We can use the chain rule to find the spatial components of U μ for μ = i = 1,2,3:īut dx i/dt is the particle's ordinary spatial velocity v = dx 1/dt, dx 2/dt, dx 3/dt = v x, v y, v z so that finally the particle's four-velocity is finally given by: Taking the derivative with respect to propert time, we can then rewrite that: To determine the components of the four-velocity vector, we recall that a process that takes a proper time ΔΤ in its own rest frame has a longer duration Δt measured by another observer moving relative to the rest frame, i.e The four-velocity of a particle is then defined as the rate of change of its four-position with respect to proper time, and is also the tangent vector to the particle's world line The vector that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space, is a four-vector and is called the Four-velocity vector.Īs we have seen in Proper time, a clock fastened to a particle moving along a world-line in four-dimensionnal spacetime will measure the particle's proper time τ and therefore it makes sense to use τ as the parameter along the path. Generally speaking mathematically, one can define a 4-vector a to be anything one wants, however for special relativity between one Inertial Frame of Reference and another, our 4-vectors are only those which transform from one inertial frame of reference to another by Lorentz transformations. These coordinates are the components of the position four-vector for the event. Also the definition x 0 = ct ensures that all the coordinates have the same units (of distance). r = r( t), this corresponds to a sequence of events as t varies. If r is a function of coordinate time t in the same frame, i.e. Where r is the three-dimensional space position vector. More precisely, a point in Minkowski space is a time and spatial position, called an Event, or sometimes the position four-vector or four-position or 4-position, described in some reference frame by a set of four coordinates: In special relativity, a four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notation The spatial velocity of the particle is a tangent vector to the path and can be written as: The three fuctions x=f(t), y=f(t), z=f(t) are called parametric equationsand give a vector whose components represent the object's spatial velocity in the three x,y,z directions. The path of a particle moving in ordinary three-dimensional Euclidean space can be described using three functions of time t, one for x, one for y and one for z. In non-relativistic physics, the velocity of an object is a three dimensional vector whose components give the object’s speed in each of three directions (the directions depend on the coordinate system).
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |