[Ambitious1999 (OP)]

Forgive me if this sounds dumb as I'm new to physics.
A lot of people say time stops or almost stops at the event horizon. If that's the case, then if it was possible to spend billions of years observing an object falling into a black hole then when it appears to stop at the event horizon, would that object remain visible there even after the black hole had evaporated billions of years later, and thus the image of the object never disappear? Would time be forever frozen at the event horizon or at the location of the event horizon after the BH had gone?

If not because space time curvature returns to normal after the BH has gone then what happens to the object that appears to have stoped at the former event horizon?

[timey]

Well firstly - black holes are predicted by the theory of general relativity.  The mathematics of the structure of the theory of general relativity imply that time slows down near a black hole.  There isn't any current means of experiment available to physics that can test what time does near a black hole, but general relativity has been confirmed by every experiment that physics has so far conducted inclusive of clocks observed to be running slower when placed in the lower gravity potential, a lower gravity potential being a position that is closer to the main body mass.  This means that a clock will tick more slowly on Earth at sea level than it will on Earth at the top of a mountain, (there are complexities involved concerning the equatorial bulge in relation to centripetal motion that apply, but for this explanation we don't need to go there), and this confirms the structure of the theory of general relativity.

The mathematics of general relativity break down at the event horizon of a black hole, which is a cause for concern for the theorist, but it is the general opinion that 'from the point of view of a remote observer', (remote meaning that the observer is not at the black hole), time grinds to a halt at the horizon.
However, it is important to note that this is 'from the point of view of the remote observer.  An observer at the black hole, which is actually a ridiculous notion because an observer at the black hole would without question not be alive, but hypothetically speaking the observer at the black hole would not observe anything different about the time there.  He would experience the time at a black hole in just the same way as he would experience time anywhere else.  It is only from the remote position that the observer will observe the black hole's time to be running slow, this being because at the remote position time is running at a faster rate.
(It is worth pointing out that the remote observer's point of view is also hypothetical, this being because physics has not been able to get an observer close enough to a black hole to observe much detail.  It is also worth pointing out that the observations that physics has been able to conduct of black holes are inclusive of superluminal jets that incorporate very high speeds, and high speeds are not conducive to the concept of slow or stopped time.  We don't need to go there either for this description, but I include the detail to indicate the fact that physics still has a lot to learn about black holes)

In asking what happens to an object at the event horizon you have touched upon an unknown.  There are theories that the object will end up inside the black hole on the other side of the event horizon, and there are theories that the object will stop outside the black hole at the event horizon.  There are theories that the objects stopped outside the black hole at the event horizon form part of the superluminal jet phenomenon that has been observed of black holes.  There are even theories that new universes are forming on the other side of the event horizon.  In fact there are no end of theories, but the truth is that physics doesn't actually know what happens to the object at the event horizon.

One thing that is a given though -  An observer who wanted to see if an object that was 'stopped' at the event horizon would still be there after the black hole had evaporated would have to wait for an exceptionally long time.    

[evan_au]

would that object remain visible there even after the black hole had evaporated billions of years later?

No. It is the extreme time dilation near a black hole that freezes the image. If the black hole evaporates, that image is no longer held there.

what happens to the object that appears to have stopped at the former event horizon?

From the viewpoint of a person falling into a black hole, they would see increasing distortion of the starfield as they approached the black hole, but they would not think that time has stopped. They would measure themselves accelerating all the way to the event horizon.
- If it is a black hole a couple of times more massive than the Sun, they would be pulled apart by the gravitational forces before they reached the event horizon.
- If it is a black hole similar to the one in the center of our galaxy, millions of times more massive than the Sun, they would reach the event horizon in a finite time, and intact (assuming they were able to miss the intense radiation associated with any accretion disk that might be present).

We can't say much about what happens inside the event horizon, but our normal concepts of space and time get rather twisted in there.

[jeffreyH]

The information about any object in the near vicinity of an event horizon is carried by photons. The speed of light is slowed relative to its distance from the horizon so the information transfer is affected accordingly. What can be said is that the direction of information flow is biased toward the hypothetical singularity at the centre of the black hole.

[timey]

Which is a tad paradoxical when one considers that the speed associated with the gravitational acceleration of an object near earth is observed to be greater than it is near the moon, suggesting that the acceleration of an object near a black hole will be greater than it is on earth.  A greater acceleration being synonymous with higher speeds, not slower.

