Homework Help: Limit computation

1. Dec 19, 2009

parton

I've the following problem. I have two four-vectors p and q where p is timelike ($$p^{2} > 0$$) and q is spacelike($$q^{2} < 0$$).
Now I should consider the quantity

$$- \dfrac{2 (pq)^{2} + p^{2} q^{2}}{q^{2}}$$

and compute the limit $$q \to 0$$.

But I don't know how to perform the limit procedure. Could anyone help me please?

I already tried to consider the problem in a special frame with $$p=(p^{0}, \vec{0})$$ but it doesn't help.

2. Dec 19, 2009

diazona

Can you show some more detail of the work you did?

3. Dec 20, 2009

parton

My attempt so far was not successfully. I considered a special frame where $$p = \left( p^{0}, \vec{0} \right)$$ which is possible, because p is timelike. Furthermore I defined $$q = (0, \epsilon, \epsilon, \epsilon)$$. This will lead to:

$$- \dfrac{2 (pq)^{2} + p^{2} q^{2}}{q^{2}} = - p_{0}^{2}$$

and for arbitrary p we should have: $$- p^{2}$$.

But somehow I don't think that I can specify q in this way. Another choice of q, e.g. $$q = (\epsilon, \epsilon, \epsilon, \epsilon)$$ would lead to a vanishing contribution $$= 0$$, so I don't know how to compute the considered quantity. Obviously it depends on the choice of q.

Any idea how to do that?

4. Dec 21, 2009

parton

I've one further information, but I don't know if it helps: $$(p-q) \in V^{+}$$.

So, I also tried to consider a special frame where
$$p-q = (p^{0} - q^{0}, \vec{0})$$.

Which leads to $$\vec{p} = \vec{q}$$ and therefore:

$$- \dfrac{2 (pq)^{2} + p^{2} q^{2}}{q^{2}} = - \dfrac{2 (p^{0} q^{0} - \vec{p} \, ^{2})^{2} + p^{2} (q_{0}^{2} - \vec{p} \, ^{2})}{q_{0} - \vec{p} \, ^{2}} \simeq 2 \vec{p} \, ^{2} - p^{2}$$

Then I rewrite the last $$\vec{p} \, ^{2}$$ into $$\vec{p} \cdot \vec{q}$$ and finally obtain (again): $$-p^{2}$$.

But it appears questionable to do the computation like this.

5. Dec 21, 2009

phsopher

Pephaps the identity (p+q)2 = p2 + q2 +2pq might be of help.

6. Dec 22, 2009

parton

...... it does not really help