Question

In a 2-dimensional Minkowski spacetime i.e.x^μ=(t,x), you can define the metric simply by the Minkowski metric,ds²=-dt²+dx², and the Christoffel symbols vanish. If you have a worldline described byγ^μ(λ)=(λ)whereλ has its usual meaning andais a real constant, how would you go about parametrizing the worldline in terms of proper time\touch that you find an expression in the form ofγ^μ(τ)?

Answer 1

To parametrize the worldline in terms of proper time τ, we can solve for τ by integrating the proper time interval and considering the initial conditions. The expression for τ in terms of the parameter λ representing the worldline is τ = -√(1 - v²)t, where v is the velocity of the particle.

Step-by-step explanation:

In order to parametrize the worldline in terms of proper time τ, we need to find an expression for τ in terms of λ, which represents the parameter along the worldline. The proper time interval is given by:

ds² = -dt² + dx² = -δt² + δx² = -dt² + λ² = -dτ².

We can solve for τ by integrating:

∫ dτ = ∫ √(-dτ²) = ∫ √(-1) ∫ √(dt² - λ²) = -i ∫ √(dt² - λ²) = -i ∫ √(dt²(1 - λ²/dt²)) = -i ∫ √(dt²(1 - v²)) = -√(1 - v²) ∫ dt = -√(1 - v²)t + C.

Since τ is the proper time of the worldline, τ = 0 when t = 0, so C = -√(1 - v²) * 0 = 0. Therefore, the expression for τ in terms of t is:

τ = -√(1 - v²)t.