Yeah! It gets bendy all right. It's one thing to accept GR results, like Mercury's perihelion precession, as a simple consequence of putting a very modest time delay onto Newton's laws. So we can say : yep, Mercury turns up ever so slighty late to some orbital point because of finite light speed delaying it's knowledge of where Jupiter is ( or whatever ). But if you then take the principles which do accurately solve that scenario to a really high degree, and ramp up the masses/speeds etc many fold, what then?
Think of being right down the gravity well, standing upon the surface of one neutron star within a really close binary sytem ( yes we'd be dead, but if that worries you then pretend to be a neutron on the surface instead ). Say not long before they are going to merge together, and you're on the 'nearside' so when you look up you'll see the other neutron star across the gap between the two. Now what you see is grossly time shifted, way worse than the lag in your favorite video game during a LAN match.
Quote:
But then, also add in time dilation effects and... Is there not going to be some unholy shear effects from time and near-light-speed effects for the near light-speed rotating surface?
I suspect that something must break or limit before such a speedup...
But the big hole in the idea is... How do you get the hole in the centre in the first place? Why would a gravitational mass transform into a toroid?
Or is the assumption that in-falling mass accumulates at the light-speed circumference that then attracts the matter from inside and out? Would you not get a super-dense toroid with a gap and then a singularity in the middle still? (All rotating.)
Ah well, firstly I should re-emphasise ( as I just assume it in my thinking and you may not know that ) that like all GR descriptions, this is with respect to some observer's viewpoint. So when I say naked singularities etc ... this is talking from the ( rather safer ) distant viewpoint.
The toroid idea comes of requiring conservation of angular momentum, so you have to have off-axis mass rotating to do that. Angular momentum is some radius times some tangential momentum, so if either are zero then total angular momentum is zero. But why you then don't get any on-axis mass I suppose is a separate issue, as angular momentum could still be conserved with sufficient toroidal mass/momentum but leave some in the exact centre. I don't know why that comes out of the solution, I'm just quoting my understanding from the reading .... the gut/everday answer is 'centrifugal force' but that is just a restatement of GR's foundations ( equivalence principle ).
As for shear effects? Well that means a differential between behaviours at nearby radii, right? But if I'm at those radii then that will be blunted by relativisitc effects ( ie. speed of light is still a constant at any observer's speed ) so I won't 'see' the adjacent matter at radii nearby me doing all that much different to me .... :-)
I think what you mean is from a more distant view, and that is best summed up as a 'maelstrom' : the further down the gurgler we look the more messy things become. The problem specifically with losing the event horizon is that the freezing of time ( as seen distantly ) on that surface is lost. So what was going to be deferred to future distant observer knowledge ( I have to wait a long time for a spaceman to wave goodbye as he gradually splats onto the horizon ) gets brought forward. In the full analysis ( which I take on trust ) you can lose cause and effect sequence. Cause ought precede the effect for all observers. The event horizon is nature's way of preserving that by deferring information coming up from the horizon into the way distant future ( in the limit infinitely so ). If we lose that then why bother? Who knows, maybe cause -> effect is our local luxury. :-) :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
There's another implicit matter with PI, the circles and our universe. A circle, however measured, is operationally defined as a locus ( set ) of points equidistant from a specific given point ( the centre ). Thus if I rotate a circle around it's centre it ought look the same, measure the same, hence with PI the same. Suppose I find by experiment this to be true wherever they are, including which day I did it, and regardless of any specific chosen radius. Does this lesson generalise to other shapes? Remember the idea of 'similiar triangles'?
Two triangles are 'similiar' if I can overlie one upon the other ( so they occupy an identical set of points in some co-ordinate system ) by some combination of (a) a change of scale, possiblity unity and (b) a rotation, possibly none and (c) a translation, again maybe none.
