# How do we find/deduce Einstein's field equation?

hugo_faurand
TL;DR Summary
How do we derive field equations ?
Hello everyone !

I'm getting into General relativity. I wonder know how we find the Einstein's field equation.

Maybe we can have an intuition with the strong equivalence principle.

So if you can enlight me ☺️☺️ please

Regards

Mentor
I wonder know how we find the Einstein's field equation.

In Misner, Thorne & Wheeler's classic textbook on GR, they describe six different ways to derive the Einstein Field Equation, from six different sets of starting assumptions.

vanhees71 and Orodruin
Mentor
Chapter 4 in Sean Carroll's online lecture notes on GR [1] also describes a derivation of the Einstein Field Equation from reasonable physical principles and the requirement for correspondence with Newtonian gravity in the low speed, weak field approximation.

[1] https://arxiv.org/pdf/gr-qc/9712019.pdf

vanhees71 and PeroK
Gold Member
John Baez, who has written a lot of articles for this site, has a nice formulation of this, which involves considering the deformation of a falling ball of liquid and applying various conservation laws. He arrives at the Einstein equations without the standard appeal to Riemannian geometry, which is hard to parse without background. Basically he is able to derive how the shape of a ball of matter deforms based on how much pressure the matter in the ball exerts, which is equivalent to the effect of the stress energy tensor on geodesics.

https://arxiv.org/pdf/gr-qc/0103044.pdf

vanhees71 and Nugatory
Staff Emeritus
Gold Member
How do we derive field equations ?

How is ##F = ma## derived?

How is the equation for Newtonian gravity,
$$F = G \frac{m_1 m_2}{r^2},$$

derived?

vanhees71
Mentor
he is able to derive how the shape of a ball of matter deforms based on how much pressure the matter in the ball exerts

A clarification: it's the energy density and pressure of the matter, not just the pressure.

hugo_faurand
Chapter 4 in Sean Carroll's online lecture notes on GR [1] also describes a derivation of the Einstein Field Equation from reasonable physical principles and the requirement for correspondence with Newtonian gravity in the low speed, weak field approximation.

[1] https://arxiv.org/pdf/gr-qc/9712019.pdf

This is exactly what I needed, first I searched something like that in Tong's course but there were nothing.

Thank you !

Mentor
Thank you !

You're welcome!

Gold Member
2022 Award
How is ##F = ma## derived?

How is the equation for Newtonian gravity,
$$F = G \frac{m_1 m_2}{r^2},$$

derived?
It's not derived but found from observations and ingenious mathematical insight. The same holds of course for GR. My favorite way is the one by Hilbert, i.e., looking for the most simple generally covariant action that can be built from the metric tensor and leads to 2nd-order partial differential equations (see Landau&Lifhitz vol. 2 or Weinberg, Gravitation).

Another alternative way, which emphasizes that gravitation is an interaction and not a priori a manifestation of a pseudo-Riemannian spacetime manifold is the way how Feynman (in the "Feynman lectures on graviation") derives it. There he makes indeed use of the strong equivalence principle. Then the geometrical reinterpretation becomes a deduced property, but the calculation is somewhat lengthy compared to the action-principle approach.

You may also like to look up Nordstrom gravity. Einstein gravity is not the unique relativistic theory of gravity compatible with Newtonian gravity. So it is not possible to uniquely derive Einstein gravity on those grounds alone, since Nordstrom gravity is also a possibility. Nordstrom gravity, however, is not compatible with the observed perihelion motion of Mercury.

vanhees71
Mentor
Nordstrom gravity, however, is not compatible with the observed perihelion motion of Mercury.

It also predicts zero light bending by the Sun and a much smaller Shapiro time delay than GR does.

vanhees71 and atyy
John Baez, who has written a lot of articles for this site, has a nice formulation of this, which involves considering the deformation of a falling ball of liquid and applying various conservation laws. He arrives at the Einstein equations without the standard appeal to Riemannian geometry, which is hard to parse without background. Basically he is able to derive how the shape of a ball of matter deforms based on how much pressure the matter in the ball exerts, which is equivalent to the effect of the stress energy tensor on geodesics.

https://arxiv.org/pdf/gr-qc/0103044.pdf
But this is not a derivation of the field equation at all. Instead, it is an elegant description of its physical content

vanhees71 and George Jones
You may also like to look up Nordstrom gravity. Einstein gravity is not the unique relativistic theory of gravity compatible with Newtonian gravity. So it is not possible to uniquely derive Einstein gravity on those grounds alone, since Nordstrom gravity is also a possibility. Nordstrom gravity, however, is not compatible with the observed perihelion motion of Mercury.
It also does not satisfy the strong equivalence principle when an EM wave packet is considered. Clifford Will has argued in several places that GR is the only known theory that does. See:

https://arxiv.org/pdf/1104.4608.pdf

which despite its abstract, notes the exception in footnote 2 on page 4.

vanhees71 and atyy
Staff Emeritus