[jeffreyH]

Which is a tad paradoxical when one considers that the speed associated with the gravitational acceleration of an object near earth is observed to be greater than it is near the moon, suggesting that the acceleration of an object near a black hole will be greater than it is on earth.  A greater acceleration being synonymous with higher speeds, not slower.

It depends upon which direction your vector is pointing.

[PmbPhy]

Forgive me if this sounds dumb as I'm new to physics.

That’s merely a mathematical abstraction which can never be realized in practice. That’s because there’s no way to place a clock at the event horizon and observe it not ticking.

[timey]

Which is a tad paradoxical when one considers that the speed associated with the gravitational acceleration of an object near earth is observed to be greater than it is near the moon, suggesting that the acceleration of an object near a black hole will be greater than it is on earth.  A greater acceleration being synonymous with higher speeds, not slower.

It depends upon which direction your vector is pointing.

Lol Jeff - I'm liking the cut of your jib!

[dutch]

Which is a tad paradoxical when one considers that the speed associated with the gravitational acceleration of an object near earth is observed to be greater than it is near the moon, suggesting that the acceleration of an object near a black hole will be greater than it is on earth.  A greater acceleration being synonymous with higher speeds, not slower.

Well the slow down in time near the Earth due to gravity is very small (around picoseconds per second). Using the equation shown below the slow down near a blackhole is very small until you're VERY close to the horizon. What happens on the Earth and moon doesn't compare to a blackhole.

t'/t = √(1 - rs/r)   where rs is the radius of the event horizon. This equation also models the Earth and Sun but these bodies are orders of magnitude larger than what their rs would be if they collapsed to a black hole.

Even at two times the radius of the event horizon the time dilation factor t'/t = .71. Time is running at 71% the rate of a far off observer.

At r = 3/2rs is the photon sphere where light orbits and nothing else could possibly loiter without using insane amounts of fuel because there are no orbits at this distance or closer. This distance is quite close to the event horizon (1/4 the diameter of the blackhole).

A lot of the accretion disk of a black hole is outside of r = 2rs. The falling matter gets to relativistic speeds (>50% c) long before gravitational time dilation becomes anywhere near zero. Look how far away the event horizon is in the NASA super computer simulation of the accretion disk below.

https://commons.wikimedia.org/w/index.php?title=File%3ANASA-led_Study_Explains_How_Black_Holes_Shine_in_Hard_X-rays.ogv

 It's easy to estimate the distance of the accretion disk is about r = 3rs at the closest point for this spinning black hole. t'/t = .82 or time is running at 82% normal rate. The frame dragging of the spinning blackhole should make this time dilation even less because the accretion disk should be spinning with the frame dragging. I'm not considering the relativistic speed of the matter because that speed becoming greater only helps show that blackholes have high speed accretion disks even with their gravitational time dilation. Including this time dilation effect for 50% light-speed is t'/t = √(1 - .5^2) = .87. Multiplying the gravitational and time dilation together only gives .71 which is still nowhere near 0.

- If it is a black hole similar to the one in the center of our galaxy, millions of times more massive than the Sun, they would reach the event horizon in a finite time, and intact (assuming they were able to miss the intense radiation associated with any accretion disk that might be present).

I wouldn't agree with this. We have no way with our current theory to know what happens as an object reaches an event horizon because we can't observe it. As to doing a coordinate transformation to the perspective of an observer falling into a black hole it prove anything as this lies outside of tested domain (Unless you've got a blackhole and you want to take a one-way trip). I don't think we can say anything scientific without some form of observation. We have pictures provided by telescopes of what looks like accretion disks and relativistic jets leaving compact dark objects. Using this we're probably pretty much right all the way down to very close to the event horizon. Extremely close to the event horizon and below we really know nothing. Sure we can use GR to make an educated guess but most physicists agree the equations break down and some (like me) are very skeptical about them working in this domain.

[jeffreyH]

Thank you for the input Dutch. You are a like a breath of fresh air.

[timey]

Which is a tad paradoxical when one considers that the speed associated with the gravitational acceleration of an object near earth is observed to be greater than it is near the moon, suggesting that the acceleration of an object near a black hole will be greater than it is on earth.  A greater acceleration being synonymous with higher speeds, not slower.

Well the slow down in time near the Earth due to gravity is very small (around picoseconds per second). Using the equation shown below the slow down near a blackhole is very small until you're VERY close to the horizon. What happens on the Earth and moon doesn't compare to a blackhole.

t'/t = √(1 - rs/r)   where rs is the radius of the event horizon. This equation also models the Earth and Sun but these bodies are orders of magnitude larger than what their rs would be if they collapsed to a black hole.

Even at two times the radius of the event horizon the time dilation factor t'/t = .71. Time is running at 71% the rate of a far off observer.

At r = 3/2rs is the photon sphere where light orbits and nothing else could possibly loiter without using insane amounts of fuel because there are no orbits at this distance or closer. This distance is quite close to the event horizon (1/4 the diameter of the blackhole).

A lot of the accretion disk of a black hole is outside of r = 2rs. The falling matter gets to relativistic speeds (>50% c) long before gravitational time dilation becomes anywhere near zero. Look how far away the event horizon is in the NASA super computer simulation of the accretion disk below.

https://commons.wikimedia.org/w/index.php?title=File%3ANASA-led_Study_Explains_How_Black_Holes_Shine_in_Hard_X-rays.ogv

 It's easy to estimate the distance of the accretion disk is about r = 3rs at the closest point for this spinning black hole. t'/t = .82 or time is running at 82% normal rate. The frame dragging of the spinning blackhole should make this time dilation even less because the accretion disk should be spinning with the frame dragging. I'm not considering the relativistic speed of the matter because that speed becoming greater only helps show that blackholes have high speed accretion disks even with their gravitational time dilation. Including this time dilation effect for 50% light-speed is t'/t = √(1 - .5^2) = .87. Multiplying the gravitational and time dilation together only gives .71 which is still nowhere near 0.

Could you perhaps make a clearer distinction between SR motion related time dilation and GR gravitational time dilation?

The gravitational acceleration caused by the blackhole will cause an object to accelerate towards the black hole much faster than an object will be accelerated towards the Earth.
The SR time dilation slowing of time effect will be (observed to be) greater for the object that is moving at these highly accelerated speeds caused by the greater gravitational field of the black hole.  But it is important that we realize that the SR time dilation that (we observe that) the object is subject to does not slow the speed that the object is moving at.  (if it did that would be a mathematical paradox)
But GR time dilation is also being slowed down as the object gets closer to the black hole.  On the basis that we are assuming that GR time dilation is slowing time down 'in' the geometry surrounding the black hole, (as opposed to SR time dilation that is slowing time for the object itself, yet cannot be slowing the objects speed without a mathematical paradox occurring) - If the GR time dilation in the space geometry near the black hole is becoming slower this 'will' affect the rate of the acceleration of the object.

The point I was making is that the mathematics are saying that the object is accelerated to relativistic speeds by the black holes gravitational field, yet that very same gravitational influence is slowing time down 'in' that field via GR...
... And that we observers can observe that an object, (be this an object free-falling towards a black hole, or superluminal jet action direction away from a black hole), can be accelerated to relativistic speeds (speeds that are held relative to our own clock) in a field where time slows down (rates of time held relative to our own clock) is somewhat paradoxical, isn't it?
Because if one, as a mathematical venture, holds 'speed' relative to seconds that are becoming longer in length (as observed by the observer), this will, when the data is transposed back to the observers rate of time, reduce the speed of the object.

We have no way with our current theory to know what happens as an object reaches an event horizon because we can't observe it. As to doing a coordinate transformation to the perspective of an observer falling into a black hole it prove anything as this lies outside of tested domain (Unless you've got a blackhole and you want to take a one-way trip). I don't think we can say anything scientific without some form of observation. We have pictures provided by telescopes of what looks like accretion disks and relativistic jets leaving compact dark objects. Using this we're probably pretty much right all the way down to very close to the event horizon. Extremely close to the event horizon and below we really know nothing. Sure we can use GR to make an educated guess but most physicists agree the equations break down and some (like me) are very skeptical about them working in this domain.

...and curious as to where the problem is?

[dutch]

... And that we observers can observe that an object, (be this an object free-falling towards a black hole, or superluminal jet action direction away from a black hole), can be accelerated to relativistic speeds (speeds that are held relative to our own clock) in a field where time slows down (rates of time held relative to our own clock) is somewhat paradoxical, isn't it?

No, it's not paradoxical. I already showed that objects move at Relativistic speeds before gravitational time dilation becomes anywhere near extreme. In fact at r = 3rs time only slows down about 18% compared to a stationary far off observer. At r = 5rs gravitational time dilation only slows things down 11%. With frame dragging this can be even less. However, in these locations objects can be moving at highly relativistic speeds well in excess of 50% light speed (and crashing into other objects).

What superluminal jets? I am not aware of anything physical going faster than light speed anywhere.