What Reimann and others found was that one can have overall geometries with scaling, rotations and translations without PI being either constant within the total geometry nor equal to the Euclid case. You can have the sum of the internal angles of a triangle not simply not equal to 180 degrees, but the sum now depends on the overall size of the triangle. A good example is triangles on the Earth's surface, which is why the airline takes the 'great circle route' for long trips ( so the convexity of the Earth is accounted for ) and not some particular constant bearing that you would quote from a 'flat' map. So if some starting point on the western coast of Morocco is exactly due east of some destination point on the US east coast, the shortest distance to fly is not along a constant exact due west course but a heading gradually changing from a bit north of west, to due west mid Atlantic, to a bit south of west toward the trip's end. On a flat map it'll be a 'bowed upwards' curve. But in sphere terms it's actually a 'straight line' defined as the shortest intervening distance.
This culminates in GR as defined by Einstein and developed by others. But the really hardest part of all, I think, is that time partakes in the geometry. So 'bendy' always means time too, and that makes all the difference. A good example was performed sometime in the 70's, I think. Make two atomic clocks, set them side by side and adjust/calibrate so that they (a) agree/synchonise on the time value at some time, say midday or whatever and (b) check they proceed in step. Then leave one in the lab and send the other back and forth across the Atlantic ( UK USA ) using the Concorde. Upon return the travelling clock is brought alongside the stay-at-home version. The traveller reports fewer ( micro ) seconds elapsed. This is a mild demonstration of the so called "Twin's Paradox", by the way.
If we had sufficiently precise clocks could we show this similiarly for day to day journeys? [ Suppose us two met in the morning, synchronised watches like commandos do, and then after a day's worth of to-ing and fro-ing had breakfast together tomorrrow. ] Actually we do, but you may not know it, directly at least. If you have a GPS then in effect this is there by default, but it's not evident because it is corrected for automatically. If it didn't then our Navman/Garmin/TomTom would drift in accuracy by about 10 kilometers per day. When the GPS unit says 'you are here ...' this is with respect to a particular frame of reference maintained and aligned by the GPS people. You can do 'differential GPS' which is effectively a shift of co-ordinate origin ( time included ).
Cheers, Mike.
( edit ) The GPS drift is predominantly attributed to the time component. Ten kilometers in distance is ( at about one English foot per nanonsecond ) 30 micro seconds of light travel time. But your hand held GPS unit will call that a distance error rather than a time error.
Differential GPS is rather interesting and roughly works like this. Go into the middle of a field and hammer a stake into the ground. Call this the origin of your local co-ordinate system. As you'll be doing some local task that only depends on matters of nearby interest then that's OK. Now move away from your stake to some other point of interest. Say you're marking out a property or construction boundary - you want to peg out a site. By using differential GPS you'll be given the difference in co-ordinates between your local origin and nearby points. In either this or some other thread I mentioned the spacetime distance ds^2 = dt^2 - (dx^2 + dy^2 + dz^2) and how this was invariant across reference systems. Now imagine this ds^2 is derived from the GPS satellites, and as they are certainly moving with respect to me ( in my construction paddock ) then there'll be some non-zero dt as well as non-zero dx, dy and dz. This ds^2 is really what is reported by GPS, however standing in my paddock I'll be calling most of this ds^2 as being due to physical displacement rather than time shift. However for each foot away from my origin point a nanosecond of light travel time is accumulated. This will hardly matter as you aren't likely to be tearing around the construction site at any speed of note.
Differential GPS is especially useful for aircraft navigation and landing systems, where we definitely want a firm answer expressed in 'differential' terms eg. how far above and short of the verge of Runway 17L am I? Because you're only feeding off a difference then the wandering about of our knowledge of some global position baseline/origin ( the previously mentioned 'drift' ) isn't important.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
T = a snap [ force types freeze out, Higgs decay, yada yada ]
Whoops, I meant 'Inflaton decay' not Higgs. An 'inflaton' is the quantum of whatever field is evolving to produce inflation. A placeholder for 'who knows what'? So having done a 'slow roll' to produce inflation the field then generates entropy, or particle production, yielding the everyday matter we eventually know and love.
Cheers, Mike.
