# I Square of the difference of four-vectors

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1. Jun 11, 2017

### Arny_Toynbee

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

Last edited: Jun 11, 2017
2. Jun 11, 2017

### 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.)

3. Jun 11, 2017

### Arny_Toynbee

p3.p4 = (p30, p3) . (p40, p4)

and p30= wp3 etc.

4. Jun 11, 2017

### 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?

5. Jun 11, 2017

### Arny_Toynbee

p3.p4 = = p30 p40 - p3.p4

6. Jun 11, 2017

### Ibix

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

7. Jun 11, 2017

### Arny_Toynbee

The question is
(p3 - p4)^2 - whether it is (wp3-wp4)2 + (p3 - p4)2

or is it

(wp3+wp4)2 - (p3 + p4)2

Hmmmmmm.....what do I get?

Last edited: Jun 11, 2017
8. Jun 11, 2017

### Staff: Mentor

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.

9. Jun 11, 2017

### Arny_Toynbee

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

10. Jun 11, 2017

### Staff: Mentor

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

11. Jun 11, 2017

### Staff: Mentor

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

12. Jun 11, 2017

### PAllen

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.

13. Jun 11, 2017

### Ibix

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?

14. Jun 12, 2017

### Arny_Toynbee

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

15. Jun 12, 2017

### Ibix

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.

16. Jun 12, 2017

### Arny_Toynbee

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.

17. Jun 12, 2017

### Ibix

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.

18. Jun 12, 2017

### PAllen

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.

19. Jun 12, 2017

### PAllen

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

20. Jun 12, 2017

### Ibix

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