What Defines Future-Directed Vectors in Physics?

[Einj]

Hello everyone,
I have a very basic question about future-directed vectors. Are they defined as those vectors whose temporal component is positive or strictly positive?

I need to check wether a certain system satisfies the null energy condition or not and I was wondering if I am allowed to take a vector $k^M$ such that $k^2=0$ and $k^t=0$.

Thanks a lot!

 

[Orodruin]

Are we talking special or general relativity here?

In SR (given a set of Minkowski coordinates), being future directed is equivalent to having a positive t-component. In GR, you have to take more care.

 

[Einj]

I was asking about general relativity and I'm particularly interested to know if we're talking about strict positivity or just positivity. Thanks!

 

[Orodruin]

In GR you first have to define what is "future". In a general coordinate system, it is not certain that there is a "time" coordinate as this may differ from event to event. Instead, you want to introduce a time-like vector field $V$, which by definition has to be non-zero everywhere ($V^2 > 0$ - with +--- convention). In a general manifold, it is not even certain that such a vector field exists, but if it does, the manifold is time-orientable. A vector $k$ is future-directed if it is non-space-like and $k\cdot V > 0$.

 

[Einj]

Ok thanks! Is there any criterion to pick $V$? For example, in AdS, can I just choose $V=(1,\vec 0)$? In this case, doesn't future-directed simply mean $k^t>0$?

 

[Orodruin]

Ok thanks! Is there any criterion to pick $V$? For example, in AdS, can I just choose $V=(1,\vec 0)$? In this case, doesn't future-directed simply mean $k^t>0$?

If you have a global coordinate system where one coordinate is time-like everywhere, then yes. You can pick that coordinate vector field as the defining one. It is not going to matter which vector field you select as different vector fields which are both time-like wrt each other will give equivalent definitions.