( edit ) Yeah it's a slow night on the graveyard shift .... :-)
With differential GPS you need two units : a base and a wanderer. Both receive satellite input ( so each knows where it's at compared to GPS Central, wherever that is ). The base and wanderer talk and compare notes to give the differential, so the light travel time between them is also known and adjusted for. The assumption is that with any given differential readout any drift that either unit suffers from GPS Central is in the same direction and of quite similiar magnitude. Why? Because the size of the differential is of quite small magnitude compared to the size of the overall GPS survey ( of Earth dimensions ). So two nearby units are going to suffer from similiarly sized ( error ) effects.
Compare this to laser surveys. These generally have a base and a wanderer, well two separate units at least. A common type has rotating mirrors that bounce back a pulse of laser light sent to it from the other module. Divide the round trip time by two, and then multiply by the speed of light, and you have the separation. Provided they have no relative movement when the reading is taken. But you'd have to run along at quite a clip to introduce noteworthy error. If you could move at one foot per second, that's still a billion times slower than lightspeed. A distance error will be of that order ie. a billionth of a foot.
Compare also with radar which just uses a different photon frequency, but the same technique of bounce and return. All these examples derive from Einstein's original concept of how to synchronise separated clocks in a reproducible and consistent sense, as part of the construction of a spacetime reference frame. I go on ( and on ... ) about the detail as relativity has this practical basis. The key to understanding the theory.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
... The toroid idea comes of requiring conservation of angular momentum, so you have to have off-axis mass rotating to do that. Angular momentum is some radius times some tangential momentum, so if either are zero then total angular momentum is zero. But why you then don't get any on-axis mass I suppose is a separate issue, as angular momentum could still be conserved with sufficient toroidal mass/momentum but leave some in the exact centre. I don't know why that comes out of the solution, I'm just quoting my understanding from the reading .... the gut/everday answer is 'centrifugal force' but that is just a restatement of GR's foundations ( equivalence principle ).
I was just thinking in terms of how such a structure might be created. You can't think in terms of 'centrifugal force' because the material is gravitationally attractive.
A good question is for what structures are stable and then consider how they might arise.
So, for a toroidal structure to have no material at the centre of the 'doughnut hole', you would need to have at least a small positive dimple in the gravity well. However, I would expect the gravity well to deepen or at least be flat across the diameter of the torus...
I could well imagine material accumulating at the light speed radius and form a torus by virtue of no longer having any radial velocity with which to fall in with.
Quote:
As for shear effects? Well that means a differential between behaviours at nearby radii, right? But if I'm at those radii then that will be blunted by relativisitc effects...
I'm just thinking of the gravitational gradient vs angular velocity at a particular radius vs relatavistic dependant mass and velocity. What happens for connected material spanning a high time gradient?!
The naive assumption is that you have a solid torus that rotates as a solid whole. However, material at the inner radius of the torus will be compelled to rotate at a different angular velocity compared to material further out... Hence the shear.
But...?
Do you get a sweet spot of gradients such that you can have a non-shearing torus?
Or are all examples of torus doomed to a shear gradient forcing the material to flow and so due to friction causing ultimately a very hot death as the angular momentum is converted to ever higher outrageous temperatures?
How hot can you go? Into another Big bang?...
Quote:
Who knows, maybe cause -> effect is our local luxury. :-) :-)
That's something I've long considered in that we are sampling our universe from just one small locale...
Hmm, here's one spin-off - seems like the material inside the event horizon would be a swirl of super-hot, super-dense, superfluid (in a bath of neutrinos and dark matter) – and the superfluid aspect would mean there's no shear force since there's no viscosity, and also with a superfluid the heat capacity becomes infinite so maybe at some point the angular momentum reaches a maximum while temperature just keeps on rising until [the limb I'm out on is now tiny branches] it begins to radiate – dark energy! Not sure how that would all balance out ... :)
Shear effects .... is an example of how far ( hopefully in the right direction! ) one can go using GR principles without solving a single equation. That is : follow the Equivalence Principle! :-)
Now in spacetime light follows the null geodesic, ds^2 = 0, 'shortest path', surface of the light-cone etc. If a body is in 'free fall' - not subject to non-gravitational forces - then over short distances and times ( remember the EP is a local law ) it'll follow that geodesic closely. The EP is a limit statement really. Down in the maelstrom our particles aren't co-incident so will be subject to tidal effects, which is a way of saying they will change their relative position as they fall because their world lines differ.
