# What is the slowest possible speed?

According to special relativity, nothing can go faster than the speed of light, and nothing can be distinguished to be in a state of absolute rest.

So it makes me wonder: is there a slowest possible speed?

Right now I'm wanting to think that there isn't. I'm pondering the idea that no matter what speed you're going, there could very well be some object (1) that is going a relativistic speed faster than an object (2) in your local reference frame, and at the same time, you going at a relativistic speed faster than another object (3). True, something moving in the opposite direction at a fast speed may appear to be going slower than you, and I'm not sure how that fact affects my hypothesis. I'd like to see others' analysis of this "no slowest possible speed" idea to see whether I'm on the right track or just way off.

It's interesting to think that there may be a fastest speed and no slowest speed, when our intuition gives us the opposite inclination.

• Surely the slowest speed would be zero, with an object being at rest. Any object can be made to have zero velocity by taking a reference frame about that object. And surely no speed can be slower than zero. – Involutius Jan 5 '15 at 4:13
• It is to be noted that via an analysis of absolute motion ( Speeds of 0 -> infinity ) that is ongoing within an absolute 4 dimensional Space-Time environment, this soon leads you to an independent discovery of Special Relativity, along with having achieved an independent creation of all of the SR equations. Thus, absolutes must be in no way excluded from reality. This includes absolute spatial rest. – Sean Jan 5 '15 at 20:21
• Your supposition that there is a slowest speed assumes that there is an absolute reference frame to measure that speed with respect to. – Chris Mueller Mar 24 '15 at 20:17

The slowest speed possible with respect to your reference frame is achieved by you, regardless of special relativity.

...in your local reference frame, and at the same time, you going at a relativistic speed faster than another object...

I'm not quite sure how to interpret this. By definition, in your local reference frame, your speed = 0.

I think you're misinterpreting special relativity.

nothing can be distinguished to be in a state of absolute rest.

This does not mean you cannot have a speed = 0 from a particular reference frame, but rather that unless you are moving at c, your velocity will change between reference frames.

• I guess I'm trying to say that while the lowest speed in a particular reference frame is going to be zero, there may not be a lowest absolute speed. Absolute speed sounds like a ridiculous concept. But light has an absolute speed since it's the same speed in every reference frame. To my knowledge, it's the only absolute speed ever measured. So I hypothesize that there's no absolute speed that you can measure where you can say, "Nothing can go slower than this." – theboombody Jan 5 '15 at 4:56
• Light does not have an absolute speed. It has the same speed in any frame of reference. These are not the same thing. – caveman Jan 5 '15 at 17:53
• Okay, I have a new question springing off of this discussion. Is it possible for an object to have the same speed in any frame of reference that's actually slower than the speed of light? – theboombody Jan 5 '15 at 18:14
• No. Unless you are moving at c, your speed changes between reference frames. That is true in Newtonian mechanics, too (without the exception for c) – robertkin Jan 6 '15 at 21:07
• This answer seems incorrect because an observer does not have a speed with respect to their own reference frame. An observer cannot move away from him/herself nor leave their own local frame of reference, and therefore cannot measure a speed with respect to her/himself or with respect to their own local frame of reference. To say that this means "speed = 0" feels very tautological. I think the spirit of the original question is to ask whether there is a smallest number beyond which no smaller number can be gathered when deriving speed from observations. – CommaToast Mar 24 '15 at 19:23

According to special relativity, nothing can go faster than the speed of light,

The speed of matter and the propagation of information is believed to be limited by the speed of light.

and nothing can be distinguished to be in a state of absolute rest.

Here you mix in the concept of absolute frames of reference.

There is also no absolute speed of 80 miles per hour. Of course there is the absoulute speed of light. :-)

So for matter it depends on the frame of reference what speed you attribute to it.

So it makes me wonder, is there a slowest possible velocity?

A speed of zero is perfectly legal for matter, as you can attach a frame of reference to it it, where it rests.

Light does not rest.

Right now I'm wanting to think that there isn't. I'm pondering the idea that no matter what speed you're going, there could very well be some object (1) that is going a relativistic speed faster than an object (2) in your local reference frame, and at the same time, you going at a relativistic speed faster than another object (3). True, something moving in the opposite direction at a fast speed may appear to be going slower than you, and I'm not sure how that fact affects my hypothesis.

