A second on Earth is different from a second on the Sun and a second next to a black hole. Therefore speed, as measured in meters per second, is different in each of these locations. Is there an accepted and universal way to measure speed without the use of time, so that it can be understood by everyone? An example of this might be by using force as a measure of speed, as in: a "speed" of 5 is what you must move a $1 \mathrm{kg}$ block in order to strike a $10 \mathrm{kg}$ cylinder that is 10 cm in length and move it by $1 \mathrm{m}$ inside a pipe that is coated with 200 grit sandpaper at which point the cylinder stops moving. I would think this would be the same regardless of where you are located, but it's only an example of what I'm thinking. Is there a better way of looking at this?
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6$\begingroup$ If you can accept that different observers measure different times, why shouldn't they also measure different speeds? $\endgroup$– d_bCommented Apr 27 at 23:58
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$\begingroup$ Neglecting special relativity, speed is still observer dependent. On the highway a car travels at 60mi/hr with respect to the ground, but at 0mi/hr with respect to the passenger. $\endgroup$– AidenCommented Apr 28 at 0:06
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$\begingroup$ The idea that two different observers do not agree on the speed of a moving object goes all the way back to Gallileo. Suppose that you and I are in space ships, crossing paths in different directions, when we both observe a rock flying by. I can use my radar to measure the velocity if the rock relative to my ship. You can measure it relative to your ship, and of course, we'll get different answers. How would you propose to define the true velocity of the rock? How could either of us determine it? $\endgroup$– Ohm's LawmanCommented Apr 28 at 0:08
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2$\begingroup$ That's not what I said. Before you can measure it, you'll need to find some way to define what you are measuring. You want to measure absolute velocity? Well, OK. What does that even mean? $\endgroup$– Ohm's LawmanCommented Apr 28 at 0:33
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4$\begingroup$ A second on Earth is different from a second on the Sun and a second next to a black hole .... Misleading, and may be a misconception. A clock in any of these places, as well as any physical process, will behave as expected in a laboratory in Earth. But an observer in a place far away viewing these places will observe different behavior (sped up or slowed down) compared to their local clocks. $\endgroup$– RC_23Commented Apr 28 at 0:42
2 Answers
A second near Earth is just as good as a second near the Sun, which is just as good as a second near a black hole. If you grab a clock, take it to that place, and let a second go by, it will all be the same thing (I dare to say this is actually even tautological). Relative to one another, they are different things, but one is not better than the other.
It doesn't really makes sense to try to define speed in a way that bypasses the relativity of time. Speed is a property of motion through spacetime, and thus it is only natural that the spacetime structure will have an impact on the definition of speed. If you really try to avoid this, you'll get to a notion of speed that does not represent what we actually understand as speed. I would guess it would even become meaningless in the relativistic case, which is precisely the situation you want to consider (since in non-relativistic physics time is absolute).
If you prefer, a fully relativistic notion of velocity is given by four-velocity. This is as invariant as velocity gets.
Edit:
There is another strong argument in favor that velocity will always depend on time: literally all measurements one can make are, deep down, made with clocks. In a relativistic spacetime, having clocks is sufficient to perform literally any physical measurement you may wish.
This is discussed in some depth in arXiv: 2311.09249 [gr-qc]. I'll give you a short argument, but check the paper for a deeper discussion. Any analog measurement you can make consists in measuring a position in a ruler (such as how far a spring has stretched, how much the block moved, what is the position of the needle in my ammeter, etc) or counting how many times something has happened (how many times the Earth has orbited the Sun, a quartz crystal has oscillated, etc). We measure forces by measuring the acceleration, which we measure by checking rates of change of position and time. We measure masses by watching deformations on a spring or whether two plates have the same position. And so on. Since relativity allows us to measure lengths using clocks (which is a consequence of the finite speed of light), all measurements are, deep down, measurements with clocks.
It is thus meaningless to try to make the notion of velocity independent of the relativity of time.
Edit 2: I also realized that, in your setup, the big block will keep moving indefinitely in the absence of friction. If you want it to move $1 \mathrm{m}$, you must specify in how much time that motion should take place.
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$\begingroup$ I had posted this under the topic of special relativity because it is commonly thought that the speed of light is the same for all observers. But look at the example of placing a mirror on the planet Mercury and flashing a laser from Earth to it and back. Now do the same thing but running the laser through a huge tunnel through the planet Venus. Obviously it would take longer for the light to return to Earth because of time dilation caused by Venus. So what I'm trying to do is find a universal way to measure speed, the same for everyone regardless of their time dilation situation. $\endgroup$ Commented Apr 28 at 0:09
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$\begingroup$ @foolishmuse I updated my question to give a stronger argument. Perhaps the most basic fact about speed is that it is not universal: even in Newtonian physics speed is relative. $\endgroup$ Commented Apr 28 at 0:11
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$\begingroup$ @foolishmuse Speed is so observer-dependent that in relativity you always have the imposition that four-velocity has unit norm. You're not allowed to change the norm of velocity, because it must be universal. This is stringkly different from acceleration, which can have arbitrary norm (with the sign being a restriction depending on your metric convention). If you define speed in an observer-independent manner, this will be so far away from the original notion of speed that you won't even be able to recognize it anymore $\endgroup$ Commented Apr 28 at 0:13
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$\begingroup$ Looking at your edited answer made me think of something that I believe would be Observer independent. What if we said that every time the Earth, Sun and a particular distant star we're lined up, that is one Galactic year. Would that not be the same for every Observer regardless of their motion or time dilation situation? Therefore one Galactic second would be, say, 1 millionth of that. That would also be the same for everyone. Is that correct thinking? $\endgroup$ Commented Apr 28 at 0:22
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3$\begingroup$ @foolishmuse No. You're forgetting about the relativity of simultaneity. Different observers might disagree that the three heavenly bodies are aligned. $\endgroup$ Commented Apr 28 at 0:25
The space-time interval is the same for all observers regardless where you are.A falling observer in a black hole measures the same amount of time with someone in flat space time , however the 2 observers will disagree that the measured time is the same but in reality it is.This is the whole point of relativity:
Let Bob and a clock be falling in a BH and let Alice and another clock be in flat space time.For Alice Bob's clock slows down the more it reaches closer to the event horizon of the BH however in their respective frames of reference the 2 clocks will measure the same time because the spacetime interval is a constant quantity.