[ The fact that from afar, they appear to 'fall' in more or less a circular path is no more mysterious than the Moon going around the Earth. ]
Now adjacent falling particles are going to affect each other. But the magnitude of gravity's effects is the huge compared to other forces for this situation.
[ In the converse to what we are used to everyday, gravity is the biggest elephant in the room for this scenario. Whereas I can normally defy the entire Earth in my office by a spot of static electricity on a paper clip picking up a piece of paper.]
Does 'usual' transfer of angular momentum outwards as mass falls inwards - in accretion discs - apply? It should, but due to time dilation - loss of photon energy coming up out of the well associated with time dilation - distant observer's won't see it quite that way. One can use a definition of temperature that relates to entropy ( change with respect to energy ), and thus area of any event horizon.
What if I'm travelling with the infalling material? 'Circular' is a word that applies to a distant perspective, but where I am everyone is going 'straight down' with not much relative velocity.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Hmm, here's one spin-off - seems like the material inside the event horizon would be a swirl of super-hot, super-dense, superfluid (in a bath of neutrinos and dark matter) – and the superfluid aspect would mean there's no shear force since there's no viscosity, and also with a superfluid the heat capacity becomes infinite so ...
Interesting idea.
Just a bit bemused by the "infinite heat capacity" bit. That would suggest that there is no possibility of any further increase in temperature...
I can appreciate the superconductivity equalising the temperature throughout the material very quickly (although some articles quote thermal fronts moving at 20m/s through superconductors due to the speed of sound)... and insulators have high heat capacities... Hence should not a superconductor have an infinitely small heat capacity?
Hence the black hole boils away into a particle soup even more quickly, (relatavistic) time permitting!
.... so I won't 'see' the adjacent matter at radii nearby me doing all that much different to me ....
My mistake, poorly deduced and phrased.
Not necessarily. With the SR velocity addition formula I could still have quite a bit of relative motion between stuff at adjacent radii, but they will barely look different from afar in terms of velocity. So if I have ( as seen from afar ) something going at 0.99c - and in it's frame something else is going at 0.99c - then as seen from afar this something else will be going at a speed between 0.99c and 1.0c ( but not equal to c ). Now for this, the kinetic energy isn't given by the low speed formula - m * v^2 / 2 - but a relativistic one. [ A SLAC upgrade in the 1980's doubled the beam energy but only increased the particle speed ( already near lightspeed in the lab frame ) by 100 km/h. ]
Now if the temperature is taken as the average kinetic energy per particle in an ensemble, and we keep piling it all down the hole, you do wind up soaking up as much as you like. There's no in-principle upper limit ......
Now 'heat capacity' is taken as the amount of heat energy change per temperature increment ( Joules per degree Kelvin say ). But if you put more stuff in then the gravitational redshift is increased - the frequency of light has more lowering at is escapes to distance - then how does this change a distant perception of temperature? Certainly if you follow Hawking - smaller black holes appear hotter - then the heat capacity is negative because adding mass/energy enlarges the hole and lowers it's effective temperature as seen by the surroundings.
Jeez Louise! No wonder the smart guys can't yet agree on this .....
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Now if the temperature is taken as the average kinetic energy per particle in an ensemble, and we keep piling it all down the hole, you do wind up soaking up as much as you like. There's no in-principle upper limit ......
Now 'heat capacity' is taken as the amount of heat energy change per temperature increment ( Joules per degree Kelvin say ). But if you put more stuff in then the gravitational redshift is increased - the frequency of light has more lowering at is escapes to distance - then how does this change a distant perception of temperature? Certainly if you follow Hawking - smaller black holes appear hotter - then the heat capacity is negative because adding mass/energy enlarges the hole and lowers it's effective temperature as seen by the surroundings.