That would be the case, if velocities $v_1$ and $v_2$ add up like $v_1 + v_2$.

However for matter that is not the case, at speeds close to the speed of light, the addition of velocities is notably different (link).

In the end matter is bound to travel at a speed $0 \le v < c$.

And for light you can do what you want, it travels at $c$ (in vacuum).

I'd like to see others' analysis of this "no slowest possible speed" idea to see whether I'm on the right track or just way off.

It's interesting to think that there may be a fastest speed and no slowest speed, when our intuition gives us the opposite inclination.

Matter can be at rest ($v=0$). No problem here.

• Yeah, but if matter is at rest, you have no possible way of telling it's at rest. For all you know, it could be moving at a constant velocity. The only things you can say are at rest are the things you assign to be at rest. So anyone who says, "There is an absolute state of rest for matter" is pretty much saying, "I know it's true, I just can't point out an example." There is no absolute rest frame. And since we can't tell what's at absolute rest and what isn't, how can we tell what's moving more slowly in the absolute picture? – theboombody Jan 5 '15 at 13:10
• First the property of matter to be at rest ($v=0$) is possible. Determining the velocity of matter is a different problem. There is no absolute rest frame, but there are relative rest frames (those with origin coinciding with the position of the object). If you know how your own frame of reference relates to such a rest frame you can start making measurements of objects and relate positions, velocities etc. between frames. You can of course compare the properties of different objects. – mvw Jan 5 '15 at 18:54
• I should maybe add that extreme large or small distances come with additional challenges. – mvw Jan 5 '15 at 19:06

Light does not have an absolute speed. It has the same speed in any frame of reference. These are not the same thing.

Regarding the other question: Is it possible for an object to have the same speed in all frames of reference that's actually different from the speed of light?

Consider this argument (modified from this page): Suppose an object A is moving with a velocity v relative to an object B, and A is moving with a velocity w relative to object C. Finally, B is moving with a velocity u relative to an object C.

What speed must v have so w = v? This will give the answer to our question.

                         v
u      -------> A
-------> B
C        w
----------------->


In non-relativistic mechanics the velocities are simply added and the answer is that A is moving with a velocity w = u+v relative to C. But in special relativity the velocities must be combined using the formula $$w = \frac{u+v}{1+\frac{uv}{c^2}}$$ We just need to set w = v, and solve for v: $$v = \frac{u+v}{1+\frac{uv}{c^2}}$$ $$v + \frac{uv^2}{c^2} = u + v$$ $$\frac{v^2}{c^2} = 1$$ $$v^2 = c^2$$ $$v = c$$ This is the only solution, and it is true for all u.

Well, not exactly, note that if u = 0, we divided by 0, so we should make this a special case. But in that case $$w = \frac{0 + v}{1 + 0} = v$$

So, the only way where an object can have the same velocity in two inertial reference frames is if the relative velocity of the reference frames is zero or where the object is traveling at the speed of light in both reference frames.

Also, note that if an object is going the speed of light in one reference frame, that it is going the speed of light in all reference frames is easily derived from this equation.

Replace v with c and see that w must also equal c: $$w = \frac{u+c}{1+\frac{uc}{c^2}}$$ $$w = \frac{u+c}{1+\frac{u}{c}} = \frac{1}{\frac{1}{c}}\cdot\frac{u+c}{c+u} = c$$

When pondering the question of the slowest possible speed, I initially found myself agreeing with all of the other given answers. There is no absolute rest velocity because there is no absolute rest frame. That means you will never find a reference frame with a velocity that can be boosted to and then boosted back from that would allow you to age faster (not sure why you'd want to). You definitely age the fastest at rest in your own inertial frame (ignoring one-way trips to other inertial frames).

But then I thought, "Maybe there's no slowest local velocity, but is there a slowest global velocity? One where even an alien race in a distant galaxy would be likely to determine their velocity wrt to it?" The answer to that question is yes. There is a universally agreeable rest velocity and that comes to us courtesy of the cosmic microwave background (CMB). See, the CMB is pretty much isotropic to any comoving observer. A comoving observer is one that is not moving wrt local space. The only motion a comoving observer has is its recessional velocity, which is due to the expansion of space and isn't really influenced by special relativity. This would, therefore, make an excellent definition of the absolutely lowest possible speed. The measure of an object's peculiar velocity, which is the velocity that it has different from a comoving object, would be something even distant alien races could agree on as a measure of how fast something is moving locally.

