A Nonaxisymmetric Solution of Einstein’s Equations Featuring Pure Radiation from a Rotating Source

Abstract

A special nonaxisymmetric solution of Einstein’s equations is derived, representing pure radiation from a rotating isolated source. The spacetime is assumed to be algebraically special having a multiple null eigenvector of the Weyl tensor forming a geodesic, shear-free, diverging, and twisting congruence 𝐤. Employing a complex null tetrad involving the vector 𝐤, the Ricci tensor, density of the radiation, divergence, and twist are calculated for the derived metric. A particular (nonaxisymmetric) subcase is shown to be flat at infinity and to contain the axisymmetric radiating Kerr metric, derived by Kramer and separately by Vaidya and Patel, as a special case. The spacetime is of Petrov type II and without Killing vectors.

1. Introduction

In this paper, we present a special solution to Einstein’s equations involving pure radiation. Pure radiation is characterised by an energy-momentum tensor of the form𝑇_𝑎𝑏=𝜙²𝑘_𝑎𝑘_𝑏,(1) where 𝐤 is a null vector. The incoherent radiation is propagated along the 𝐤 lines and 𝜙²is its energy density, which must, therefore, be positive. For our purpose, it will be assumed that the 𝐤 vector field forms a geodesic, shear-free congruence, that 𝐤 is a multiple eigenvector of the Weyl tensor, and that the metric is algebraically special. In addition, the congruence will be taken to be diverging and twisting. The consequences of these assumptions will be detailed in Section 2. In particular, they lead to a form of the metric originally derived by I. Robinson and J. R. Robinson [1] and Robinson et al. [2].

To obtain solutions of Einstein’s equations representing pure radiation, authors have in the main assumed some form of symmetry. An early one of spherical symmetry by Vaidya [3] featured pure radiation from a central Schwarzschild body, with the mass being consequently a function 𝑚(𝑢)of the time coordinate 𝑢. This was followed by the axisymmetric radiating Kerr metric by Kramer [4], and also a different version of this by Vaidya and Patel [5], where in the latter setting a certain parameter to zero regained the Kerr metric in its original form [6]. These two versions represent pure radiation from a rotating central source. Herlt [7] found the complete solution for pure radiation in axisymmetric Kerr-Schild metrics (cf. Debney et al. [8]).

More recently, pure radiation solutions have been derived by exploiting possible symmetry groups in the hyperspace 𝑟=const (see (8)). These authors [9–12] employed the theory of Cauchy-Riemann structures (see, e.g., Robinson and Trautman [13]). An axisymmetric solution of some generality, involving pure radiation from a rotating source, has been derived up to a linear second-order ordinary differential equation by Kramer and Hähner [14].

A valuable method of generating algebraically special pure radiation fields from vacuum cases was devised by Stephani, thereby obtaining some explicit axisymmetric solutions [15]. In this paper, we shall employ Stephani’s device but will generate a nonaxisymmetric solution, which has the radiating Kerr metric as a special case. If a star can catastrophically collapse through its event horizon and if its ultimate state is a rotating axisymmetric Kerr black hole, then the time-dependent solution given here may be interpreted as a theoretical example of a nonaxisymmetric stage in that process.

2. The Basic Framework

Because of our assumptions in Section 1, we can write the following as properties of the 𝐤 congruence:(i)since 𝐤 is geodesic, 𝜅=0⇒𝑘_[𝑎𝑘_𝑏];𝑐𝑘^𝑐=0, or via an affine parameter 𝑘_𝑎;𝑏𝑘^𝑏=0;(ii)𝐤 is shear-free: 𝜎=0⟹𝑘_(𝑎;𝑏)𝑘^𝑎;𝑏−12𝑘^𝑎;𝑎²=0;(2)(iii)𝐤 is diverging and twisting: Θ=(1/2)𝑘^𝑎;𝑎>0,𝜔²=(1/2)𝑘_[𝑎;𝑏]𝑘^𝑎;𝑏>0;(iv)𝐤 is a multiple Weyl eigenvector: 𝜓₀=𝜓₁=0.

We shall refer the spacetime to a complex null tetrad 𝐦,𝐦𝐛𝐚𝐫,𝐧, and 𝐤 with labels 1, 2, 3, and 4, respectively, with the fourth member being the 𝐤 of the above context. In relation to this tetrad, in order to display the form of the metric we adopt coordinates 𝑥^𝑖=𝑥,𝑦,𝑟, and 𝑢 for 𝑖=1,2,3, and 4, respectively. The spacelike coordinate 𝑥 is complex, 𝑦 its conjugate, 𝑟 an affine parameter along the 𝐤 lines, and 𝑢 a retarded time. The Newman-Penrose spin coefficients 𝜅,𝜎 and 𝜌=−(Θ+𝑖𝜔) are expressible in terms of the tetrad by the relations:𝜅=−𝜌𝑘=_𝑎;𝑏−𝑚^𝑎(𝑘Θ^𝑏+=𝑖0𝜔,)=−𝜎𝑘=_𝑎;𝑏−𝑘_𝑎;𝑏𝑚^𝑎𝑚^𝑏=0,𝑚^𝑎𝑚^𝑏≠0.(3)