Jeez Louise! No wonder the smart guys can't yet agree on this .....
Are 'black holes' in effect a perpetual energy machine? One view is that you throw in matter and that matter gains energy as heat and yet that same matter then contributes to attracting yet more matter to give away yet more energy as heat!
We calculate potential energy with 'background space' at a zero potential and gravitational masses at a negative potential. As they gain mass, their potential decreases yet further...
Is there something spooky about how the universe's potential energy budget works? How do you allow for the effects of 'black holes'? That aspect in itself requires an absolute mass for a black hole. Yet, assuming a 'singularity' with infinities of density and gravitation, can you really gain an infinity of potential energy for the infalling matter?...
RE: Phew, that's a mind
)
Yeah! It gets bendy all right. It's one thing to accept GR results, like Mercury's perihelion precession, as a simple consequence of putting a very modest time delay onto Newton's laws. So we can say : yep, Mercury turns up ever so slighty late to some orbital point because of finite light speed delaying it's knowledge of where Jupiter is ( or whatever ). But if you then take the principles which do accurately solve that scenario to a really high degree, and ramp up the masses/speeds etc many fold, what then?
Think of being right down the gravity well, standing upon the surface of one neutron star within a really close binary sytem ( yes we'd be dead, but if that worries you then pretend to be a neutron on the surface instead ). Say not long before they are going to merge together, and you're on the 'nearside' so when you look up you'll see the other neutron star across the gap between the two. Now what you see is grossly time shifted, way worse than the lag in your favorite video game during a LAN match.
Ah well, firstly I should re-emphasise ( as I just assume it in my thinking and you may not know that ) that like all GR descriptions, this is with respect to some observer's viewpoint. So when I say naked singularities etc ... this is talking from the ( rather safer ) distant viewpoint.
The toroid idea comes of requiring conservation of angular momentum, so you have to have off-axis mass rotating to do that. Angular momentum is some radius times some tangential momentum, so if either are zero then total angular momentum is zero. But why you then don't get any on-axis mass I suppose is a separate issue, as angular momentum could still be conserved with sufficient toroidal mass/momentum but leave some in the exact centre. I don't know why that comes out of the solution, I'm just quoting my understanding from the reading .... the gut/everday answer is 'centrifugal force' but that is just a restatement of GR's foundations ( equivalence principle ).
As for shear effects? Well that means a differential between behaviours at nearby radii, right? But if I'm at those radii then that will be blunted by relativisitc effects ( ie. speed of light is still a constant at any observer's speed ) so I won't 'see' the adjacent matter at radii nearby me doing all that much different to me .... :-)
I think what you mean is from a more distant view, and that is best summed up as a 'maelstrom' : the further down the gurgler we look the more messy things become. The problem specifically with losing the event horizon is that the freezing of time ( as seen distantly ) on that surface is lost. So what was going to be deferred to future distant observer knowledge ( I have to wait a long time for a spaceman to wave goodbye as he gradually splats onto the horizon ) gets brought forward. In the full analysis ( which I take on trust ) you can lose cause and effect sequence. Cause ought precede the effect for all observers. The event horizon is nature's way of preserving that by deferring information coming up from the horizon into the way distant future ( in the limit infinitely so ). If we lose that then why bother? Who knows, maybe cause -> effect is our local luxury. :-) :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: Why should a universe
)
There's another implicit matter with PI, the circles and our universe. A circle, however measured, is operationally defined as a locus ( set ) of points equidistant from a specific given point ( the centre ). Thus if I rotate a circle around it's centre it ought look the same, measure the same, hence with PI the same. Suppose I find by experiment this to be true wherever they are, including which day I did it, and regardless of any specific chosen radius. Does this lesson generalise to other shapes? Remember the idea of 'similiar triangles'?
Two triangles are 'similiar' if I can overlie one upon the other ( so they occupy an identical set of points in some co-ordinate system ) by some combination of (a) a change of scale, possiblity unity and (b) a rotation, possibly none and (c) a translation, again maybe none.
