The Minimum Speed in the Universe: An Exploration of the Cosmic Limit

[Nick666]

What is the minimum speed in the universe ?

 

[ulianjay]

0 m/s

 

[Ibix]

Zero. But not everyone will agree what is zero speed.

The point about the speed of light is that it is invariant. Everyone agrees its value relative to them, so you don't need to specify relative to what. That's not true for any other speed.

 

[wabbit]

But might there be a smallest measurable velocity from quantum gravity arguments ?
There is a natural time scale from the cosmological constant, so I am wondering whether perphaps one Planck length per cosmological time unit might provide a lower limit. Maybe some principle about how velocity can be measured, the curvature effect of lambda... Or maybe not : )

 

[ghwellsjr]

Zero. But not everyone will agree what is zero speed.

The point about the speed of light is that it is invariant. Everyone agrees its value relative to them, so you don't need to specify relative to what. That's not true for any other speed.

How can everyone agree that the value of the speed of light is relative to them and not agree what is zero speed or what it is relative to? I think it would be clearer if you said the speed of light is defined to be the same value in any inertial reference frame but the speed of any massive object can be different in different inertial reference frames.

Also, to address the question, any object at rest in an inertial reference frame has a speed of zero according to that reference frame. It doesn't matter if there are also any observers at rest in that reference frame.

 

[HallsofIvy]

If you are going to use "relativity", either "special" nor "general", you should certainly know that speed is relative! Given any object in the universe, there exist a coordinate system in which it has speed 0. Since "speed" (unlike "velocity") is never negative, the "minimum speed" possible is obviously 0.

If you meant "velocity" rather than "speed", which is a vector quantity, then you can always set up a coordinate system in which something (a light beam, say) is moving down the x-axis with velocity, relative to you, of -c. In that sense, the "minimum velocity" is -c.

 

[rede96]

the "minimum speed" possible is obviously 0.

Relatively speaking, is it really possible for something to have zero speed? Wouldn't that imply that if an object has zero speed it can not change position, which would mean it is at absolute rest?

 

[ghwellsjr]

Relatively speaking, is it really possible for something to have zero speed? Wouldn't that imply that if an object has zero speed it can not change position, which would mean it is at absolute rest?

As I said in post #5, an object at rest in an inertial reference frame has a speed of zero in that reference frame. Having a speed of zero according to an inertial reference frame and being at rest in that reference frame mean the same thing. But just because it has zero speed in one inertial reference frame doesn't mean or imply that it is at absolute rest because we can transform to another reference frame moving at some speed with respect to the first reference frame and then the object has that same speed in the new reference frame. No reference frame is any better than any other. That's what relativity says.

 

[rede96]

As I said in post #5, an object at rest in an inertial reference frame has a speed of zero in that reference frame. Having a speed of zero according to an inertial reference frame and being at rest in that reference frame mean the same thing. But just because it has zero speed in one inertial reference frame doesn't mean or imply that it is at absolute rest because we can transform to another reference frame moving at some speed with respect to the first reference frame and then the object has that same speed in the new reference frame. No reference frame is any better than any other. That's what relativity says.

Yes of course, no argument there. But I was thinking of the OP's question in 'absolute' terms, if you will. As it is not possible to say an object can be at absolute rest, then by the same logic it can not be possible to say an object has an absolute speed of zero, or any absolute speed for that matter.

So the minimum speed 'in the universe' can not be determined. Only the minimum speed in an inertial reference frame.

 

[ghwellsjr]

Yes of course, no argument there. But I was thinking of the OP's question in 'absolute' terms, if you will. As it is not possible to say an object can be at absolute rest, then by the same logic it can not be possible to say an object has an absolute speed of zero, or any absolute speed for that matter.

If it's not possible to talk about absolute rest or speeds, then why are you thinking in those terms?

So the minimum speed 'in the universe' can not be determined. Only the minimum speed in an inertial reference frame.

All speeds require an inertial reference frame according to Special Relativity, not just a minimum speed.

 

[rede96]

If it's not possible to talk about absolute rest or speeds, then why are you thinking in those terms?

Was just trying to answer the op's question of course :)

All speeds require an inertial reference frame according to Special Relativity, not just a minimum speed.

Yes hence why I said

or any absolute speed for that matter.