It now becomes possible to write the metric in the following form [1, 2] (we use mainly the notation of [16]):d𝑠²=2𝜔¹𝜔²−2𝜔³𝜔⁴,(4) where𝜔¹=−d𝑥𝜔₂𝑃𝜌,=−d𝑦𝜔𝑃𝜌₃,𝜔=d𝑢+𝐿d𝑥+𝐿d𝑦,₄=d𝑟+𝑊d𝑥+𝑊d𝑦+𝐻𝜔³.(5) The dual frame is then, relative to the 𝑥^𝑖 system,𝑚^𝑖=−𝑃𝜌,0,𝑃𝑊𝜌,𝑃𝐿𝜌,𝑚^𝑖=𝑛0,−𝑃𝜌,𝑃𝑊𝜌,𝑃𝐿𝜌_𝑖,𝑘=(0,0,−𝐻,1),_𝑖=(0,0,1,0).(6) Here𝜌=−1Σ=−𝑖𝑃₂(𝑟+𝑖Σ),𝜕𝐿−𝜕𝐿𝑊=𝐿_,𝑢2,𝜕≡𝜕𝜌_𝑥+𝑖𝜕Σ,−𝐿𝜕_𝑢,(7) and the metric will take the following form:d𝑠²=−2+d𝐻𝑢dd𝑢𝑟²++𝐿(d𝑊𝑥+d𝑟2+𝐻𝐿)d𝑢d𝑥+𝑊+2𝐻𝐿d𝑢d𝑦+𝐿𝑊+𝐻𝐿₂𝐿d𝑦d𝑟d𝑥²+𝐿𝑊+𝐻𝐿²d𝑦²−1𝑃²𝜌𝜌−𝐿𝑊−𝐿𝑊−2𝐻𝐿𝐿d𝑥d𝑦.(8) In particular, the coefficient of d𝑢² is −2𝐻where𝐻=𝐾2−𝑟𝑃⁻¹𝑃_,𝑢−(𝑚𝑟+𝑀Σ)𝐾=2𝑃₂𝑟²+Σ²,Re𝜕𝑃⁻¹𝜕𝑃−𝐿_,𝑢,(9) and 𝑀 will be defined by (11).

In these relations 𝑃, 𝐾, 𝑚, and 𝑀 are real functions, and 𝐿 is a complex function, of 𝑥, 𝑦, and 𝑢. 𝐻 is a real function and 𝑊 a complex function of 𝑥, 𝑦, 𝑟, and 𝑢. A subscript indicates partial differentiation.

For pure radiation, the two equations to be satisfied are (cf. [16])3𝐿_,𝑢−𝜕(𝑚+𝑖𝑀)=0,(10)𝑀=Σ𝐾+𝑃²Re𝜕𝜕Σ−2𝐿_,𝑢𝜕Σ−Σ𝜕_𝑢𝜕𝐿.(11) Then 𝜙²is given by the equation𝑃⁴𝜕−2𝐿_,𝑢+2𝑃⁻¹𝜕𝑃𝜕𝐶−𝑃³𝑃⁻³(𝑚+𝑖𝑀)_,𝑢=𝜙²2𝜌𝜌,(12) where𝐶=𝜕𝑃⁻¹𝜕𝑃−𝐿_,𝑢+𝑃⁻¹𝜕𝑃−𝐿_,𝑢².(13)

3. A Nonaxisymmetric Pure Radiation Rotation Field

Referring to (1), we have from the energy equation 𝑇^{𝑎𝑏;𝑏}=0, using the properties listed in (2), namely, 𝑘_𝑎;𝑏𝑘^𝑏=0, Θ=(1/2)𝑘^𝑎;𝑎=−(1/2)(𝜌+𝜌), and noting from (6) that 𝐤=𝜕_𝑟,that 𝜙² has the form𝜙²=𝜓(𝑥,𝑦,𝑢)𝜌𝜌.(14) For the functions 𝐿 and 𝑃, we initially choose𝐿=𝑃𝐿=₀𝑃₀=𝑎𝑥𝑦+𝑏(𝑥+𝑦)+𝑐,=−𝑖𝑛⁰+𝑛₁𝑦+𝑛₂𝑦²(𝑎𝑥𝑦+𝑏(𝑥+𝑦)+𝑐)²𝑚=𝑚₀,,(15) where𝑎, 𝑏, 𝑐, 𝑛₀, 𝑛₁, 𝑛₂, and 𝑚₀(>0) are real constants.

We now derive from the definitions in Section 2 that𝑀=𝑀₀=0.(16) Henceforth, we shall calculate tensor components relative to the complex null tetrad. In that frame, the metric coefficients are𝑔_𝑎𝑏=⎛⎜⎜⎜⎜⎜⎜⎝01001000−0001−0001⎞⎟⎟⎟⎟⎟⎟⎠.(17) In particular, we find for the Ricci tensor𝑅_𝑎𝑏=0.(18) We, therefore, have a vacuum spacetime. Accordingly, we shall now adopt the method of Stephani [15] to generate a pure radiation field.