What Reimann and others found was that one can have overall geometries with scaling, rotations and translations without PI being either constant within the total geometry nor equal to the Euclid case. You can have the sum of the internal angles of a triangle not simply not equal to 180 degrees, but the sum now depends on the overall size of the triangle. A good example is triangles on the Earth's surface, which is why the airline takes the 'great circle route' for long trips ( so the convexity of the Earth is accounted for ) and not some particular constant bearing that you would quote from a 'flat' map. So if some starting point on the western coast of Morocco is exactly due east of some destination point on the US east coast, the shortest distance to fly is not along a constant exact due west course but a heading gradually changing from a bit north of west, to due west mid Atlantic, to a bit south of west toward the trip's end. On a flat map it'll be a 'bowed upwards' curve. But in sphere terms it's actually a 'straight line' defined as the shortest intervening distance.
This culminates in GR as defined by Einstein and developed by others. But the really hardest part of all, I think, is that time partakes in the geometry. So 'bendy' always means time too, and that makes all the difference. A good example was performed sometime in the 70's, I think. Make two atomic clocks, set them side by side and adjust/calibrate so that they (a) agree/synchonise on the time value at some time, say midday or whatever and (b) check they proceed in step. Then leave one in the lab and send the other back and forth across the Atlantic ( UK USA ) using the Concorde. Upon return the travelling clock is brought alongside the stay-at-home version. The traveller reports fewer ( micro ) seconds elapsed. This is a mild demonstration of the so called "Twin's Paradox", by the way.
If we had sufficiently precise clocks could we show this similiarly for day to day journeys? [ Suppose us two met in the morning, synchronised watches like commandos do, and then after a day's worth of to-ing and fro-ing had breakfast together tomorrrow. ] Actually we do, but you may not know it, directly at least. If you have a GPS then in effect this is there by default, but it's not evident because it is corrected for automatically. If it didn't then our Navman/Garmin/TomTom would drift in accuracy by about 10 kilometers per day. When the GPS unit says 'you are here ...' this is with respect to a particular frame of reference maintained and aligned by the GPS people. You can do 'differential GPS' which is effectively a shift of co-ordinate origin ( time included ).
Cheers, Mike.
( edit ) The GPS drift is predominantly attributed to the time component. Ten kilometers in distance is ( at about one English foot per nanonsecond ) 30 micro seconds of light travel time. But your hand held GPS unit will call that a distance error rather than a time error.
Differential GPS is rather interesting and roughly works like this. Go into the middle of a field and hammer a stake into the ground. Call this the origin of your local co-ordinate system. As you'll be doing some local task that only depends on matters of nearby interest then that's OK. Now move away from your stake to some other point of interest. Say you're marking out a property or construction boundary - you want to peg out a site. By using differential GPS you'll be given the difference in co-ordinates between your local origin and nearby points. In either this or some other thread I mentioned the spacetime distance ds^2 = dt^2 - (dx^2 + dy^2 + dz^2) and how this was invariant across reference systems. Now imagine this ds^2 is derived from the GPS satellites, and as they are certainly moving with respect to me ( in my construction paddock ) then there'll be some non-zero dt as well as non-zero dx, dy and dz. This ds^2 is really what is reported by GPS, however standing in my paddock I'll be calling most of this ds^2 as being due to physical displacement rather than time shift. However for each foot away from my origin point a nanosecond of light travel time is accumulated. This will hardly matter as you aren't likely to be tearing around the construction site at any speed of note.
Differential GPS is especially useful for aircraft navigation and landing systems, where we definitely want a firm answer expressed in 'differential' terms eg. how far above and short of the verge of Runway 17L am I? Because you're only feeding off a difference then the wandering about of our knowledge of some global position baseline/origin ( the previously mentioned 'drift' ) isn't important.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: T = a snap [ force
)
Whoops, I meant 'Inflaton decay' not Higgs. An 'inflaton' is the quantum of whatever field is evolving to produce inflation. A placeholder for 'who knows what'? So having done a 'slow roll' to produce inflation the field then generates entropy, or particle production, yielding the everyday matter we eventually know and love.
