Now we want to figure out how to write the laws of physics in the presence of a gravitational field. In other words, we want to find a “recipe” to apply the Principle of General Covariance: we know the mathematical expression describing a certain non-gravitational phenomenon in the Minkowski spacetime in Cartesian coordinates and we want to write the same physical law in the presence of a gravitational field in arbitrary coordinates. The case of non-inertial reference frames in flat spacetime will be automatically included.
As it will be more clear from the examples below, there are some ambiguities when we want to translate the laws of physics from flat to curved spacetimes. This means that purely theoretical arguments are not enough to do it and experiments have to confirm if the new equations are right or not.
As the first step to solve this problem, we can start considering how to write a physical law valid in flat spacetime and Cartesian coordinates for the case of flat spacetime and non-Cartesian coordinates. This has been partially discussed in the previous chapters. We start from manifestly Lorentz-invariant expressions and we make the following substitutions:
1. we replace the Minkowski metric in Cartesian coordinates with the metric in the new coordinates:ημν→gμν;
2. partial derivatives become covariant derivatives:∂μ→ ∇μ;
3. if we have an integral over the whole spacetime (e.g. an action), we have to integrate over the correct volume elementd4Ω:d4x→ |J|d4x, whereJ is the Jacobian. We will see below thatJ =√−g, whereg is the determinant of the metric tensor.
The new equations are written in terms of tensors and therefore they are manifestly covariant. We could apply the same recipe to write the physical laws in curved space- times. For the time being, experiments confirm that we obtain the right equations.
For the point 3 above, we know that, if we move from an inertial reference frame with the Minkowski metricημνand the Cartesian coordinates{xμ}to another refer- ence frame with the coordinates{xμ}, the new metric is
gμν= ∂xα
∂xμ
∂xβ
∂xνηαβ, (6.41)
and therefore the determinant is
6.7 Non-gravitational Phenomena in Curved Spacetimes 119
detgμν=det ∂xα
∂xμ
∂xβ
∂xνηαβ =det
∂xα
∂xμ det
∂xβ
∂xν
detηαβ. (6.42)
Note that2
detgμν= −g, det ∂xα
∂xμ
= J, detηαβ=1, (6.43) whereJis the Jacobian of the coordinate transformationxμ→xμ. From Eq. (6.42), we see thatJ =√−g, and therefore we find that we need the following substitution when we have an integral and we move to an arbitrary reference frame
d4x→ √
−g d4x. (6.44)
Let us consider the electromagnetic field. The fundamental variable is the 4- potential Aμ. In the Faraday tensorFμν, we replace the partial derivatives with the covariant ones, but there is no difference
Fμν = ∇μAν− ∇νAμ=∂μAν−Γμνσ Aσ−∂νAμ+Γνμσ Aσ
=∂μAν−∂νAμ. (6.45)
The Maxwell equations in the manifestly Lorentz-invariant form, in flat spacetime, and in Cartesian coordinates are given in Eqs. (4.40) and (4.48). We replace the partial derivatives with the covariant ones and we have
∇μFνρ+ ∇νFρμ+ ∇ρFμν =0, (6.46)
∇μFμν = −4π
c Jν. (6.47)
Let us now talk about possible ambiguities of our recipe. First, we have to replace partial derivatives with covariant ones. However, partial derivatives commute while covariant derivatives do not. This may be a problem, but in practical examples it is not because it is possible to find a way to figure out the right order.
Second, in general we may expect that tensors that vanish in flat spacetime may af- fect the physical phenomenon when gravity is turned on. For example, the Lagrangian density of a real scalar field in Minkowski spacetime in Cartesian coordinates is
L = − 2ημν
∂μφ
(∂νφ)−1 2
m2c2
φ2, (6.48)
2We indicate withgthe determinant of the metricgμν . Sinceg<0 in a (3+1)-dimensional space- time, the absolute value of the determinant ofgμνis detgμν= −g.
