Four-Vectors: Definition Issues & Exam Prep

[Bigfoots mum]

Im having some four-vector definition issues. I have a relativity exam coming up and they quite often ask about 4-vectors.

1) Does this definition sound ok?
'A four-vector is 4 numbers, say X=(X⁰, X¹, X², X³), used to describe an event in minkowski space. The 'zeroth' is the time component, while the other 3 components are the spatial components of a 3-vector. A four-vector differs from a 3-dimensional vector in that it can undergo a lorentz transformation and remain a four-vector. '
2) How do i show that a vector is actually a four vector?
Do i just show that it remains a valid four vector under a lorentz transformation?

Any help is greatly appreciated
thanks

 

[peterjaybee]

Hi, the best way to define 4 vectors are by their transformation properties, which is essentially what you have said. In equation form it is

$$\widehat{x}^{\mu}=\Lambda^{\mu}_{\nu}x^{\nu}$$

Here Lambda is the mu, nu component of the lorentz matrix, and the einstein summation convention is used.

 

[diazona]

2) How do i show that a vector is actually a four vector?
Do i just show that it remains a valid four vector under a lorentz transformation?

Yep, that works. I think it's also possible to show that it's a four-vector by demonstrating that if you contract it with another four-vector, the result is Lorentz-invariant (i.e. is a scalar). Sometimes that might be easier.