Cheers, Mike.
( edit ) Yeah it's a slow night on the graveyard shift .... :-)
With differential GPS you need two units : a base and a wanderer. Both receive satellite input ( so each knows where it's at compared to GPS Central, wherever that is ). The base and wanderer talk and compare notes to give the differential, so the light travel time between them is also known and adjusted for. The assumption is that with any given differential readout any drift that either unit suffers from GPS Central is in the same direction and of quite similiar magnitude. Why? Because the size of the differential is of quite small magnitude compared to the size of the overall GPS survey ( of Earth dimensions ). So two nearby units are going to suffer from similiarly sized ( error ) effects.
Compare this to laser surveys. These generally have a base and a wanderer, well two separate units at least. A common type has rotating mirrors that bounce back a pulse of laser light sent to it from the other module. Divide the round trip time by two, and then multiply by the speed of light, and you have the separation. Provided they have no relative movement when the reading is taken. But you'd have to run along at quite a clip to introduce noteworthy error. If you could move at one foot per second, that's still a billion times slower than lightspeed. A distance error will be of that order ie. a billionth of a foot.
Compare also with radar which just uses a different photon frequency, but the same technique of bounce and return. All these examples derive from Einstein's original concept of how to synchronise separated clocks in a reproducible and consistent sense, as part of the construction of a spacetime reference frame. I go on ( and on ... ) about the detail as relativity has this practical basis. The key to understanding the theory.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: RE: Phew, that's a
)
And that's without any beer!
I was just thinking in terms of how such a structure might be created. You can't think in terms of 'centrifugal force' because the material is gravitationally attractive.
A good question is for what structures are stable and then consider how they might arise.
So, for a toroidal structure to have no material at the centre of the 'doughnut hole', you would need to have at least a small positive dimple in the gravity well. However, I would expect the gravity well to deepen or at least be flat across the diameter of the torus...
I could well imagine material accumulating at the light speed radius and form a torus by virtue of no longer having any radial velocity with which to fall in with.
I'm just thinking of the gravitational gradient vs angular velocity at a particular radius vs relatavistic dependant mass and velocity. What happens for connected material spanning a high time gradient?!
The naive assumption is that you have a solid torus that rotates as a solid whole. However, material at the inner radius of the torus will be compelled to rotate at a different angular velocity compared to material further out... Hence the shear.
But...?
Do you get a sweet spot of gradients such that you can have a non-shearing torus?
Or are all examples of torus doomed to a shear gradient forcing the material to flow and so due to friction causing ultimately a very hot death as the angular momentum is converted to ever higher outrageous temperatures?
How hot can you go? Into another Big bang?...
That's something I've long considered in that we are sampling our universe from just one small locale...
Keep searchin',
Martin
(Other spin-offs to be considered later!)
See new freedom: Mageia Linux
Take a look for yourself: Linux Format
The Future is what We all make IT (GPLv3)
Hmm, here's one spin-off -
)
Hmm, here's one spin-off - seems like the material inside the event horizon would be a swirl of super-hot, super-dense, superfluid (in a bath of neutrinos and dark matter) – and the superfluid aspect would mean there's no shear force since there's no viscosity, and also with a superfluid the heat capacity becomes infinite so maybe at some point the angular momentum reaches a maximum while temperature just keeps on rising until [the limb I'm out on is now tiny branches] it begins to radiate – dark energy! Not sure how that would all balance out ... :)
Shear effects .... is an
)
Shear effects .... is an example of how far ( hopefully in the right direction! ) one can go using GR principles without solving a single equation. That is : follow the Equivalence Principle! :-)
Now in spacetime light follows the null geodesic, ds^2 = 0, 'shortest path', surface of the light-cone etc. If a body is in 'free fall' - not subject to non-gravitational forces - then over short distances and times ( remember the EP is a local law ) it'll follow that geodesic closely. The EP is a limit statement really. Down in the maelstrom our particles aren't co-incident so will be subject to tidal effects, which is a way of saying they will change their relative position as they fall because their world lines differ.
