How is velocity defined in the context of relativity?

[RK1992]

Sorry, I'm very new to this topic, this is going to annoy you I expect.

How do we define velocity in relativity? There's no fixed aether to measure against.

So in the lorentz transformation, when we say v²/c²...

What does v mean? If we measure against a photon the photon will shoot past at c, so it's obviously not that.

If we measure against something else then we get a velocity relative to something else. Am I right in thinking this is how we measure velocity? And that what we draw from this measured velocity when we use the lorentz transformation is how time dilates in our frame compared to how time dilates in the other frame which we see as "at rest"?

And then the rest mass of something also confuses me...

If there is no definite substance to measure against, how can we say that it has a certain rest mass when for all we know it is moving and therefore the rest mass we emasure is dilated?

I suspect I'm being dumb and that this question is what relativity is based on...

Apologies and thanks in advance.

 

[tiny-tim]

Hi RK1992!

We use photons, but we bounce them off the object, radar-style.

If the photon leaves us at time a and returns to us at time b (measured by our clock), then we say that the distance of the object when the photon hit it was (b+a)/2, and that that was at time (b-a)/2.

The formula for speed I'll leave to you.

(and I'm sorry, i don't understand your question about rest-mass )

 

[Rasalhague]

Sorry, I'm very new to this topic, this is going to annoy you I expect.

How do we define velocity in relativity? There's no fixed aether to measure against.

So in the lorentz transformation, when we say v²/c²...

What does v mean?

These equations transform the space and time coordinates of events from one spacetime coordinate system, call it F, to another, call it F'.

v is the velocity of the space coordinates of F' relative to those of F. In other words, v is the velocity measured in F of anything which is not moving with respect to F'.

And then the rest mass of something also confuses me...

If there is no definite substance to measure against, how can we say that it has a certain rest mass when for all we know it is moving and therefore the rest mass we measure is dilated?

The rest mass of an object is its energy divided by c² as measured in any spacetime coordinate system (reference frame) in which the object is at rest. Such a coordinate system is called the "rest frame" of the object.

There is no absolute notion of "moving" or "not moving". These words have no meaning on their own. If you say an object is moving, you have to say what it's moving relative to.

 

[Altabeh]

Sorry, I'm very new to this topic, this is going to annoy you I expect.

How do we define velocity in relativity? There's no fixed aether to measure against.

I can smell like you're talking about the implication of velocity in Special Relativity so that I continue in this zone. This may be done by calculating the 4-velocity

u^a=dx^a/d\tau

where d\tau is the infinitesimal time interval elapsed on the clock of the traveling object and dx^a being the infinitesiaml space-time components of distance as measured by an inertial observer using his own clock and ruler. The norm of 4-velocity u^a=(u^0,...,u^3) is called "Proper Velocity" or "celerity" which is not frame-dependent i.e. would be the same for all inertial observers measuring it and has the following expression in general:

u=\tanh^{-1}v/c

where v is the coordinate velocity of the moving object as measured by an stationary observer. This latter notion of velocity is coordinate-dependent and is to be defined, in its vector form, by

v^a=dx^a/dt

where now dt represents the infinitesimal time interval elapsed on the clock of an stationary observer and dx^a is calculated in the frame of this observer.

So in the lorentz transformation, when we say v²/c²...

What does v mean? If we measure against a photon the photon will shoot past at c, so it's obviously not that.

v here is the relative speed; i.e. the speed at which one frame moves along the other relative to it. The direction of motion is usually, for simplicity, taken to be that of the x-axis.

If we measure against something else then we get a velocity relative to something else. Am I right in thinking this is how we measure velocity?

Yes.

And that what we draw from this measured velocity when we use the lorentz transformation is how time dilates in our frame compared to how time dilates in the other frame which we see as "at rest"?

Yes.

If there is no definite substance to measure against, how can we say that it has a certain rest mass when for all we know it is moving and therefore the rest mass we emasure is dilated?

Rest mass actually does "look" increased in a relative motion when I'm on a spaceship traveling to moon at a speed close to c, my mass m_0 would seem increased by an amount of \gamma m_0 when observed by an observer at rest in a lab on Earth.

AB

 

[Rasalhague]

I can smell like you're talking about the implication of velocity in Special Relativity so that I continue in this zone. This may be done by calculating the 4-velocity

RK1992, if the hyperbolic tangent function, abstract index notation and the concept of four dimensional vectors in spacetime, etc. are new to you, Altabeh's answer may not make much sense yet, but don't be daunted by it! You'll get there eventually. For now, just bear in mind that 4-velocity is not the same thing as the kind of velocity we're all familiar with, which is 3-velocity. You don't need to know what 4-velocity is to understand that the v in equations such as x' = x(x-vt)/sqrt(1-(v/c)² is the velocity, in the ordinary, everyday sense, of the new coordinate system with respect to the old.

Rest mass actually does "look" increased in a relative motion when I'm on a spaceship traveling to moon at a speed close to c, my mass m_0 would seem increased by an amount of \gamma m_0 when observed by an observer at rest in a lab on Earth.

Here m_0 is what's called "rest mass", and \gamma m_0 "relativistic mass". Some people prefer to use the name "mass" for "rest mass" (and the plain old letter m), and avoid the term "relativistic mass". But conventions vary, so watch out for that.

Oh, and note that even when you're on a spaceship "travelling to moon at a speed close to c" in some coordinate system, such as one in which the moon is taken as motionless, you're still not moving at all in your own rest frame, and (strange as it seems) light still always travels at c relative to you (i.e. as measured in your rest frame).

 

[DrGreg]

I agree with Rasalhague that Altabeh's response will be meaningless to the questioner who is very much a beginner. But for the benefit of anyone else reading it, I must correct some mistakes as follows:

The norm of 4-velocity u^a=(u^0,...,u^3) is called "Proper Velocity" or "celerity" which is not frame-dependent i.e. would be the same for all inertial observers measuring it and has the following expression in general:

u=\tanh^{-1}v/c

where v is the coordinate velocity of the moving object as measured by an stationary observer.

The norm of 4-velocity is actually called "the speed of light". tanh^-1(v/c) is called "rapidity", not "celerity". Rapidity and celerity(=proper velocity) are both frame-dependent. Altabeh, if you want to discuss this further I suggest a new thread as this won't help the questioner.

 

[RK1992]

Yes, I'm afraid that I'm only 17 with A level maths knowledge - I'm going into second year of 6th form and will be doing further maths which includes hyperbolic trig functions, but as of yet they mean nothing to me :( But thanks for the explanations, I can't wait until I have the maths skills to help me understand the effects :)

 

[Altabeh]

I agree with Rasalhague that Altabeh's response will be meaningless to the questioner who is very much a beginner. But for the benefit of anyone else reading it, I must correct some mistakes as follows:
The norm of 4-velocity is actually called "the speed of light". tanh^-1(v/c) is called "rapidity", not "celerity". Rapidity and celerity(=proper velocity) are both frame-dependent. Altabeh, if you want to discuss this further I suggest a new thread as this won't help the questioner.

I must correct myself this way and if you find anything questionable then we can start a new thread and discuss it:

1- The norm of 4-velocity is actually called "the speed of light" as the formula \eta_{ab}u^au^b=c^2 in Minkowskian spacetime shows. I don't know why I wrote the norm equals "proper velocity" and I'm sorry for this. (Maybe for sending this at 04:20 AM -local time-).

2- I said "Proper Velocity" or "celerity" which means they are the same thing.

3-"Rapidity" is defined to be u=\tanh^{-1}v/c and since it contains a coordinate velocity, so it is frame-dependent.

4- I want to support your argument that proper velocity or celerity is frame-dependent since it is the magnitude of a frame-dependent 3-velocity.

AB

 

[Altabeh]

RK1992, if the hyperbolic tangent function, abstract index notation and the concept of four dimensional vectors in spacetime, etc. are new to you, Altabeh's answer may not make much sense yet, but don't be daunted by it! You'll get there eventually. For now, just bear in mind that 4-velocity is not the same thing as the kind of velocity we're all familiar with, which is 3-velocity. You don't need to know what 4-velocity is to understand that the v in equations such as x' = x(x-vt)/sqrt(1-(v/c)² is the velocity, in the ordinary, everyday sense, of the new coordinate system with respect to the old.

Here m_0 is what's called "rest mass", and \gamma m_0 "relativistic mass". Some people prefer to use the name "mass" for "rest mass" (and the plain old letter m), and avoid the term "relativistic mass". But conventions vary, so watch out for that.

Oh, and note that even when you're on a spaceship "travelling to moon at a speed close to c" in some coordinate system, such as one in which the moon is taken as motionless, you're still not moving at all in your own rest frame, and (strange as it seems) light still always travels at c relative to you (i.e. as measured in your rest frame).

Thank you for your clarifications.

AB