Integrating to get r^2 = l^2 + a^2

  • Thread starter Lapidus
  • Start date
  • #1
Lapidus
343
11
I'm reading a physics book( "Einstein's gravity in a nutshell" Zee) and at page 126 the author says:

(dr/dl)^2 + a^2 / r^2 = 1

gives r^2 = l^2 + a^2

where we absorbed an integration constant into l by setting l=0 when r=a.

Can someone explain what's going on here? How do I have to "massage" the first equation to make an integration that gives the second equation? What does Zee mean with absorbing an integration constant?

Thanks for any respons!
 

Answers and Replies

  • #2
36,216
8,199
I'm reading a physics book( "Einstein's gravity in a nutshell" Zee) and at page 126 the author says:

(dr/dl)^2 + a^2 / r^2 = 1

gives r^2 = l^2 + a^2

where we absorbed an integration constant into l by setting l=0 when r=a.

Can someone explain what's going on here? How do I have to "massage" the first equation to make an integration that gives the second equation? What does Zee mean with absorbing an integration constant?

Thanks for any respons!
Rewrite the DE as dr/dl = ##\pm \sqrt{1 - \frac{a^2}{r^2}} = \pm \frac{\sqrt{r^2 - a^2}}{|r|}##
This equation is separable, with a fairly easy integration. The work is simpler if you can assume that r > 0 and dr/dl > 0.
 
  • #3
Lapidus
343
11
Rewrite the DE as dr/dl = ##\pm \sqrt{1 - \frac{a^2}{r^2}} = \pm \frac{\sqrt{r^2 - a^2}}{|r|}##
This equation is separable, with a fairly easy integration. The work is simpler if you can assume that r > 0 and dr/dl > 0.

Thanks, Mark. But I am afraid, I do not follow. For example, how does the l^2 emerge?
 
Last edited:
  • #4
36,216
8,199
If it's reasonable to assume that dr/dl > 0 and r > 0, we can write the DE as ##dr/dl = \frac{\sqrt{r^2 - a^2}}{r}##.

Separating this, we get ##\frac{r dr}{\sqrt{r^2 - a^2}} = dl##.

Now integrate both sides.
 
  • Like
Likes Lapidus

Suggested for: Integrating to get r^2 = l^2 + a^2

Replies
6
Views
2K
Replies
3
Views
2K
Replies
2
Views
2K
  • Last Post
Replies
8
Views
3K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
6
Views
5K
  • Last Post
Replies
1
Views
1K
Replies
3
Views
5K
Replies
2
Views
2K
Top