# 8π in Einstein Equations

1. Mar 10, 2013

### NicosM

Hi!

I would like to ask where the factor of 8π comes from in the Einstein Equations. I understand that it is augmented in order to comply with the Newtonian limit. But why did Einstein decide to place 8π and not for example 8.1π or 3π-3π/e=8.27 or whatever around this value (which would give almost the same correspondence with the experiment)?

Thanks

2. Mar 10, 2013

### kevinferreira

It comes from the Newtonian limit, as you said, and the Newtonian formula for the potential $\Delta\Phi=4\pi\rho$, which gets multiplied by 2 somewhere. This $4\pi$ factor here ultimately comes from the spherical symmetry of the Newtonian gravitation theory.

3. Mar 10, 2013

### NicosM

Thanks for your reply. I understand the 4π factor of the Newtonian formula. Can you tell me where the 2 comes from, in case you know? Thanks again

4. Mar 10, 2013

### Ben Niehoff

Ultimately, the constant in Einstein's equations is arbitrary and can be set to any value with an appropriate choice of units. I typically use

$$R_{\mu\nu} = \frac12 T_{\mu\nu}$$
because I do theoretical physics and I don't care about units. I guess I'm working with $\hbar = 1, c = 1, G = 1/16\pi$.

5. Mar 10, 2013

### NicosM

Thanks for your reply too. Then my question should be restated as why G/h=1/8 ? :)

6. Mar 10, 2013

### Ben Niehoff

It doesn't mean anything at all. G and Planck's constant have different units, so G/h also has units, and can be set to any value we choose by choosing the right system of units.

The only meaningful numbers in physics are dimensionless (ironically enough). The rest is convention.

7. Mar 11, 2013

### PAllen

Isn't it because Einstein enjoyed pie more than Newton?

8. Mar 11, 2013

### NicosM

Actually Archimedes enjoyed that more than both of them :)

9. Mar 11, 2013

### rbj

there is a very good and very old discussion of this $8 \pi$ thing in an old sci.physics.research thread.

from John Baez:
from Daryl McCullough:
i think i transcribed the math into LaTeX accurately.

i found this paper to be very useful:

http://math.ucr.edu/home/baez/einstein/einstein.html][/PLAIN] [Broken] The Meaning of Einstein's Equation

and the main statement in the paper:
Baez and Bunn tell us that this statement is equivalent to Einstein's

$$G_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu}$$

the constant of proportionality in the Baez-Bunn statement is, i believe, $\frac{4 \pi G}{c^2}$ not $\frac{8 \pi G}{c^2}$. i believe it is:

$${\ddot V\over V} \Bigr\vert _{t = 0} = - \frac{4 \pi G}{c^2} \left(\rho c^2 + P_x + P_y + P_z \right)$$

this reinforces my opinion that normalizing $4 \pi G = 1$ rather than $8 \pi G = 1$ is more natural in defining the most "natural" units. i think Planck missed it by a factor of $4 \pi$ regarding $G$ and i think that subsequently, they should have normalized $\epsilon_0$ rather than $4 \pi \epsilon_0$ regarding the electric charge in either cgs or in Planck units..

Last edited by a moderator: May 6, 2017
10. Mar 11, 2013

### NicosM

thanks rbj

11. Mar 11, 2013

### rbj

yer welcome. can't say that i can explain it.