 

[BiGyElLoWhAt]

I think it would be clearer if you said the speed of light is defined to be the same value in any inertial reference frame but the speed of any massive object can be different in different inertial reference frames.

I don't know how I feel about that word, defined. We defined the speed of light to be the same? I thought we just noticed that it was.

 

[Ibix]

How can everyone agree that the value of the speed of light is relative to them and not agree what is zero speed or what it is relative to? I think it would be clearer if you said the speed of light is defined to be the same value in any inertial reference frame but the speed of any massive object can be different in different inertial reference frames.

A light pulse will pass me and you at c by our own measurements whether we are at rest with respect to each other or not. However an object traveling at any other speed (possibly stationary by my measurements) will not in general have the same speed by yours.

There's a qualitative difference between the concept of a "maximum speed" (which everyone agrees on) and the concept of "zero speed" (which is a frame-dependent quantity), which is what I was getting at, and I think what you are saying too. I didn't think I was being too cryptic, but maybe I was.

 

[Cruz Martinez]

I don't know how I feel about that word, defined. We defined the speed of light to be the same? I thought we just noticed that it was.

Well, you need a synchronization convention to measure one-way speeds, and the standard synchronization method uses the one-way speed of light, this speed is necessarily a defInition.

 

[wabbit]

Well, you need a synchronization convention to measure one-way speeds, and the standard synchronization method uses the one-way speed of light, this speed is necessarily a defInition.

Why ? Doesn't the standard method use light because it is already known experimentally to travel at the invariant speed?

But actually I don't see where you need synchronization to measure a velocity. One observer can do that alone, he doesn't need to talk to anyone else. And he can also compare relative (to him) velocity of waves emitting by two other sources without ever synchronizing with those sources - and see whether that velocity is independent of the motion of those sources relative to him.

 

[thedaybefore]

What is the minimum speed in the universe ?

The normal concept of speed doesn't apply to the universe, because we don't know anything that it would be moving with respect to, unless there are other universes we don't konw about.

.You could talk about the speed of expansion or contraction of the universe.

 

[Cruz Martinez]

Why ? Doesn't the standard method use light because it is already known experimentally to travel at the invariant speed?

But actually I don't see where you need synchronization to measure a velocity. One observer can do that alone, he doesn't need to talk to anyone else. And he can also compare relative (to him) velocity of waves emitting by two other sources without ever synchronizing with those sources - and see whether that velocity is independent of the motion of those sources relative to him.

But the mere definition of velocity is distance traveled/time employed, how would you now the time employed without sychronizing? This time employed is a difference between the readings of two distant clocks.

 

[wabbit]

I don't think so, you are measuring the wave as it passes near you, using your own clock in your own lab.

Or for a moving object you can use radar ranging, again involving only your own clock.

I can't think of a velocity measurement done using a distant clock(*). The velocity of an object relative to me does not depend on that object having a clock of its own.

(*) well yes redshift of light emitted by that object does that, but in GR this measures a combination of velocity and gravity - in SR it does measure velocity but you can also use reflection (radar) redshift, which does not depend on the object internal clock I think. The measurements you do are defined by your (local) apparatus, not by someone else's.

 

[Nugatory]

But actually I don't see where you need synchronization to measure a velocity.

You need synchronization to measure a one-way velocity. You calculate the one-way speed by dividing the distance traveled by the difference between the start and end times - and you need a synchronization convention to determine at least one of those times.

However, none of this has any bearing on the original question, for which the answer is of course "zero".

 

[Cruz Martinez]

I don't think so, you are measuring the wave as it passes near you, using your own clock in your own lab.

Or for a moving object you can use radar ranging, again involving only your own clock.

I can't think of a velocity measurement done using a distant clock(*). The velocity of an object relative to me does not depend on that object having a clock of its own.

(*) well yes redshift of light emitted by that object does that, but in GR this measures a combination of velocity and gravity - in SR it does measure velocity but you can also use reflection (radar) redshift, which does not depend on the object internal clock I think. The measurements you do are defined by your (local) apparatus, not by someone else's.

Sure you can see the wave passing without the need of synchronization, but seeing the wave passing is not a measurement of velocity.

On the other hand, radar involves the two-way speed of light, so you have to define its one-way speed anyways.

 

[BiGyElLoWhAt]

On the other hand, radar involves the two-way speed of light, so you have to define its one-way speed anyways.

