- #1

- 99

- 0

Thanks!

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter copernicus1
- Start date

- #1

- 99

- 0

Thanks!

- #2

- 33,757

- 12,123

I'm having a hard time showing this from the general expression for the 4-acceleration.

What general expression are you starting from?

- #3

- 99

- 0

http://en.m.wikipedia.org/wiki/Four-acceleration

- #4

- 33,757

- 12,123

I'm starting from the expression on this page

Yes, that will work. How are you getting from that to the second expression you gave in your OP?

- #5

Chestermiller

Mentor

- 21,046

- 4,641

Chet

- #6

Bill_K

Science Advisor

- 4,155

- 201

He's right, you know! In the first and second equation,ais the same in the third equation as in the first and second equations. If you want to be working with the same 4 acceleration, you have to apply the Lorentz Transformation to its components.

Chet

- #7

- 99

- 0

This is the way I went about it:

The 4-acceleration is: $$a^\mu=\left(\gamma^4\frac{{\bf v}\cdot{\bf a}}c,\gamma^2{\bf a}+\gamma^4\frac{\bf v\cdot a}{c^2}{\bf v}\right).$$

The magnitude should be given by $$a_\mu a^\mu=\gamma^8\frac{\left|{\bf v\cdot a}\right|^2}{c^2}-\gamma^4|{\bf a}|^2-\gamma^8\frac{|{\bf v\cdot a}|^2}{c^4}|{\bf v}|^2-2\gamma^6\frac{|{\bf v\cdot a}|^2}{c^2}.$$

The first and third terms can be combined to give $$\left(1-\frac{|{\bf v}|^2}{c^2}\right)\gamma^8\frac{\left|{\bf v\cdot a}\right|^2}{c^2}=\gamma^6\frac{|{\bf v\cdot a}|^2}{c^2}.$$ So we now have $$a_\mu a^\mu=-\gamma^4|{\bf a}|^2-\gamma^6\frac{|{\bf v\cdot a}|^2}{c^2}.$$

- #8

- 99

- 0

- #9

- 1,948

- 200

You can still improve it as

[tex]a_\mu a^\mu=-\gamma^4|{\bf a}|^2-\gamma^6\frac{|{\bf v\cdot a}|^2}{c^2}.[/tex]

[tex]a_\mu a^\mu=-\gamma^6[\gamma^{-2}|{\bf a}|^2-\frac{|{\bf v\cdot a}|^2}{c^2}].[/tex]

[tex]a_\mu a^\mu=-\gamma^6[(1-\frac{|{\bf v}|^2}{c^2})|{\bf a}|^2-\frac{|{\bf v}|^2|{\bf a}|^2(cos \theta)^2}{c^2}].[/tex]

[tex]a_\mu a^\mu=-\gamma^6 |{\bf a}|^2[1-\frac{|{\bf v}|^2}{c^2}(1+ (cos \theta)^2)].[/tex]

That is, if that is an improvement...

[tex]a_\mu a^\mu=-\gamma^4|{\bf a}|^2-\gamma^6\frac{|{\bf v\cdot a}|^2}{c^2}.[/tex]

[tex]a_\mu a^\mu=-\gamma^6[\gamma^{-2}|{\bf a}|^2-\frac{|{\bf v\cdot a}|^2}{c^2}].[/tex]

[tex]a_\mu a^\mu=-\gamma^6[(1-\frac{|{\bf v}|^2}{c^2})|{\bf a}|^2-\frac{|{\bf v}|^2|{\bf a}|^2(cos \theta)^2}{c^2}].[/tex]

[tex]a_\mu a^\mu=-\gamma^6 |{\bf a}|^2[1-\frac{|{\bf v}|^2}{c^2}(1+ (cos \theta)^2)].[/tex]

That is, if that is an improvement...

Last edited:

Share: