Compute Limit of 4-Vectors: p and q

In summary, the problem is to compute the quantity -2(pq)^2/q^2 in the limit q->0, given that p is a timelike four-vector and q is a spacelike four-vector. The attempt to consider a special frame with p=(p^0,0) and q=(0,ε,ε,ε) did not lead to a successful solution. Another attempt to consider a special frame where p-q=(p^0-q^0,0) also did not provide a clear solution. The identity (p+q)^2=p^2+q^2+2pq was suggested, but it does not seem to be of much help in this case.
  • #1
parton
83
1
I've the following problem. I have two four-vectors p and q where p is timelike ([tex] p^{2} > 0 [/tex]) and q is spacelike([tex] q^{2} < 0 [/tex]).
Now I should consider the quantity

[tex] - \dfrac{2 (pq)^{2} + p^{2} q^{2}}{q^{2}} [/tex]

and compute the limit [tex] q \to 0 [/tex].

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 [tex] p=(p^{0}, \vec{0}) [/tex] but it doesn't help.
 
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  • #2
Can you show some more detail of the work you did?
 
  • #3
My attempt so far was not successfully. I considered a special frame where [tex] p = \left( p^{0}, \vec{0} \right) [/tex] which is possible, because p is timelike. Furthermore I defined [tex] q = (0, \epsilon, \epsilon, \epsilon) [/tex]. This will lead to:

[tex] - \dfrac{2 (pq)^{2} + p^{2} q^{2}}{q^{2}} = - p_{0}^{2} [/tex]

and for arbitrary p we should have: [tex] - p^{2} [/tex].

But somehow I don't think that I can specify q in this way. Another choice of q, e.g. [tex] q = (\epsilon, \epsilon, \epsilon, \epsilon) [/tex] would lead to a vanishing contribution [tex] = 0 [/tex], 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
I've one further information, but I don't know if it helps: [tex] (p-q) \in V^{+} [/tex].

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

Which leads to [tex] \vec{p} = \vec{q} [/tex] and therefore:

[tex] - \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} [/tex]

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

But it appears questionable to do the computation like this.

Could anyone help me please?
 
  • #5
Pephaps the identity (p+q)2 = p2 + q2 +2pq might be of help.
 
  • #6
... it does not really help
 

Related to Compute Limit of 4-Vectors: p and q

What is a 4-vector?

A 4-vector is a mathematical concept used in physics and mathematics to describe quantities that have both magnitude and direction in four-dimensional spacetime.

How do you compute the limit of 4-vectors p and q?

To compute the limit of 4-vectors p and q, you need to first determine the components of the vectors in four-dimensional spacetime. Then, you can use mathematical techniques such as vector addition and subtraction, dot product, and cross product to manipulate the vectors and eventually find the limit.

What is the significance of computing the limit of 4-vectors p and q?

Computing the limit of 4-vectors p and q can provide valuable information about the behavior and properties of physical systems in four-dimensional spacetime. It can also help to predict and analyze the outcomes of various physical phenomena.

What are some applications of computing the limit of 4-vectors p and q?

Some applications of computing the limit of 4-vectors p and q include studying the motion of particles in special relativity, analyzing the behavior of electromagnetic fields, and predicting the dynamics of quantum systems.

Are there any limitations to computing the limit of 4-vectors p and q?

Yes, there are certain limitations to computing the limit of 4-vectors p and q. For example, the calculations can become very complex and difficult to solve for systems with a large number of interacting particles. Additionally, the accuracy of the results may be affected by experimental errors or uncertainties in the measurements of the 4-vectors.

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