By its construction, the latter monitors the evolution of a cross-sectional area perpendicular to the direction of the field lines. The first indicates the evolution of a cross-sectional area perpendicular to the direction of the magnetic lines of force. The relation Dbua=hbdhac∇duc and the definition of the projection tensor hab can easily lead to the relation.
Specifically, the volume scalar determines the expansion (Θ>0) or contraction (Θ<0) of the timelike congruence.
The gravitational field
6R(gacgbd−gadgbc), (2.7) where Rab=Rcacbis is the (symmetric) Ricci tensor (where R = Ra represents the scalar Ricci) and Cabcd is the Weyl tensor, which satisfies the same symmetries as the Riemann tensor and moreover, it is trace-free in all its indices, namely Ccacb= 0. The Riemann tensor decomposition in (2.7) shows that the gravitational field splits into a local (Ricci) part and a long-range part (Weyl). The local gravitational field, which is present due to the distribution of matter at a given point and the local curvature of spacetime it causes at the same point, is described by the Ricci tensorRab and the associated scalar R in Einstein's field equations ( see Eq.
On the other hand, the non-local long-range gravitational field, which is present due to the curvature caused by the local matter at another point of the spacetime, describes tidal forces and gravitational waves and is governed by the Weyl tensorCabcd.
Matter fields and conservation laws
For a perfect fluid, where the energy and momentum tensor is given by (2.11), these conservation laws reduce to. The law of conservation of momentum for a perfect fluid implies that the quantity ρ+p determines its total inertial mass.
Kinematics of timelike congruences
Focusing Theorem for timelike congruences
We also assume that the congruence is non-rotating, that is, ωab = 0, and so the last term on the right-hand side of the above equation vanishes. In general, the singularity in Θ implied by the 'Concentration Theorem' simply represents a singularity in congruence, not a singularity in the structure of spacetime. It simply says that the caustic will develop into a congruence if convergence occurs at every point on a geodesic in congruence.
Spatial curvature
3Θ(σab+ωab) +σchaσcbi−ωchaωcbi+ 2σc[aωcb], (2.34) which is also known as the Gauss-Codazzi equation. In the above, R= 2(κρ−13Θ2+σ2− ω2+Λ) is the 3-Ricci scalar and Eab defines the electric field experienced by the observer. If the fluid we are studying (as it happens in our case) is only a magnetic field, then ρ=ρB = B22 and πab = Πab = B32hab−BaBb, where Ba defines the magnetic field vector and B2 =BaBa.
It should also be noted that all terms on the right, except the first, are trace-free. Note that when the host spacetime is non-rotating (i.e. ωab = 0), the 3-Ricci tensor becomes symmetric, namely Rab=R(ab), as its antisymmetric part vanishes. The vorticity tensor in the above equation only appears in rotating spaces where each observer's 4-velocity vector is orthogonal to its own 3D rest space and therefore no simultaneity hypersurface can be formed.
For non-rotating spaces (ie ωab= 0) the 3-Riemann tensor satisfies all the symmetries of its 4-D counterpart. As a result, this 'matter' also contributes to the right-hand side of Einstein's field equations (see Eq. The electromagnetic field is described by the Faraday tensor Fab, which is antisymmetric and is - relative to fundamental observers - given by the relation .
Fab = 2u[aEb]+abcBc, (3.1) where Ea = Fabub and Ba = abcFbc/2 represent the electric and magnetic fields of the observer and are also spatial, so that Eaua= 0 =Baua. Finally, it should be noted that the trailing part of (3.2) leads to the condition Ta(em)a = 0, which is perfectly consistent with the nature of the traceless electromagnetic radiation.
Maxwell’s equations
Therefore, in addition to the expected 'curl' and 'divergence'. 3.6) and (3.7) have additional terms that appear due to the relative motion of neighboring observers.
Conservation laws
Ohm’s law
Therefore, when the substance we are working with is highly conductive, so that ς → ∞, the electric field can be completely neglected. Specifically, if the field of matter we are working with is a magnetic field, the latter will be treated as a highly conductive medium that is "frozen" in the matter so that the electric field can disappear.
Magnetic fields-Anisotropy
The 1+1+2 covariant formulation of general relativity introduces a covariant 1+2 splitting of a 3D space, along and parallel to a preferred direction, which in our case is the direction of the magnetic field. This approach uses (the same) irreducible variables, as in the case of the 1+3 formalism, to describe the relative motion of a congruence of (space-like) worldlines in the observer's 2D rest space, perpendicular to the chosen direction. -tie. Using the fields na and ˜hab we can achieve a 1+2 split of the 3D space in a spatial direction along na and a 2D space-like surface perpendicular to na.
By employingua, hab,na and ˜hab we thus have an overall 1+1+2 division of the 4-D spacetime in a temporal direction (parallel toua), along a preferred spatial direction (parallel to na) and on a 2-D space-like surface orthogonal to both of the above vectors. Then, in complete analogy with the 1+3 case (see definitions (2.1)), the derivatives parallel and orthogonal to the na-field of a general tensor field are given by. As in the 1+3 formulation, we can follow a similar process and decompose the 3-D covariant derivative of thena vector into a spatial part orthogonal tona and a part along the vector na.