But GR time dilation is also being slowed down as the object gets closer to the black hole.  On the basis that we are assuming that GR time dilation is slowing time down 'in' the geometry surrounding the black hole, (as opposed to SR time dilation that is slowing time for the object itself, yet cannot be slowing the objects speed without a mathematical paradox occurring) - If the GR time dilation in the space geometry near the black hole is becoming slower this 'will' affect the rate of the acceleration of the object.

Relative to whom? To me way outside of the blackhole it looks like things are slowing way down. If something took a trajectory that brushed close to the event horizon and then was ejected back out it would likely think its trip was rather quick. However, from my perspective well outside the event horizon it would take considerable time. BTW how are you measuring speed and acceleration? Whose time are you using?

What you seem to not be getting is that you can move to highly relativistic speeds at a few factors of rs. However, at 3 to 5 rs gravitational time dilation will be relatively low but speeds very high.

What about the exact center of the Earth? There is no acceleration there yet time is running a little slow relative to some outside observer. In theory you could surround yourself with a shell of matter large enough that it almost causes time to stop relative to some outside observer but yet there still is no acceleration provided the shell in uniform.

[timey]

What superluminal jets? I am not aware of anything physical going faster than light speed anywhere.

http://www.stsci.edu/ftp/science/m87/press.txt

I already showed that objects move at Relativistic speeds before gravitational time dilation becomes anywhere near extreme. In fact at r = 3rs time only slows down about 18% compared to a stationary far off observer. At r = 5rs gravitational time dilation only slows things down 11%. With frame dragging this can be even less. However, in these locations objects can be moving at highly relativistic speeds well in excess of 50% light speed (and crashing into other objects).

BTW how are you measuring speed and acceleration? Whose time are you using?

Exactly.  I stipulated that it 'is' the remote observers time that we are using.  But if the remote observer is observing objects at speeds that are 98% of the speed of light, (see link) but is also observing, at r = 3rs, an 18% (as you suggested) reduction of 'gravitational' time dilation as a slower rate of time - then won't the gravitational time dilation be slowing the speed of objects at r = 3rs by 18%?

[evan_au]

What superluminal jets?

From certain angles, the jets emitted by black hole/quasar accretion disks can seem to be travelling faster than light.
See: https://en.wikipedia.org/wiki/Superluminal_motion

By the way, the power source for jets from black hole/neutron star accretion disks has been a bit of a mystery. Magnetohydrodynamics is a very complex subject to model on a supercomputer.

I heard a recent interview with a couple of plasma physicists who suggested that with an accretion disk in close orbit, you can get strong coupling between protons and the electric & magnetic field of the plasma, causing them to be ejected. Apparently, electrons do not couple so well to these fields, otherwise much more of the energy would be emitted as visible light/X-rays, rather than as matter jets.

See: http://omegataupodcast.net/249-plasma-physics/

[dutch]

Exactly.  I stipulated that it 'is' the remote observers time that we are using.  But if the remote observer is observing objects at speeds that are 98% of the speed of light, (see link) but is also observing, at r = 3rs, an 18% (as you suggested) reduction of 'gravitational' time dilation as a slower rate of time - then won't the gravitational time dilation be slowing the speed of objects at r = 3rs by 18%?

Well t'/t = √(1 - rs/r)  and c(r)/c =  1 - rs/r so it's actually a little worse than 18% for c(r) (1 - .82^2 =.32) from the observer's perspective at infinity. However, this isn't the issue. There's still possibly trillions of tons of matter moving around other matter at relativistic speeds from either perspective (infinity or local). There's also frame dragging, angular momentum, linear momentum, and potentially large/powerful magnetic fields. The source of the jets moving at close to light speed is thought to be large magnetic fields accelerating the matter and/or frame dragging (tapping into the angular momentum of the blackhole).

A lot of matter falls into the blackhole and quite a bit can eject back out in huge high-speed jets. Considering how complicated accretion disks and the area a few rs out of the blackhole is I don't think near light-speed jets are a paradox even with time dilation becoming significant close to rs.

From certain angles, the jets emitted by black hole/quasar accretion disks can seem to be travelling faster than light.

I've heard of certain observations that look faster than light. The point I was trying to make is no one has clear evidence anything physical goes faster than light.

By the way, the power source for jets from black hole/neutron star accretion disks has been a bit of a mystery. Magnetohydrodynamics is a very complex subject to model on a supercomputer.