I still don't like that word. You mean define c=c? Or less arbitrarily, c =2.99...x10^8m/s? I don't see how that is a definition, and not just an observation. When I think of a definition, I think of $\Phi=B\cdot A$, the definition of magnetic flux. We made that quantity up, it did not exist before someone defined it. c, however existed always. The speed of light in a vacuum has always been the same, even before we took our first measurements of it.

 

[Cruz Martinez]

I still don't like that word. You mean define c=c? Or less arbitrarily, c =2.99...x10^8m/s? I don't see how that is a definition, and not just an observation. When I think of a definition, I think of $\Phi=B\cdot A$, the definition of magnetic flux. We made that quantity up, it did not exist before someone defined it. c, however existed always. The speed of light in a vacuum has always been the same, even before we took our first measurements of it.

We made up the quantity "speed" as well, that's not the problem.
The thing is you run into a circular argument when you try to set up things so that you can masure this quantity, especifically with synchronization.

 

[BiGyElLoWhAt]

The quantity, not the value.

 

[wabbit]

You need synchronization to measure a one-way velocity. You calculate the one-way speed by dividing the distance traveled by the difference between the start and end times - and you need a synchronization convention to determine at least one of those time

Oh OK I missed the significance of that "one way" mention. For that you need a choice of coordinates (which implies a choice of simultaneity), I understand - but still your own clock is enough : you can measure the two events with your local equipment, and for instance assign them radar coordinates, and from this, differences in time and a distance.
But if the point is that some coordinates depend on remote clocks, of course I won't argue with that.Edit - I probably misunderstood the original comment, I thought it said you always needed to synchronize two clocks before you could measure a (one way) velocilty - I am not disagreeing with with what you say, that it implies a simultaneity convention.

 

[Cruz Martinez]

The quantity, not the value.

Yes, the two way speed of light is observed, but the one-way speed is defined necessarily, it can't be other way. With this I mean the actual value has to be defined.
I'm not sure I understand your point actually, do you mean the actual real number c has always existed?

 

[BiGyElLoWhAt]

Ahh, I see. I still probably wouldn't use the word "defined" there, but I see where you're coming from, now. Me personally, I would say calculated, but that's just me.

 

[nitsuj]

I don't know how I feel about that word, defined. We defined the speed of light to be the same? I thought we just noticed that it was.

It's postulated to be invariant. The assertion of it being defined may have been referring the 1/2 a round trip (average speed) = c calculation. (i.e. c defined as 1/2 of a round trip, though have never seen it "defined" as being that)

 

[newjerseyrunner]

Speed is relative to who is observing. Two protons moving perfectly parallel to each other, no matter how fast they are moving to us, are moving at 0m/s in relation to each other.

 

[ghwellsjr]

Oh OK I missed the significance of that "one way" mention. For that you need a choice of coordinates (which implies a choice of simultaneity), I understand - but still your own clock is enough : you can measure the two events with your local equipment, and for instance assign them radar coordinates, and from this, differences in time and a distance.

Yes, you can use just your own clock and radar to establish the time and distance to remote events but during that process, you must apply Einstein's second postulate that each radar signal takes the same amount of time to reach its target as it takes for the return echo to get back to you. This is where you define what simultaneity means between a local clock and a remote event.

Look at the first article of Einstein's 1905 paper entitled "Definition of Simultaneity" where he uses the word "defined" or "definition" 13 times.

 

[ghwellsjr]

It's postulated to be invariant. The assertion of it being defined may have been referring the 1/2 a round trip (average speed) = c calculation. (i.e. c defined as 1/2 of a round trip, though have never seen it "defined" as being that)

See the link in my previous post and you will see where it is "defined".

 

[wabbit]

Yes, you can use just your own clock and radar to establish the time and distance to remote events but during that process, you must apply Einstein's second postulate that each radar signal takes the same amount of time to reach its target as it takes for the return echo to get back to you. This is where you define what simultaneity means between a local clock and a remote event.

Look at the first article of Einstein's 1905 paper entitled "Definition of Simultaneity" where he uses the word "defined" or "definition" 13 times.

Sure, You define coordinates, so you define simultaneity which means equal time coordinate.

 

[ghwellsjr]

Sure, You define coordinates, so you define simultaneity which means equal time coordinate.

Does this mean you are ready to retract your statements from post #15?

Well, you need a synchronization convention to measure one-way speeds, and the standard synchronization method uses the one-way speed of light, this speed is necessarily a defInition.

Why ? Doesn't the standard method use light because it is already known experimentally to travel at the invariant speed?

But actually I don't see where you need synchronization to measure a velocity. One observer can do that alone, he doesn't need to talk to anyone else. And he can also compare relative (to him) velocity of waves emitting by two other sources without ever synchronizing with those sources - and see whether that velocity is independent of the motion of those sources relative to him.

 

[wabbit]

As I made clear in this statement, I was referring to synchronisation as a process involving two clocks. Since you mean something else by this (namely defining simultaneity from an observer's viewpoint), my statement stands but it doesn't contradict yours.

 

[nitsuj]

See the link in my previous post and you will see where it is "defined".

pulled from the paper "We have not defined a common “time” for A and B, for the latter cannot be defined at all unless we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A."


Is that what you were referring to as being "invariance of c defined"? I really thought it is postulated to be invariant, by the SR postulate.

... the speed of light is defined to be the same value in any inertial reference frame ...

to me the above is about invariance something I assume is true for every observer, below is about defining the value, something I can measure and calculate..

Well, you need a synchronization convention to measure one-way speeds, and the standard synchronization method uses the one-way speed of light, this speed is necessarily a defInition.

 

[nitsuj]

I still don't like that word. You mean define c=c? Or less arbitrarily, c =2.99...x10^8m/s? I don't see how that is a definition, and not just an observation. When I think of a definition, I think of $\Phi=B\cdot A$, the definition of magnetic flux. We made that quantity up, it did not exist before someone defined it. c, however existed always. The speed of light in a vacuum has always been the same, even before we took our first measurements of it.

To what level of precision? the concept of "speed" is somewhat* different in the case of an invariant one. who cares the "speed" (value)

*understatement

 

[ghwellsjr]

As I made clear in this statement, I was referring to synchronisation as a process involving two clocks. Since you mean something else by this (namely defining simultaneity from an observer's viewpoint), my statement stands but it doesn't contradict yours.

Please don't get my comments mixed up with others. I never discussed synchronization and I didn't even bring up simultaneity. I just agreed with your comment regarding a single observer's clock and radar. My point is that it is not possible to measure the propagation of light, instead it is defined to be c in every Inertial Reference Frame.

I object to your statement that light "is already known experimentally to travel at the invariant speed". You are claiming that it is possible to measure the propagation of light rather than recognizing that it is defined or postulated or stipulated or assumed. We measure the invariant speed, c, with a two way or roundtrip setup. Do you see the difference?

 

[bahamagreen]

Speed c is a maximum and an "absolute" speed in the sense that it will be found to be the same in every IRF... the word "absolute" being operative to the OP's question about a minimum speed... I'm supposing that the OP is looking at SR and I think the full version of the OP's question might be:

If all IRFs have a maximum speed of c,
and all IRFs agree on the value of c,
how is it that individual IRFs each have a minimum speed,
but those minimum speeds do not agree when compared (mutually observed)?

The implication of the question seems to be that if absolute maximum speeds agree, so should the absolute minimums, yet they don't; how can there be an absolute (universal) maximum speed but not a corresponding universal absolute minimum speed?

This is the kind of question where the logical baggage of the word "absolute" may be getting lost during the flight. We have two words that seem to mean the same thing: speed and motion.
Per Wiki:
Motion: "In physics, motion is a change in position of an object with respect to time."
Speed: "...the speed of an object is the magnitude of its velocity (the rate of change of its position);..."

I can't explain it, but what is very weird is if the word "speed" is replaced with "motion", then the role of the word "absolute" seems to take on a different sense. If one is saying "speed" but thinking of "motion" in a natural verbal-logical fashion, there is a seemingly logical argument that has confused many...

Two non-accelerating objects in relative motion...
Verbal-logically;
- both are in absolute (certain) relative motion with respect to the other
- therefore, at least one of the two objects must be in absolute (certain) motion without having to specify that the motion is relative; relative motion is motion, thus it implies the existence of absolute motion without further stipulation of reference
- therefore, there is an absolute rest with respect to absolute motion
- one of the objects might be at absolute rest, but not both; both may be in absolute motion

This kind of logic finds absolute rest by finding absolute motion from relative motion. Using "motion", it seems by verbal-logic that any relative motion demonstrates absolute motion without having to identify an at-rest motion of no magnitude - it is implied from absolute motion in spite of not being able to assign a magnitude to either the absolute motion or absolute rest.

