a gravitational center, any two objects should orbit at the same speed. This speed could be seen as a natural characteristic of the particular space-time path followed by the orbiting-satellite, no? I am even inclined to call an object moving at such a speed as to achieve perpetual orbit "at rest" in its own frame, but this may be going too far, I'm not sure.

Anyway, the issue I'm trying to get at is that an object/satellite orbiting a black hole would have to travel at a speed sufficient to avoid descending into the gravity well of the black hole. Therefore, at the Zwartschild radius distance from the black hole's center, I would guess that the velocity needed to orbit the black hole without descending further into its gravity well would be the speed of light.

However, I believe that mass/momentum/energy of objects approaching the speed of light must increase infinitely for them to reach the speed of light. Therefore an object orbiting a black hole at its Zwartschild radius would attain maximum energy, wouldn't it? Attaining maximum energy would, I think, have to involve conversion of matter to energy obey the law of conservation of matter-energy.

The other possibility I can imagine is that length contraction and space-time dilation would absorb the increasing energy. In that case, the object/satellite approaching the speed of light would shrink according to its acceleration, which would be accompanied by a corresponding dilation of space-time (outside the black hole) from its perspective. Inside the black hole, I think that space-time would contract proportionately to the length-contraction of the object, and as such the object would perceive itself as not changing size relative to its surroundings.

However, from the perspective of an observer outside the black hole, the object would appear to shrink to the point of disappearance along with some red-shift, I think. The reason I throw in "redshift" is because I wonder if very distant galaxies, which appear to be exiting viewing range because of the universe expanding, are not actually exhibiting black-hole entrance behavior.

The reason I consider this plausible is because velocity is always relative to a point of reference. Therefore it seems that a galaxy's apparent velocity due to the universe expanding over a long distance is no different than its actual velocity relative to the point of observation. So, for example, the disappearance of an object that would be caused by traveling away from it at the speed of light should have the same effect on the object left behind as it does on the object leaving. I.e. both objects appear to each other as if they were accelerating beyond the speed of light, which is the same thing as crossing the Zwartschild radius of a black hole.

I'll stop here since I've probably already lost most readers with such a long post. Sorry if this is viewed as inappropriate for this sub-forum. I realize that the end of this post begins to sound like a crackpot theory, but I'm just trying to elaborate some modeling assumptions that seem to fit with these notions of relativity and get feedback about how I may be misinterpreting the theories. Thanks in advance for any insights.

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tiny-tim
Homework Helper
Welcome to PF!

Hi brainstorm! Welcome to PF!

(btw, it's Schwarzschild, meaning "black shield" )
… Therefore, at the Zwartschild radius distance from the black hole's center, I would guess that the velocity needed to orbit the black hole without descending further into its gravity well would be the speed of light.
No, at just outside the Schwarzschild radius (r = 2M), it would need to be travelling at the speed of light directly outward (or very nearly so).

The smallest orbit for light is on the photon sphere , at r = 3M.

The smallest orbit for ordinary matter (with non-zero rest-mass) is at r = 6M.

Nothing can orbit a black hole at less than r = 3M.

Thanks for the welcome, tiny tim. I realize my post was a heap to deal with, but any related information helps advance my thinking on the subject.

Do you know of some logical explanation why light can orbit at r = 3m but only disappears completely at r = 2m? Intuitively I would think that the Schwarzschild radius would be a single limit outside of which light would orbit. I suppose there must be some region where light is being pulled in but not yet incapable of escaping. I just can't think of any reason why or how it could no longer orbit but still be considered outside the event horizon.

Likewise, why would ordinary matter be unable to orbit below r=6m? These seem like arbitrary numbers to me, probably since I'm not familiar with the method that generated them.

tiny-tim
Homework Helper
At any point, there is a "cone" of directions in which light can go without crashing into the black hole.

Very close to the event horizon, the cone is very narrow, and light has to be pointing more or less upward, otherwise it will curve round and crash into the black hole.

Further away, the cone gets larger, and at r = 3M, it has flattened out to cover half of space …

light going slightly "down" will crash, light going slightly "up" will escape completely, and light going exactly "sideways" will "almost" escape, since it will stay at the same distance, in circular orbit for ever.

Ar r > 3M, even light going slightly "down" will still escape (and light going "sideways" will easily escape, and so cannot orbit).

These are the cones you see in books about black holes …

which book are you using?​

At any point, there is a "cone" of directions in which light can go without crashing into the black hole.

Very close to the event horizon, the cone is very narrow, and light has to be pointing more or less upward, otherwise it will curve round and crash into the black hole.

Further away, the cone gets larger, and at r = 3M, it has flattened out to cover half of space …

light going slightly "down" will crash, light going slightly "up" will escape completely, and light going exactly "sideways" will "almost" escape, since it will stay at the same distance, in circular orbit for ever.

Ar r > 3M, even light going slightly "down" will still escape (and light going "sideways" will easily escape, and so cannot orbit).

These are the cones you see in books about black holes …

which book are you using?​
No book. I have gathered various info from websites, forum discussions, etc. I'm actually only interested in black-holes to the extent that I'm trying to understand space-time dilation, ST curvature, and length contraction.

Most recently, I am trying to figure out if there could be a relationship between length-contraction and ST dilation/contraction caused by velocity and that caused by gravity. The reason I think there could be a connection is that objects moving through space-time are always moving relative to other gravitational fields which have affected their achieved velocities.

Likewise, it is impossible for an object to increase velocity without increasing altitude within whatever gravity-well(s) it is located relative to. So it seems to me that velocity is relative to gravitation, which may seem quite obvious from the perspective that ST dilates/contracts.

Black holes come into the picture for me when I consider how an object could accelerate closer to the speed of light and how it could approach the speed of light and what would happen. Spiraling into a black hole seems to represent the best scenario for a material object approaching the speed of light, because the object could theoretically continue to orbit and accelerate unimpeded by any matter or energy, since both are contained within the BH.

Because length contraction and ST dilation result from acceleration nearing C, I am inclined to think that black holes are a relative appearance and that dilation and length-contraction for objects approaching them equates to expansion of space-time relative to the object as it approaches and accelerates relative to the black hole.

I actually wonder if the hubble radius is not equivalent to the expanded space-time inside a black hole. I say this because the red-shift and disappearance of some galaxies that appear to be expanding away from Earth may actually perceive the Milky Way as disappearing into a black-hole, in that its light is no longer able to overcome the relative acceleration between the galaxies.

Basically I'm speculating if the hubble radius couldn't be the same thing as a Zwarschild radius, where what appears as universal expansion from one perspective appears as universal contraction from another. The question would be whether wavelength discrepancies seen in distant galaxies exhibit a pattern similar to what would be seen as a result of the variable "cones" of light nearing a black hole.

I know this sounds crankish, since black holes are conceptualized as local phenomena without much volume, but with regard to space-time dilation caused by gravity and velocity, I think that some similarity if not convergence can be hypothesized between black-holes from the perspective of an object entering one and expanding space-time from the perspective of an object located within a hubble-radius.