[ The fact that from afar, they appear to 'fall' in more or less a circular path is no more mysterious than the Moon going around the Earth. ]
Now adjacent falling particles are going to affect each other. But the magnitude of gravity's effects is the huge compared to other forces for this situation.
[ In the converse to what we are used to everyday, gravity is the biggest elephant in the room for this scenario. Whereas I can normally defy the entire Earth in my office by a spot of static electricity on a paper clip picking up a piece of paper.]
Does 'usual' transfer of angular momentum outwards as mass falls inwards - in accretion discs - apply? It should, but due to time dilation - loss of photon energy coming up out of the well associated with time dilation - distant observer's won't see it quite that way. One can use a definition of temperature that relates to entropy ( change with respect to energy ), and thus area of any event horizon.
What if I'm travelling with the infalling material? 'Circular' is a word that applies to a distant perspective, but where I am everyone is going 'straight down' with not much relative velocity.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: Hmm, here's one
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Interesting idea.
Just a bit bemused by the "infinite heat capacity" bit. That would suggest that there is no possibility of any further increase in temperature...
I can appreciate the superconductivity equalising the temperature throughout the material very quickly (although some articles quote thermal fronts moving at 20m/s through superconductors due to the speed of sound)... and insulators have high heat capacities... Hence should not a superconductor have an infinitely small heat capacity?
Hence the black hole boils away into a particle soup even more quickly, (relatavistic) time permitting!
Keep searchin',
Martin
See new freedom: Mageia Linux
Take a look for yourself: Linux Format
The Future is what We all make IT (GPLv3)
RE: .... so I won't 'see'
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My mistake, poorly deduced and phrased.
Not necessarily. With the SR velocity addition formula I could still have quite a bit of relative motion between stuff at adjacent radii, but they will barely look different from afar in terms of velocity. So if I have ( as seen from afar ) something going at 0.99c - and in it's frame something else is going at 0.99c - then as seen from afar this something else will be going at a speed between 0.99c and 1.0c ( but not equal to c ). Now for this, the kinetic energy isn't given by the low speed formula - m * v^2 / 2 - but a relativistic one. [ A SLAC upgrade in the 1980's doubled the beam energy but only increased the particle speed ( already near lightspeed in the lab frame ) by 100 km/h. ]
Now if the temperature is taken as the average kinetic energy per particle in an ensemble, and we keep piling it all down the hole, you do wind up soaking up as much as you like. There's no in-principle upper limit ......
Now 'heat capacity' is taken as the amount of heat energy change per temperature increment ( Joules per degree Kelvin say ). But if you put more stuff in then the gravitational redshift is increased - the frequency of light has more lowering at is escapes to distance - then how does this change a distant perception of temperature? Certainly if you follow Hawking - smaller black holes appear hotter - then the heat capacity is negative because adding mass/energy enlarges the hole and lowers it's effective temperature as seen by the surroundings.
Jeez Louise! No wonder the smart guys can't yet agree on this .....
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: Now if the temperature
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Are 'black holes' in effect a perpetual energy machine? One view is that you throw in matter and that matter gains energy as heat and yet that same matter then contributes to attracting yet more matter to give away yet more energy as heat!
We calculate potential energy with 'background space' at a zero potential and gravitational masses at a negative potential. As they gain mass, their potential decreases yet further...
Is there something spooky about how the universe's potential energy budget works? How do you allow for the effects of 'black holes'? That aspect in itself requires an absolute mass for a black hole. Yet, assuming a 'singularity' with infinities of density and gravitation, can you really gain an infinity of potential energy for the infalling matter?...
Keep searchin',
Martin
See new freedom: Mageia Linux
Take a look for yourself: Linux Format
The Future is what We all make IT (GPLv3)
All our assumptions about
)
All our assumptions about black holes assume that they always are spinning.
Are there, or can there be, any black holes that are not rotating?
Keep searchin',
Martin
See new freedom: Mageia Linux
Take a look for yourself: Linux Format
The Future is what We all make IT (GPLv3)