Free particle in Minkowski spacetime

[coopre]

Homework Statement

A free particle is moving in the x direction through Minkowski spacetime,
and has velocity V as measured by a stationary observer at x = 0; t = 0. Express
the particle's world-line parametrically in terms of V , parametrized by the particle's
proper time 

Homework Equations

unknown

The Attempt at a Solution

Help I have been stuck on this problem for hours now, and what I keep getting is wrong so idk where to start now please help!

 

[tiny-tim]

welcome to pf!

hi coopre! welcome to pf!

you're looking for a parametrisation (x,t) = (x(τ),t(τ)), where τ is the proper time of the particle

(y and z will be constant)

you know (x,t) = (Vt,t) …

so find τ as a function of t (and V), then invert it to get t as a function of τ (and V) …

what do you get? ​

 

[camron_m21]

I was getting confused on this one as well; it seemed too easy. My solution turned out to be:

t = τγ, and x = Vτ/γ, by using dx/dt = γ dx/dτ.

Hopefully this is correct, helpful, or both.

 

[tiny-tim]

hi camron_m21!

t = τγ, and x = Vτ/γ,

one of them is right!

 

[paranormal]

I understand your frustration and I am happy to help you with this problem. First, let's define some terms. Minkowski spacetime is a mathematical model that combines space and time into a single four-dimensional continuum. In this model, the x direction represents space and the t direction represents time. A free particle is a particle that is not subject to any external forces, meaning it is moving at a constant velocity.

Now, let's look at the problem. We are given that the particle is moving in the x direction and has a velocity V as measured by a stationary observer at x = 0; t = 0. This means that at the initial time t = 0, the particle is located at x = 0 and is moving with a constant velocity V in the positive x direction.

To express the particle's world-line parametrically in terms of V, we can use the following equations:

x = Vt
t = τ

where x represents the position of the particle in the x direction, t represents time, and τ represents the particle's proper time. The proper time of a particle is the time experienced by the particle itself, as opposed to the time measured by an external observer.

Substituting t = τ into the equation x = Vt, we get:

x = Vτ

This is the parametric equation for the particle's world-line. It shows how the particle's position in the x direction changes as a function of its proper time. We can see that as the proper time increases, the particle's position in the x direction also increases at a constant rate of V.

I hope this helps you understand the problem better and gives you a starting point to solve it. Remember to always define your variables and use the appropriate equations to solve the problem. Good luck!