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3 and 4 velocity and acceleration

  1. Jul 1, 2008 #1

    could someone clarify some concepts and definitions for me, concerning how velocity and acceleration are measured?

    Please consider the following setup (to keep it simple for me :biggrin:)

    We have a linear accelerator in a lab. A small mass (maybe a proton) is accelerated to 0.8c in one second. The accelerator is designed to accelerate the mass with constant proper acceleration. All motion is in a straight line and I only want to consider motion along the x axis and time t.

    0.8c is the velocity measured in the lab frame in the normal way as v=x/t. (Thats the 3 velocity right?)

    From the relativistic rocket equations http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html the proper acceleration (a) can be found from the equation:

    [tex] a = \frac{v}{t\sqrt{1-(v/c)^2}}[/tex]

    which with t=1 second and v=0.8c as measured in the lab the value of the constant proper acceleration works out as 1.3333

    So after 1 second in the lab frame with constant proper acceleration of 1.3333 what is the:

    Proper velocity of the particle? (I assume zero?)
    3 velocity of the particle in the lab frame? (I assume 0.8c?)
    4 velocity of the particle? (c?)

    Proper acceleration of the particle? (1.3333 assumed)
    3 acceleration of the particle in the lab frame? (I assume 1.3333/gamma^3)?
    4 acceleration of the particle?

    where gamma is 1/0.6 calculated from the instaneous velocity of 0.8c

    Also, what is the:

    Proper force acting on the particle?
    3 force acting on the particle?
    4 force acting on the particle?

    After 2 seconds with constant acceleration the velocity of the particle in the lab would be

    [tex] v = \frac{at}{\sqrt{1+(at/c)^2}} = 0.93633c [/tex]

    what would the corresponding proper, 3 and 4 vector quantities be after 2 seconds?

    If you can give any additional insight as to what the terms mean in a practical sense that would be very much appreciated :smile:

    It is often mentioned that acceleration and the force causing the acceleration are not always parallel in relativity. I assume that is not he case in this particular example?
  2. jcsd
  3. Jul 1, 2008 #2


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    I haven't heard the term "proper velocity" before. It seems like a useless concept if it's defined so that it's always zero. :smile:
    Of course.
    You know it's (1 0) in the co-moving frame, so it must be

    [tex]\gamma\begin{pmatrix}1 && v\\v && 1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}[/tex]

    where v=0.8. Its Minkowski space norm is always 1 (i.e. =c) though.
    Yes, your calculation looks correct.

    That's what I get too.
    It's components are (0 a) in the co-moving frame, so use the same Lorentz transformation as for the four-velocity.
    No time to think about the forces now. Maybe later.
  4. Jul 1, 2008 #3
    The term proper velocity does not make any sense in relativity. As velocity is a relative not an absolute concept.

    Having said that I would not be shocked if cosmologists were to find a good meaning for it. ;)
  5. Jul 1, 2008 #4


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    Although the name "proper velocity" is probably not appropriate,
    I believe it refers to the "spatial-component of the 4-velocity of an object",
    with magnitude [tex]\sinh\theta[/tex] (in natural units) where [tex]\theta[/tex] is the rapidity.
    ([tex]\sinh\theta[/tex] is sometimes called the "celerity".) In more three-dimensional terms, it is [tex]\gamma \vec v[/tex].

    It can probably be thought of as the "relativistic 3-momentum divided by the rest mass".
    Last edited: Jul 1, 2008
  6. Jul 1, 2008 #5
    Some texts, such as the SR text by Sears and Brehme, use the term proper velocity as a synonym for four-velocity.

  7. Jul 1, 2008 #6
    The term "Proper Distance" is widely used basic terminology in SR and GR literature. Taylor & Wheeler give a clear explanation of Proper Distance on the 4th page of their textbook "Exploring Black Holes." Here's a brief excerpt:

    "...proper distance between [two events in spacetime is more formally called] the spacelike spacetime interval. All free-float observers agree on the value of the proper distance - the proper distance is an invariant. In contrast, the value of t and the value of s between these events typically differ, respectively, as measured in different frames. Proper distance can be used to describe the separation between any pair of events for which s is greater than t. It tells the observer in any frame what the [proper distance] is between the events as measured in a frame in which they occur at the same time."

    Here is a brief excerpt from the definition of Proper Distance and Proper Velocity from the paper "Coordinate Confusion in Conformal Cosmology", 7/07, by Lewis, Francis, Barnes & James, in the context of the FLRW metric. Proper Velocity is defined the way you would expect it to be: The derivative of Proper Distance as a function of Proper Time in the coordinate system being used.

    "A fundamental definition of distance in general relativity is the proper distance, defined as the spatial separation between two points along a hypersurface of constant time. Given the form of the FLRW metric the radial distance from the origin to a coordinate x along a hypersurface of constant t is;

    Dp = a(t) x

    Taking the derivative with respect to coordinate time [which is synchronous for all comoving observers (fixed x) and is equivalent to their proper time [tex]\tau[/tex]] we obtain what we will refer to as the proper velocity .... [equations] ...

    Last edited: Jul 1, 2008
  8. Jul 1, 2008 #7
    So already in the above posting there are two completely different definitions of proper distance. :)
  9. Jul 1, 2008 #8
    Hi MeJennifer,
    Don't follow your point. There is a basic concept of proper distance in SR, and then there is an equation stated for applying it to an expanding FLRW metric. Same exact concept, applied to two different metrics. As you know better than most, the metric for an expanding FLRW universe is different from an empty SR universe.

  10. Jul 1, 2008 #9
    One definition pertains to a distance in spacetime while the other definition pertains to a distance in space. Two entirely different things.
  11. Jul 1, 2008 #10
    Hi MeJennifer,

    I don't understand how a "spacelike spacetime distance" is different from "the spatial separation between two points along a hypersurface of constant time", as long as the elapsed time is zero in both cases, which is the whole point of the measurement. The only difference I can see is that the Taylor & Wheeler definition is broader to encompass situations where the elapsed time isn't zero, but that broader aspect isn't relevant if one wants simply to measure physical distance.

    Last edited: Jul 1, 2008
  12. Jul 1, 2008 #11
    Hi all,

    thanks for the answers so far, but if anything I am more confused than before I asked the question(s) :P

    I put "proper velocity" in the list of questions mainly for completeness (not wanting to take anything for granted) and partly because some texts seem to use it loosely to mean 4 velocity.

    I think I understand the part about the definition of proper distance in an accelerating context where an observer with constant proper acceleration in a rocket or in a gravitational field considers herself to remain at at a constant distance from the origin of their own coordinate system, in the same way we consider our distance from the centre of the Earth to remain constant even though we experience constant acceleration.

    Anyone want to take a shot at the force definitions?
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