2.2 The Lagrangian, evolution equations and conserved quantities

The EKG evolution equations can be derived from the action [219]
( ) ∫ 1 √ --- 4 𝒮 = -----R + ℒℳ , − gd x (7 ) 16πG
where R is the Ricci scalar of the spacetime represented by the metric gab, and its determinant √ −-g. The term ℒℳ describes the matter, which here is that of a complex scalar field, ϕ
1[ ab ¯ ( 2)] ℒ ℳ = − 2 g ∇a ϕ ∇bϕ + V |ϕ | , (8 )
where ¯ ϕ is the complex conjugate of the field and 2 V (|ϕ| ) a potential depending only on the magnitude of the scalar field, consistent with the U (1) invariance of the field in the complex plane.

Variation of the action in Eq. (7View Equation) with respect to the metric gab leads to the well-known Einstein equations

R- Rab − 2 gab = 8πGTab (9 ) 1[ ] 1 [ ( ) ] Tab = --∇a ¯ϕ ∇b ϕ + ∇a ϕ∇b ¯ϕ − --gab gcd∇c ¯ϕ ∇d ϕ + V |ϕ|2 , (10 ) 2 2
where Rab is the Ricci tensor and Tab is the real stress-energy tensor. Eqs. (9View Equation) form a system of 10 non-linear partial differential equations for the spacetime metric components gab coupled to the scalar field via the stress-energy tensor given in Eq. (10View Equation).

On the other hand, the variation of the action in Eq. (7View Equation) with respect to the scalar field ϕ, leads to the Klein–Gordon (KG) equation

ab -dV-- g ∇a ∇bϕ = d|ϕ|2ϕ. (11 )
An equivalent equation is obtained when varying the action with respect to the complex conjugate ¯ϕ. The simplest potential leading to boson stars is the so-called free field case, where the potential takes the form
V (|ϕ|2) = m2 |ϕ|2, (12 )
with m a parameter that can be identified with the bare mass of the field theory.

According to Noether’s theorem, the invariance of the Klein–Gordon Lagrangian in Eq. (8View Equation) under global U(1) transformations ϕ → ϕeiφ implies the existence of a conserved current

i( ) Ja = --ϕ¯∇a ϕ − ϕ ∇a ¯ϕ , (13 ) 2
satisfying the conservation law
∇ J a = √-1--∂ (√ −-g gabJ ) = 0. (14 ) a − g a b
The spatial integral of the time component of this current defines the conserved Noether charge, given by
∫ 0b √ --- 3 N = g Jb − g dx , (15 )
which can be associated with the total number of bosonic particles [188].
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