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I don't wan't to go in a big mathematical approach here but more of a kind understand some implications of special relativity.

Since a object from a steady observer is time dilated and length contracted, is the velocity then still the same from the observers perspective? I mean when an object approaching a black hole it will "appear" to slow down when it approaches the event horizon and at a time it will freeze at get red shifted into oblivion. But if it appears to be slowing when the velocity is increasing, would an object accelerated from rest (in your frame), say to 0.6c, actually move with a that velocity when it gets to that speed. If you watch it accelerate with constant acceleration it would not appear as constant acceleration but acceleration which gets a bit slower until it reaches the velocity? That is what i mean by a velocity that is implicit dilated from the consequences of that constant acceleration from a steady observer isn't possible and therefore reaches a velocity that is a bit slower than it should be. Would that imply that an object moving 0.9999c is appearing to move slow?

Update

Okay, so i'll try to reformulate some of the relevant things in the question. First okay so because relative velocity is reflexive we can agree on that velocity is same between to reference frames, (v or -v). Though from a steady observer watching a moving object, the viewer will calculate how big the time dilation and length contraction is for the moving object from the rest perspective. Then the velocity must still be the same even though the length of 1 meter is a bit shorter and the duration of a second is a little longer. Now we proceed to why the example with the event horison is relevant.

If an object is approaching the speed of light at the event horisont, which could be imitated with a crazy rocket (and therefore still be SR), the viewer would see it slowing down. If the object shuts of its engines close to the speed of light, would it look like it is travelling that fast then?. Wouldn't its motion almost freeze and therefore have another velocity?

Okay, so i'll try to reformulate some of the relevant things in the question. First okay so because relative velocity is reflexive we can agree on that velocity is same between to reference frames, (v or -v). Though from a steady observer watching a moving object, the viewer will calculate how big the time dilation and length contraction is for the moving object from the rest perspective. Then the velocity must still be the same even though the length of 1 meter is a bit shorter and the duration of a second is a little longer. Now we proceed to why the example with the event horison is relevant.

If an object is approaching the speed of light at the event horisont, which could be imitated with a crazy rocket (and therefore still be SR), the viewer would see it slowing down. If the object shuts of its engines close to the speed of light, would it look like it is travelling that fast then?. Wouldn't its motion almost freeze and therefore have another velocity?

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  • $\begingroup$ velocity is velocity (three-velocity) - the only one you, in the lab-frame, can observe, and defined relative to you. Acceleration is more subtle. Constant acceleration for you in the lab-frame, or constant acceleration in the instantaneous rest-frame of the accelerated object? First one is impractical (need ever-growing force to maintain acceleration), second one will lead to Hyperbolic motion $\endgroup$
    – Cryo
    Commented Jan 13, 2021 at 1:24
  • $\begingroup$ I'm a little confused as what the question is? But you should remove the references to black holes if you're wishing to only talk about special relativity. Similarly, it may be more straight forward to leave out 'acceleration' if the question is just about relative velocities, unless I've misunderstood. $\endgroup$
    – Eletie
    Commented Jan 13, 2021 at 1:26
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    $\begingroup$ I agree with Eletie: forget about event horizons & other GR things until you're totally clear on how this stuff works in plain SR. As I explain here velocity is reflexive, so if you observe my velocity to be ${\bf v}$, I observe your velocity to be ${\bf -v}$. You may find The Relativistic Rocket helpful. $\endgroup$
    – PM 2Ring
    Commented Jan 13, 2021 at 3:19
  • $\begingroup$ Okay read me answer which is just a reformulation. $\endgroup$
    – JohanL
    Commented Jan 13, 2021 at 10:35

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You are confusing yourself by mixing up a few concepts, so I will try to untangle them for you.

When we measure the speed of an object, we measure how far it moves in a given time, where the distance and the time are as measured in our reference frame, so time dilation and length contraction play no part in the measurement.

Secondly, there is a difference between where an object actually is at a given time and where it appears to be, because in order to see an object we have to wait for light from the object to reach us. If an object reaches the horizon of a black hole, it might seem to slow down because the light from it takes longer and longer to reach us, but that does not mean that it actually has slowed down.

You therefore need to maintain a distinction between the real velocity of an object and the velocity it might appear to have if you view it from a distance.

What I have described is not just true of SR. There are analogous effects in everyday life. If a distant object directly crosses your line of sight it will seem to be moving faster than a distant object travelling at the same speed but at a smaller angle to your line of sight. So the apparent speed viewed from a distance might seem less than the real speed.

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