Yes, it's extremely complicated. The simulation I showed was only simulating the relativistic speeds achieved by matter circulating a spinning blackhole. It also shows the accretion disk ending at a little over 3rs. In any case there is a whole lot of high-speed movement occurring and potentially very complex magnetic fields etc.

[timey]

I've heard of certain observations that look faster than light. The point I was trying to make is no one has clear evidence anything physical goes faster than light.

I got that...  Which was why I was making the point about the observations of 'apparent superluminal motion', that after angular and magnetic considerations are taken into account (see link I provided few posts ago) are estimated to be travelling at 98% of the speed of light.
If we are saying that we are observing matter at say rs=3, (or rs=5), travelling at 98% of the speed of light, but we are also saying that time at rs=3 is 18% slower than the time with which we are measuring (or rs=5 is 11% slower), then what 'actual speed' is the matter really travelling at?

Think second that is 18% longer at that location.
Think observed motion that is (after angular and magnetic considerations are accounted for) 98% of the speed of light, at that location...

[dutch]

If we are saying that we are observing matter at say rs=3, (or rs=5), travelling at 98% of the speed of light, but we are also saying that time at rs=3 is 18% slower than the time with which we are measuring (or rs=5 is 11% slower), then what 'actual speed' is the matter really travelling at?

Think second that is 18% longer at that location.
Think observed motion that is (after angular and magnetic considerations are accounted for) 98% of the speed of light, at that location...

Well the jets are often many light-years long (even thousands of light-years plus) and the time dilation is insignificant at this distance. The jets would essentially move at 98% c relative to us (little gravitational time dilation exists). We haven't really zoomed in to see a few rs (although there is an experiment soon to be done that may achieve this). If a simple non-spinning blackhole was the only thing in the picture these jets would likely not occur. A non-rotating gravity well accelerates matter in and does not accelerate it away. There is clearly something else involved in the process.

The angular momentum/frame dragging of a spinning blackhole, potentially immense magnetic fields compressed by the blackhole, and matter crashing into matter at relativistic speeds (ionizing and heating) seems to have the ability to produce these jets. Blackholes should have the ability to have quite strong magnetic fields (or compress already present fields) and these fields could potentially accelerate matter away from the blackhole (something clearly does). Frame dragging also has the ability to accelerate matter away from the blackhole as described by Roger Penrose (Penrose Process).

Just because the average speed of the matter is say 50% the speed of light in a region where time dilation is 18% is not evidence against blackholes accelerating away a some of the matter at a significant speed or evidence against time dilation. There's a lot of very high-speed matter and other effects going on. There's Magnetohydrodynamics and extreme gravity.

Think second that is 18% longer at that location.
Think observed motion that is (after angular and magnetic considerations are accounted for) 98% of the speed of light, at that location...

I'm not sure I understand your point you're trying to make anymore.

[timey]

Think second that is 18% longer at that location.
Think observed motion that is (after angular and magnetic considerations are accounted for) 98% of the speed of light, at that location...

I'm not sure I understand your point you're trying to make anymore.

If the observer is registering the matter in motion at 98% of the speed of light (after angular and magnetic considerations have been deducted), but time at the location the observer observes the matter in motion is running at between 18% and 11% between rs=3 & rs=5, then the motion that the observer is oberveing between rs=3 & rs=5 is moving between 18% and 11% slower.
But the observer is observing the motion of the matter between rs=3 & rs=5 (when angular and magnetic considerations are deducted) as being 98% of the speed of light, and he is measuring this speed by the rate of time of his own clock that is running between 18% and 11% faster than the time is where the motion is occurring.  Translating this slower time back to the observers rate of time, the motion the observer is observing must be 98% of the speed of light + 18% (at rs=3) of 98% of the speed of light.
This places the motion of matter occurring at rs=3 as being over 100% of the speed of light...

This cannot be so - because nothing can move faster than the speed of light.

Fortunately General Relativity tells us that the speed of light is differring in differring GP's.  Just make the speed of light relative to the length of second at  rs=3 and now we can translate this speed of motion back to the observers rate of time and say that the 'actual' speed of motion occurring at rs=3 is 98% of the speed of light minus 18% of 98%.  Now there is no violation of the speed of light at rs=3.

A guy here at the forum called Mike Gale tried to incorporate this variable speed of light (VLS) into the Schwartzschild metric, but he ran into problems with distance and length contraction.