Applying (4.3 b) tona and considering the definition of the projection tensor ˜hab, we indeed obtain. 4.5) Now that we remember that the 2D covariant derivative of the aftervector is a second-rank (spatial) tensor, we can split it into its symmetric, antisymmetric and symmetric and trace-free part, viz. Θ˜˜hab+ ˜σab+ ˜ωab+n0anb, (4.7) where ˜Θ = Dana = ˜Dana is the scalar surface expansion/contraction, ˜σab = ˜Dhbnai the 2-D symmetric and trace-free shear tensor, ˜ωab = ˜ D[bna] is the 2-D antisymmetric vorticity tensor, and n0a is the acceleration vector. The last three of these variables are orthogonal in construction to the thena field and satisfy the constraints: n0ana= 0.
The physical/geometric meaning of precedent irreducible kinematic variables is analogous to that of their 3-D counterparts (see §2.4) as they are used to describe the relative motion of a family of observers in their 2-D spatial rest space. orthogonal to the field na. Then, positive/negative values of the area scalar indicate divergence/convergence of the aforementioned curves and a corresponding expansion/contraction of the S cross section.
Kinematics of spacelike congruences
Irreducible kinematic evolution
In particular, suppose that the after-field is tangent to a congruence of spacelike curves and consider a 2-D cross-section S of this congruence. The symmetric and trace-free 2-tensor ˜σab monitors the shape changes of S under constant area. Substituting decomposition (4.7) into the left-hand side of the relation above and recalling (4.4 a), we lead to. 4.12) Finally, by projecting the latter orthogonally onto the after-field and taking (2.33 a) into account, we arrive at
This term governs the development of the space-like congruence tangent to the unitary na field, along the latter's (spatial) direction. More specifically, the track, the projected symmetric trackless and the projected antisymmetric components in (4.12) give the evolution equations of the area scalar (˜Θ), of the 2-D shear tensor (˜σab) and of the 2-D vorticity tensor (˜ωab), respectively .
Raychaudhuri’s equation for spacelike congruences
It should be noted that Rab=Rcacb defines the 3-Ricci tensor and ˜σ2 = ˜σabσ˜ab/2, ˜ω2 = ˜ωabω˜ab/2 define the scalar magnitudes of the shear and vorticity 2 tensors, respectively. In complete analogy to the 1+3 case, the positive terms on the right-hand side of (4.14) lead our spatial congruence to divergence, while the negative ones lead to its convergence. We should note the similar form of Raychaudhuri's equation for temporal and spatial congruences (see (2.20) and (4.14) respectively).
Clearly, if the curves of our (space-like) compliance are space-like geodesics, the derivative n0a cancels out and so the Raychaudhuri equation reduces to. 2ωabn˙anb, (4.15) Furthermore, when there is no 3-D rotation, the last term of the above equation cancels and the 3-Ricci tensor becomes symmetric (see §2.5).
Shear and vorticity evolution
We should note that the formulas derived so far are purely geometric and depend solely on the space-time structure, without making any assumptions about its material content.
Raychaudhuri’s equation for magnetic forcelines
Dan0a−n0an0a+ 2ωabn˙anb, (4.21) which represents the magnetic analogue of the Raychaudhuri equation in 3 dimensions and is the key equation for the following discussion on the focusing of the magnetic lines of force.
Focusing of magnetic forcelines
- First scenario
- Second scenario
- Third scenario
- Last scenario
At this point we see a big difference from the time-like case where the focusing of the lines could not be prevented, even if they were initially parallel, due to the negative term of the (conventional) matter −12κ(ρ+ 3p) on the right of (2.27). In this case, the fate of the magnetic lines is well predictable, since the terms on the right side of the previous equation have a well-defined sign. Shear-like deformations of the 2D space will contribute to the contraction and final focus of the congruence, while vorticity will tend to arrest the collapse.
2 = ˜Dan0a−n0an0a (4.24) In this case, the focusing of the lines is monitored by the first term on the right-hand side of (4.24), since the termn0an0a is obviously positive and (with the minus sign) leads to contraction . Apart from the first term on the right-hand side of (4.25), which is clearly positive, none of the remaining terms have a particular sign. However, using the 1+1+2 formalism of the previous chapter, Maxwell's equations can be further broken down into pure 2-D and 3-D parts that describe the evolution of the electromagnetic field in the observers' 2-D space-like rest surface. .
Moreover, all terms of the electric 3-currents and the electric charge density disappear, because only magnetic fields are present. We are now ready to use the 1+1+2 decomposition from the previous chapter to obtain the respective 1+1+2 forms of equations. The first equation implies that the evolution of the scalar magnitude of the magnetic field depends only on the scalar surface area of the fluid element, always within the framework of the ideal MHD approximation.
In this work we used covariant methods and in particular the 1+1+2 spacetime decomposition to arrive at the magnetic analogue of the Raychaudhuri equation in 3 dimensions. The former is the key equation that monitors the expansion/contraction of a 2-D cross-sectional area orthogonal to the field lines and the corresponding focusing of the magnetic field lines. The deformation of the lines is determined by the 3-D spatial curvature along the direction of the magnetic field, namely by the term Rabnanb.
The definition of the 2-D covariant derivative, the 2-D projection tensor, and the magnetic field vector Ba=Bna (with nana= 1→nan0a= 0) lead to.