From a Classical / Euclidean standpoint, the certainty of relative motion, hence the certainty of motion itself, hence the certainty of absolute motion and therefore absolute rest seems pretty strong. The weakest link may be in the step where the certainty of relative motion leads to the certainty of motion itself, leading from the certainty of motion itself, to that there be certainty of absolute motion prior to knowing (and not having to know) what reference that absolute motion uses as its absolute rest... but this still seems pretty strong.

Yet, when the argument is shifted to "speed" rather than motion, even before evoking SR, the sense of the implications don't hold the same. The subsequent steps of the argument are now flimsy and confused because relative speed seems to take the argument differently than relative motion... the natural language sense of motion must be a little different from that of speed?

 

[ghwellsjr]

Speed c is a maximum and an "absolute" speed in the sense that it will be found to be the same in every IRF...

Speed c is "found to be the same in every IRF"? No, it is defined to be c in every IRF.

The remainder of your post appears to me to be at odds with the stated purpose of this forum--to teach relativity--it's not to try to understand the confusion that the OP is suffering from or to try to second-guess what his real concern is. If he doesn't want to come back and clarify, there's no point in trying to go beyond the obvious and simple answer that has been provided multiple times: zero.

 

[Khashishi]

In quantum mechanics, Heisenberg uncertainty principle puts a minimum on $\Delta p \Delta x$ which sort of puts a limit on speed. It's not a barrier like c, but anything below that limit is smeared out.

 

[Nick666]

If he doesn't want to come back and clarify, there's no point in trying to go beyond the obvious and simple answer that has been provided multiple times: zero.

I read all the replies, and I'm hoping I'm the one that will get clarification

The idea of minimum speed just accidentaly popped in my mind the other days when reading threads here about the (maximum) speed of light etc.

And I googled to see what I get and I got this result.

http://arxiv.org/pdf/1211.3612v1.pdf

"This research aims to consider a new principle of symmetry in
the space-time by means of the
elimination of the classical idea of rest including a
universal minimum limit of speed in the subatomic
world. Such a limit, unattainable by the particles, represents
a preferred inertial reference frame
associated with a cosmological background field that breaks
Lorentz symmetry. So there emerges
a modified relativistic dynamics with a minimum speed related
to the Planck length of the early
universe, which leads us to search for a quantum gravity at low energies"

 

[nitsuj]

Please don't get my comments mixed up with others. I never discussed synchronization and I didn't even bring up simultaneity. I just agreed with your comment regarding a single observer's clock and radar. My point is that it is not possible to measure the propagation of light, instead it is defined to be c in every Inertial Reference Frame.

I object to your statement that light "is already known experimentally to travel at the invariant speed". You are claiming that it is possible to measure the propagation of light rather than recognizing that it is defined or postulated or stipulated or assumed. We measure the invariant speed, c, with a two way or roundtrip setup. Do you see the difference?

You referenced a section of the paper titled Definition of Simultaneity that defines c to be 1/2 a round trip, circumnavigating the simultaneity issue of measuring c one way. That says nothing (directly) about invariance, which is postulated; not defined, implicit is it's assumed always true...not sure how stipulated can fit in there.

It is already postulated to travel at an invariant speed, which is known experimentally to still be true.

I don't object to their statement; it says nothing about measuring "the propagation of light", which of course isn't possible...but measuring-calculating the one way speed is, as you pointed out that process is defined.

Invariance does not mean maximum.

 

[ghwellsjr]

You referenced a section of the paper titled Definition of Simultaneity that defines c to be 1/2 a round trip, circumnavigating the simultaneity issue of measuring c one way.

No it doesn't, the definition of c is given for the full round trip, not 1/2:

That says nothing (directly) about invariance,

That says nothing about invariance, either directly or indirectly. It's simply a statement about the experimentally derived fact of the universally measured round trip speed of light.

which is postulated; not defined, implicit is it's assumed always true...not sure how stipulated can fit in there.