[dutch]

Where did I say the speed of light was faster than c? The equations from the point of view of the observer at infinity below are straight out of General Relativity and derived directly from the Schwartzchild Metric. It's important to note that this is a non-local perspective and the speed of light is always measured as c locally. Gravitational time dilation does cause objects to linger close to a blackhole longer than they would in flat space when viewed by the far off observer. This includes light. And yes you do have to incorporate space as that also changes (in fact differently in the radial dimension as compared to the other two spatial dimensions). Gravitational time dilation does indeed slow down clocks (and everything else including light) from the perspective of the far off observer. It does not change the speed of light locally.

t'/t = √(1 - rs/r)  and c(r)/c =  1 - rs/r   the equation for space is d'/d = 1 / √(1 - rs/r)  where d is in the radial direction.

Read about the equations here. http://www.mathpages.com/rr/s6-01/6-01.htm

Translating this slower time back to the observers rate of time, the motion the observer is observing must be 98% of the speed of light + 18% (at rs=3) of 98% of the speed of light.
This places the motion of matter occurring at rs=3 as being over 100% of the speed of light...

How? I'm not following this logic.

A guy here at the forum called Mike Gale tried to incorporate this variable speed of light (VLS) into the Schwartzschild metric, but he ran into problems with distance and length contraction.

c(r)/c =  1 - rs/r is the apparent speed of a radial beam of light at r leaving a non-rotating blackhole at the perspective of r = infinity (or far enough away so space is flat). This factor includes both time dilation and length contraction effects. This is in full compliance with GR and Local Lorentz Covariance.

All this implies matter has needs an energy source to accelerate out of a gravity well. Something like frame dragging and/or electromagnetic fields must accelerate the matter. It is weird and scientists don't fully understand how the matter accelerates away from the blackhole in the jets. We have good ideas on the source but don't know exactly how it works.

[timey]

Where did I say the speed of light was faster than c?

Nowhere.  Why do you think I am under the impression that you did?

The equations from the point of view of the observer at infinity below are straight out of General Relativity and derived directly from the Schwartzchild Metric. It's important to note that this is a non-local perspective and the speed of light is always measured as c 'locally'.

So when the speed of light is always measured as c 'locally', which local's rate of time is the observer holding 299 792 458 m/s relative to?  The observer observes (hypothetically) that the rate of time is occurring 18% slower at rs=3.  If a second is 18% longer at rs=3, then at the local of rs=3 the speed of light will be 299 792 458 m/second that is 18% longer than the observer's length of second.

Gravitational time dilation does indeed slow down clocks (and everything else including light) from the perspective of the far off observer. It does not change the speed of light locally.

So the speed of light is the same distance traveled held relative to the time period of a second.  If a second in one part of the universe, lets say rs=3 for instance, can be 18% longer, then the speed of light at rs=3 is 299 792 458m/second that is 18% longer that the speed of light at the observers location, which is 299 792 458 m/s.  At both locals the speed of light travels the same distance in the time period of a second, but each local is inherent with seconds that have an 18% difference in length.

Translating this slower time back to the observers rate of time, the motion the observer is observing must be 98% of the speed of light + 18% (at rs=3) of 98% of the speed of light.
This places the motion of matter occurring at rs=3 as being over 100% of the speed of light...

How? I'm not following this logic.

If the observer observes the matter moving at superluminal speeds and works out, after angular considerations, that the motion between rs=3 and rs=5 is 98% of the speed of light, who's speed of light is he holding his measurements relative to?

If the observer holds the speed of light relative to the time period of a second in his own local where the 'observed" motion is 98% of the speed of light, then when he takes into consideration that it is taking the matter between 18% and 11% longer to travel from rs=3 to rs=5 in that local than it is in his own local, he will realize that the matter must be travelling between 18% and 11% faster from rs=3 to rs=5 than 98% of the speed of light of his own local...
If it has taken the matter longer to travel between point A and point B, then clearly the matter must be moving faster than 98% of the speed of light.  The matter will be moving between 18% and 11% faster than 98% of the speed of light held relative to the time period of a second of the local of the observer.

If the observer makes a calculation of the motion of the matter between rs=3 and rs=5 holding the speed of light relative to the time period of the second at rs=3, the time period of a second at rs=5, and the time periods of the seconds in-between those locals, then this is the upside down of the previous consideration.  The matter will be moving between 18% and 11% slower than 98% of the speed of light held relative to the time period of a second of the local of the observer.

Sorry, I'm not well enough versed in math to present my observations in that format, but hope the lingual explanation suffices.  In any case I am only 'having a natter', and am certainly not accusing you of saying, or not saying anything.  I like discussing the parts of physics that are unknown is all.