General Relativity. Note 7

[ home ]

Hilbert action for the gravitational field

General Relativity. Note: « 7 »

Unnamed Web page

Hilbert action for the gravitational field

The most successful action for the gravitational field is the Hilbert action S_g = −¹/_2κ ∫ R√[−g]dΩ, where κ is a constant (Einstein's constant). Its variation δS_g under a variation δg^ab is δS_g = − ¹/_2κ ∫ (R_ab−¹/₂g_abR) δg^ab√[−g]dΩ.

Gravitational field equations (Einstein equations)

From the least action principle δS_g+δS_m = 0, where the variation of the matter action δS_m is δS_m = ¹/₂ ∫ T_ab δg^ab√[−g]dΩ, we find the gravitational field equations R_ab−¹/₂g_abR = κ T_ab.

Exercises

From the variational principle δ S=0 derive the corresponding equations of motion by directly calculating the variation of the action for the following systems (flat space unless stated otherwise):
- Nonrelativistic particle in a potential V: S=∫(¹/₂mv²−V(r))dt [answer: m^d²r/_dt²=−^dV/_dr]
- Free relativistic particle in a curved space: S_p=−m∫ ds [answer: ^du_a/_ds=¹/₂g_bc,au^bu^c]
- Free electromagnetic (EM) field: S_em=−¹/_8π∫ A^a_,b A_a^,bdΩ [answer: A^a,b_,b=0]
- EM field created by a given current j_a: S =−∫ A^aj_adΩ+S_em [answer: A^a,b_,b=4πj^a]
- A particle with charge q in an EM field A^a: S = −mc∫ ds − q∫ A^adx_a [answer: m^{du^a}/_ds=F^abu_b]