In the introduction, Einstein states his second postulate concerning the one-way speed of light "that light is always propagated in empty space with a definite velocity c". In the first article, he goes into detail and uses the terms "defined" and "definition". These are not in conflict with his first term. I use the additional terms "assumed" and "stipulated" in contrast to "measured" or "experimentally determined".

It is already postulated to travel at an invariant speed, which is known experimentally to still be true.

There is no experiment to show that light propagates at c, only that it is compatible with the second postulate.

I don't object to their statement; it says nothing about measuring "the propagation of light", which of course isn't possible...but measuring-calculating the one way speed is, as you pointed out that process is defined.

Agreed.

Invariance does not mean maximum.

I didn't say it did.

 

[Nugatory]

The idea of minimum speed just accidentaly popped in my mind the other days when reading threads here about the (maximum) speed of light etc.

And I googled to see what I get and I got this result.

http://arxiv.org/pdf/1211.3612v1.pdf

Stuff like this is the reason that not all arxiv papers are acceptable references here. It's an interesting speculation, but it's still far from part of accepted scientific thinking. Unless and until these ideas can be leveraged to solve some currently unsolved problem, explain some otherwise unexplainable experimental observation, or improve our current solutions they will remain that way.

 

[nitsuj]

No it doesn't, the definition of c is given for the full round trip, not 1/2:

That says nothing about invariance, either directly or indirectly. It's simply a statement about the experimentally derived fact of the universally measured round trip speed of light.

The idea it has to be measured by a defined method, is because it is invariant. otherwise there would be no simultaneity issues with moving clocks

In the introduction, Einstein states his second postulate concerning the one-way speed of light "that light is always propagated in empty space with a definite velocity c". In the first article, he goes into detail and uses the terms "defined" and "definition". These are not in conflict with his first term. I use the additional terms "assumed" and "stipulated" in contrast to "measured" or "experimentally determined".

definite is not the same word as defined. "always" & "definite velocity" is about invariance...not what ever the value maybe or how to measure it.

There is no experiment to show that light propagates at c, only that it is compatible with the second postulate.

I don't know what you mean with " no experiment to show that light propagates at c". A method to measure the DEFINITE speed is defined. The fact that it's invariant, is still observed to be true.

Here is a link with a table of the history of calculations of the speed of light, first of which was in 1676 using predicted & observed orbits.

Here is another wording, speaking to the definition of a second and the implication..."This defines the speed of light in vacuum to be exactly 299,792,458 m/s." taken from here, which is different from the defined method to measure light traveling that speed.

 

[LTadam]

Hi all. I think this is an interesting question. I think that minimum speed is so hard to define because there are no true inertial reference frames without first defining an inertial reference frame.

 

[PWiz]

there are no true inertial reference

Theoretically, they exist. Practically, they don't.

without first defining an inertial reference frame.

An inertial frame of reference is a well defined concept. The minimum speed which can exist will always be 0.

 

[William Turley]

What is the minimum speed in the universe ?

why not consider negative speed, i.e. where were you prior to now?

 

[Nugatory]

why not consider negative speed, i.e. where were you prior to now?

People often say "speed" when they mean "velocity" or vice versa, but when we're being careful with words "speed" refers to the magnitude of a velocity vector and therefore is never negative. That's why the speedometer of a car starts at zero and doesn't have any room for negative speeds - if I'm driving north at 30 km/hr my velocity is 30 km/hr north, and if I'm driving south my velocity is -30 km/hr, but either way the speedometer reads 30 km/hr and that's my speed relative to the surface of the earth.

 

[nitsuj]

People often say "speed" when they mean "velocity" or vice versa, but when we're being careful with words "speed" refers to the magnitude of a velocity vector and therefore is never negative. That's why the speedometer of a car starts at zero and doesn't have any room for negative speeds - if I'm driving north at 30 km/hr my velocity is 30 km/hr north, and if I'm driving south my velocity is -30 km/hr, but either way the speedometer reads 30 km/hr and that's my speed relative to the surface of the earth.

Or maybe more interestingly...how does one "envision" a "negative" speed, to your point of it being directionless. And I'm not getting much from

i.e. where were you prior to now?

to know which way is which (assuming velocity was intended meaning, as you inferred).

 

[theodoros.mihos]

I think there is no confuse if we make a 2D Minkowski's diagram. For all frames, even for angles up to 45 degrees (any that can mean) we can define speed and velocity both well. And for all, the minimum speed is zero.