Square of the difference of four-vectors

[Arny_Toynbee]

What is the correct way to expand (p₃-p₄)² where p₃ and p₄ are 4-vectors, with metric g_{mu nu}=diag[1,-1,-1,-1], p = [w_p, p], where p is 3-vector, and w_p= (p²+m²)^(1/2)

 

[Ibix]

Do you know how to take the dot product of two vectors when the metric isn't trivial? (If I'm allowed to phrase it that way.)

 

[Arny_Toynbee]

p₃.p₄ = (p₃⁰, p₃) . (p₄⁰, p₄)

and p₃⁰= w_p3 etc.

 

[Ibix]

And do you know how to expand that? In Euclidean 3-space $\vec a.\vec b=a_xb_x+a_yb_y+a_zb_z$. Do you know the equivalent Minkowski 4-space expression?

 

[Arny_Toynbee]

And do you know how to expand that? In Euclidean 3-space $\vec a.\vec b=a_xb_x+a_yb_y+a_zb_z$. Do you know the equivalent Minkowski 4-space expression?

p₃.p₄ = = p₃⁰ p₄⁰ - p₃.p₄

 

[Ibix]

Right. So when you replace the vector with a difference of two vectors what do you get?

 

[Arny_Toynbee]

Right. So when you replace the vector with a difference of two vectors what do you get?

The question is
(p₃ - p₄)^2 - whether it is (w_p3-w_p4)² + (p₃ - p₄)²

or is it

(w_p3+w_p4)² - (p₃ + p₄)²

Hmmmmmm.....what do I get?

 

[PeterDonis]

.what do I get?

Instead of guessing, write it out explicitly: $(p_3 - p_4)^2 = (p_3 - p_4) \cdot (p_3 - p_4)$, and then just expand out the product and do the algebra.

 

[Arny_Toynbee]

Instead of guessing, write it out explicitly: $(p_3 - p_4)^2 = (p_3 - p_4) \cdot (p_3 - p_4)$, and then just expand out the product and do the algebra.

How would you write out (p₃ - p₄)?

There's a bit of subtlety here, and not guessing, see for e.g. Equation 46.29 here

 

[jtbell]

How would you write out (p₃ - p₄)?

You don't need to do that. The 4-vector product distributes over addition and subtraction just like the ordinary scalar product does.

 

[PeterDonis]

How would you write out (p3 - p4)?

see for e.g. Equation 46.29

You do realize that that equation answers the question you just asked?

 

[PAllen]

How would you write out (p₃ - p₄)?

There's a bit of subtlety here, and not guessing, see for e.g. Equation 46.29 here

First, vector subtraction against an orthonormal basis is always just what you would expect.

Second, you don't need to do this anyway because dot product distributes over vector addition/subtraction.

 

[Ibix]

The question is
(p₃ - p₄)^2 - whether it is (w_p3-w_p4)² + (p₃ - p₄)²

or is it

(w_p3+w_p4)² - (p₃ + p₄)²

Hmmmmmm.....what do I get?

You wrote down an expression for the dot product of two vectors in terms of the components. So if the vectors, instead of $p_3$ and $p_4$, are both $p$, and $p=p_3-p_4$ what do you get?

 

[Arny_Toynbee]

You do realize that that equation answers the question you just asked?

Actually, you may not realize, there might be a surprise on the way!

 

[Ibix]

Actually, you may not realize, there might be a surprise on the way!

Perhaps it would be helpful if you either completed the algebra we've been discussing or said what you think is wrong with 4.29 in the pdf you linked.

 

[Arny_Toynbee]

Perhaps it would be helpful if you either completed the algebra we've been discussing or said what you think is wrong with 4.29 in the pdf you linked.

The link provided may be using a different convention. Until now, I have provided all the algebra and the question. It was suggested that "...realize that the link you provided...has the solution...?" The original question has two alternatives, and it is useful to see what the forum members come up with. This is not a homework that is to be submitted to the forum in all its completeness, for it to be graded.

 

[Ibix]

The link provided may be using a different convention. Until now, I have provided all the algebra and the question. It was suggested that "...realize that the link you provided...has the solution...?" The original question has two alternatives

You mean your expressions in #7? Neither appears to be correct, as PeterDonis said in #8. How did you get them? I haven't tried to replicate 4.29 from your link, but it looks plausible at a quick glance.

 

[PAllen]

The link provided may be using a different convention. Until now, I have provided all the algebra and the question. It was suggested that "...realize that the link you provided...has the solution...?" The original question has two alternatives, and it is useful to see what the forum members come up with. This is not a homework that is to be submitted to the forum in all its completeness, for it to be graded.

The issue is that for everyone who has responded on this thread so far, your problem is trivial, yet you seem to have misunderstandings. We feel it is much more useful to help you resolve these than simply give you the answer.

 

[PAllen]

You mean your expressions in #7? Neither appears to be correct, as PeterDonis said in #8. How did you get them? I haven't tried to replicate 4.29 from your link, but it looks plausible at a quick glance.

I did replicate that equation, just doing the algebra in my head.

 

[Ibix]

I did replicate that equation, just doing the algebra in my head.

Thank you. I was confident up to sign errors on the cross terms - it's been a long day...

 

[PeterDonis]

Until now, I have provided all the algebra

Have you done the algebra I asked for in post #8? If so, post it. If not, do it and post it.

The original question has two alternatives

Do you mean the question you asked in post #7? Neither of those alternatives is correct.

Now go do the algebra I asked you to do in post #8, so you can compute what the right answer is instead of guessing.

This is not a homework that is to be submitted to the forum in all its completeness, for it to be graded

No, but that doesn't mean we're just going to tell you an answer that you could derive for yourself by doing basic algebra. PF is not an oracle that you can feed guesses to and get yes/no answers back.

 

[Arny_Toynbee]

Thank you. I was confident up to sign errors on the cross terms - it's been a long day...

Sorry, sign errors in which equation? By the way, the way it is derived in the link is not necessarily consistent.

What is the correct answer (of the alternatives posted)?

 

[PeterDonis]

What is the correct answer (of the alternatives posted)?

Do the algebra yourself and find out what the correct answer is. As I said, PF is not an oracle.

Thread closed.