Covariant Characterization of Causality in Continuum: T^{ik}v_k

[Emil_M]

Hi!

Let $T^{ik}$ be the stress-energy-tensor, and $v_k$ some future-pointing, time-like four vector.

How can I see that the object $T^{ik}v_k$ is future-pointing and not space-like?

Thank you for your help!

 

[robphy]

What is the dot product of two future pointing timelike vectors in your signature convention?

 

[Emil_M]

$v_k v^k$

 

[PeterDonis]

How can I see that the object $T^{ik}v_k$ is future-pointing and not space-like?

Compute its squared length. What do you get?

 

[Emil_M]

Compute its squared length. What do you get?

The thing is, I am not really sure how to do that :)

What I'd do is the following:

$T^{ik}v_k T_{il}v^l=T^{ik}v_k \eta_{im}\eta_{ln}T^{mn} \eta^{lj}v_j$ but I'm not sure if that leads anywhere...

 

[PeterDonis]

What I'd do is the following

This is correct as an expression for the squared length of the vector, yes. If that vector is timelike, then the sign of its squared length must be the same as the sign of $v_k v^k$. How would you go about comparing the signs of the two?

(One hint: you should find that, in order for the signs of the two to be the same, you have to impose conditions on the components of $T^{ij}$; i.e., the vector you're looking at is not always timelike, it only is if $T$ satisfies certain conditions.)

 

[Emil_M]

If that vector is timelike, then the sign of its squared length must be the same as the sign of $v_k v^k$. How would you go about comparing the signs of the two?

(One hint: you should find that, in order for the signs of the two to be the same, you have to impose conditions on the components of $T^{ij}$; i.e., the vector you're looking at is not always timelike, it only is if $T$ satisfies certain conditions.)

Thanks for your help.

According to my calculations $$T^{ik}v_k T_{ij}v^j=(T^{00}v_0+T^{01}v_1+T^{02}v_2+T^{03}v_3)^2-(T^{10}v_0+T^{11}v_1+T^{12}v_2+T^{13}v_3)^2-(T^{20}v_0+T^{21}v_1+T^{22}v_2+T^{23}v_3)^2-(T^{30}v_0+T^{31}v_1+T^{32}v_2+T^{33}v_3)^2.$$

However, I am struggling to find conditions for $T^{ik}$ from this...

 

[PeterDonis]

According to my calculations

These don't look right. The expression $T^{ik} v_k T_{ij} v^j$ should expand to terms that look like $( T^{00} v_0 + T^{01} v_1 + T^{02} v_2 + T^{03} v_3 ) ( T_{00} v^0 + T_{01} v^1 + T_{02} v^2 + T_{03} v^3 )$, with the index positions on $T$ and $v$ switching from one factor to the other.