When something has a non-zero peculiar velocity, that causes the CMB that it observes to doppler shift. We can also measure the peculiar velocity of distant objects by finding the difference in the redshift of light from them from what we'd expect a comoving object at that distance to have. This means we can measure our velocity relative to local background space and the velocity of distant objects relative to the distant local background space and this number is something everyone would agree on (our peculiar velocity is around 0.0012c).

So while it's true you could always find a frame of reference where something else is moving slower than an object with 0 peculiar velocity, I daresay you won't find such a universally agreeable frame of reference to use. As long as we can assume the CMB is isotropic for a comoving observer, then this becomes a very natural frame of reference to use to say "the slowest possible velocity is one with no peculiar velocity". Even if that's just imposing a preferred frame of reference, it's an agreeable one to impose.

TL;DR There's not really an absolute rest velocity, but there is a universally agreeable rest velocity that we can use just the same.

• In what way do you define the frame known as the universe. Is not an expanding universe at rest relative to an expanding universe, meaning at rest relative to itself? – Sean Jan 5 '15 at 20:25
• @Sean I figure when asking what something is at rest relative to, I have found the root object when I reach an infinite loop. An expanding universe is at rest relative to itself, which is at rest relative to itself, which is at rest..... You get the idea. If the natural tendency is to define something's velocity relative to itself, then it's probably something that most would accept as a more or less absolute reference frame – Jim Jan 5 '15 at 20:30
• My point? No one will say "the universe is moving a speed v wrt this object in it" for any practical purposes. So it might as well be the reference for our "absolute" frame – Jim Jan 5 '15 at 20:33

I feel strongly there is no way to prove that such thing as a "slowest possible non-zero speed" exists. However there are realistic limits regarding the degree to which slower and slower speeds can reliably be measured.

Example (purely hypothetical)

Consider a pair of neutrinos, ν1 and ν2, emitted from the sun, one right after the other, at times T0 and T1, where:

T1 - T0 = tp (1 Planck time)

Now consider that they travel along the exact same path over the course of 500 billion trillion quadrillion years, and eventually, they arrive at some distant galaxy, where some aliens have a neutrino detector set up. When they get there, they arrive at times T0' and T1', where:

T1' - T0' = 2tp (2 Plank times)

What is the relative velocity difference between the two neutrinos when they were emitted from the sun? It's a really darn small number.

This number could continue getting smaller the farther away the alien detector was, assuming T1' - T0' remained 2tp. Therefore there does not seem to be a minimum possible speed.

Discussion

Clearly this hypothetical example is ridiculous. I made it that way to illustrate a point, which is that determining a velocity requires two measurements, and it requires being able to identify something as being the same thing you observed both times.

In our example the aliens would have no way to know if the neutrinos both came from exactly the same place or not. Even if they did they'd have no way to observe when the neutrinos were emitted relative to one another.

I have a very strong feeling that the minimum speed that it is possible to observe depends heavily upon the detection equipment, and even then, there are realistic limits to consider.

For example, are we talking about trying to measure the relative speed difference between two objects that are moving at relativistic speeds compared to us? Or are we staring at an apparently immobile object and waiting a million years for it to move a single plank length?

I think there comes a time when an observer would either give up and say, "It ain't moving," or they would die.

Furthermore, instruments used to detect speed are generally calibrated to units of time such as meters per second with a limited degree of accuracy. Something moving 1 plank length per hundred googolplex years would not be considered "in motion," would it?

To put it another way the minimum possible measurable non-zero speed is lp/∂T, where ∂T is the change in time from the first point the object was measured at to the second point the object was measured at, and lp is a planck length.

• By this argument, zero is the smallest velocity difference. You seem to be saying that the slowest speed is zero, or at rest. But that is neither here nor there as this whole answer is an opinion and not based on any physics or scientific expertise – Jim Mar 24 '15 at 19:45
• That's not what I'm trying to say. I added some elucidation that, I hope, might help. My point is that while I don't think there is a smallest possible non-zero speed, there will be a smallest possible measurable non-zero speed, based on many factors. – CommaToast Mar 24 '15 at 19:49

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