and the action is
S= 1 c
Ld4x. (6.49)
If we apply our recipe, the Lagrangian density describing our real scalar field in curved spacetime should be (we remind that, for scalars,∇νreduces to∂ν)
L = − 2gμν
∂μφ
(∂νφ)−1 2
m2c2
φ2. (6.50)
The action is
S =1 c
L√
−g d4x. (6.51)
We say that such a scalar field isminimally coupledbecause we have applied our recipe that holds for both non-inertial reference frames and gravitational fields. How- ever, we cannot exclude that the Lagrangian density in curved spacetime is
L = − 2gμν
∂μφ
(∂νφ)−1 2
m2c2
φ2+ξRφ2, (6.52)
where ξ is a coupling constant. In this case, we say that the scalar field isnon- minimally coupledbecause we have a term coupling the scalar field with the gravita- tional field. The Lagrangian density in Eq. (6.52) reduces to Eq. (6.48) when gravity is turned off. There are no theoretical reasons (such as the violation of some fun- damental principle) to rule out the expression in Eq. (6.52), which is instead more general than the Lagrangian density in (6.50) and therefore more “natural” – in the- oretical physics, it is common to follow the principle according to which whatever is not forbidden is allowed. For the time being, there are no experiments capable of testing the presence of the termξRφ2, in the sense that it is only possible to constrain the parameterξ to be below some huge unnatural value.
Up to now, there are no experiments that require that some physical law requires a non-minimal coupling when we move from flat to curved spacetimes. However, we have to consider that the effects of gravitational fields via non-minimal coupling are extremely weak in comparison to non-gravitational interactions. This is because the Planck mass MPl=√
c/GN=1.2·1019GeV is huge in comparison with the energy scales in particle physics and there are no environments in the Universe today where gravity is “strong” in terms of the Planck mass.
Problems 121
Problems
6.1 In the Schwarzschild metric, the line element reads ds2 = −f(r)c2dt2+ dr2
f(r)+r2dθ2+r2sin2θdφ2, (6.53) where
f(r)=1−rSch
r , (6.54)
andrSch=2GNM/c2. The Schwarzschild metric describes the spacetime around a static and spherically symmetric object of massM. Write the vierbeins to move to a locally Minkowski reference frame.
6.2 In the Kerr metric, the line element reads ds2 = −
1−rSchr
c2dt2+
Δdr2+dθ2 +
r2+a2+rSchr a2sin2θ
sin2θdφ2−2rSchr asin2θ
cdt dφ , (6.55) where
=r2+a2cos2θ , Δ=r2−rSchr+a2, (6.56) rSch=2GNM/c2, anda =J/(Mc). The Kerr metric describes the spacetime around a stationary and axisymmetric black hole of mass M and spin angular momentum J. Write the relation between the proper time of an observer with constant spatial coordinates and the time coordinatet.
6.3 In Sect.4.3, we met the equation of the conservation of the electric current in flat spacetime in Cartesian coordinates:∂μJμ=0. Rewrite this equation in:(i)a general reference frame, and(ii)flat spacetime in spherical coordinates.
6.4 The field equation for the scalar fieldφthat we obtain from the Lagrangian density (6.48) in Cartesian coordinates is the Klein–Gordon equation
∂μ∂μ−m2c2 2
φ=0. (6.57)
Write the Klein–Gordon equation in curved spacetime, first assuming minimal cou- pling and then non-minimal coupling.
6.5 Write the Lagrangian in (6.52) and the associated Klein–Gordon equation in the case of a metric with signature(+ − −−).
6.6 The energy-momentum tensor of a perfect fluid in flat spacetime and Cartesian coordinates is given in Eq. (3.104). Write its expression for a general reference frame.
Chapter 7
Einstein’s Gravity
In Chap.6, we discussed non-gravitational phenomena in curved spacetimes for a generic metric theory of gravity. General covariance is the basic principle: once we have the metric of the spacetime, we can describe non-gravitational phenomena.
A different issue is the calculation of the metric of the spacetime. The Principle of General Covariance is not enough to determine the metric. We need the field equations of a specific gravity theory. With Einstein’s gravity, we refer to the gravity theory described by the Einstein–Hilbert action or, equivalently, by the Einstein equations.