Stephani’s result can be stated: given 𝑚=𝑚₀(=const),𝑀₀(=0),𝐿₀,𝑃₀specifying an algebraically special vacuum solution to Einstein’s equations, then if we change (only) 𝑃 from 𝑃₀ to 𝑃₀𝐴(𝑥,𝑦,𝑢)and if (10) and (11) remain valid (with 𝑀=𝑀₀=0), then the resultant spacetime represents a field of pure radiation.

For our case, starting with (15) except for the modification of 𝑃, we find that Stephani’s conditions are satisfied if we adopt for 𝐴(𝑥,𝑦,𝑢)the value𝐴(𝑥,𝑦,𝑢)=𝑢,(19) provided we set𝑛₂=𝑏𝑛¹−𝑎𝑛₀𝑐.(20)

Σ and 𝐾 will be altered accordingly in (7) and (9), with 𝐿=𝐿₀ remaining unchanged.

We may verify that with the revised value of 𝑃 that 𝑀=𝑀₀=0,and that the equations for pure radiation, (10) and (11), are then satisfied.

Calculation of the Ricci tensor in the null tetrad yields the result:𝑅_𝑎𝑏=6𝑚⁰𝑢𝜌𝜌𝛿^3𝑎𝛿^3𝑏.(21) It, therefore, follows from (1) and (21), or from (12), that 𝜙²=(6𝑚₀/𝑢)𝜌𝜌. From the definition of 𝜌in (7), and also as −(Θ+𝑖𝜔), we obtain for the expansion and twist of the 𝐤 congruence:Θ=𝑐²𝑟[𝑎𝑥𝑦+𝑏(𝑥+𝑦)+𝑐]²𝜔=−𝑢₂𝐷,×2𝑐𝑛[₀𝑎𝑥𝑦+𝑏(𝑥+𝑦)+𝑐]−𝑛₁[𝑎𝑏𝑥𝑦+𝑎𝑐(𝑥+𝑦)+𝑏𝑐]𝐷=𝑐₂2𝑏²−𝑎𝑐𝑥𝑦+𝑏𝑐(𝑥+𝑦)+𝑐²/𝐷,𝑟²[𝑎𝑥𝑦+𝑏(𝑥+𝑦)+𝑐]²+𝑢⁴−𝑛2₁𝑛₀[𝑎𝑏𝑥𝑦+𝑎𝑐(𝑥+𝑦)+𝑏𝑐]2𝑏²−𝑎𝑐𝑥𝑦+𝑏𝑐(𝑥+𝑦)+𝑐²².(22) For the energy density of the pure radiation, we can now write𝜙²=6𝑚⁰𝑐²[𝑎𝑥𝑦+𝑏(𝑥+𝑦)+𝑐]²𝑢𝐷,(23) which is >0, as required, and→0as either 𝑟 or 𝑢→∞. Because of the radiation emission, the central isolated mass will steadily decrease.

The Ricci tensor being given by (21), we have the invariants:𝑅=0,𝑅_𝑎𝑏𝑅^𝑎𝑏=0.(24)

4. Behaviour at Infinity

Our spacetime becomes axisymmetric if we set 𝑏=0. But if we go further and adopt a subset of our previous assignments:𝑏=0,𝑎=1𝑃(𝑥,𝑦,𝑢)=1+𝑥𝑦2,𝑐=1,2,𝐿(𝑥,𝑦)=−(1+𝑖𝑛𝑥𝑦𝑦/2)²,(25) and then make the transformation𝑥=√2tan𝜃2𝑒^𝑖𝜉,𝑦=√2tan𝜃2𝑒^−𝑖𝜉𝑟=𝑟,𝑢=𝑡,(26) we obtain a form of the Kerr metric:d𝑠²=𝑟²+𝑛²cos(𝜃)²d𝜃²+sin(𝜃)²𝑟²+𝑛²+2𝑛²𝑚₀𝑟sin(𝜃)²𝑟²+𝑛²cos(𝜃)²d𝜉²−2𝑛sin(𝜃)²d𝜉d𝑟−2d𝑟d𝑡+4𝑛𝑚⁰𝑟sin(𝜃)²𝑟²+𝑛²cos(𝜃)²−1−2𝑚₀d𝜉d𝑡𝑟𝑟²+𝑛²cos(𝜃)²d𝑡².(27) As is well known, the Kerr metric is flat at infinity. For any stationary metric associated with an isolated mass and which is flat at infinity, expansion in powers of 1/𝑟 has the following form [17, page 452]:d𝑠²=−1−2𝑀𝑟+𝑂𝑟1²d𝑡²−4𝜀_𝑗𝑘𝑚𝑆^𝑘𝑋^𝑚𝑟³+𝑂1𝑟³d𝑋^𝑗+1+2d𝑡𝑀𝑟+𝑂𝑟1²𝛿^𝑗𝑘+(vanishingterms)𝛿_𝑗𝑘d𝑋^𝑗d𝑋^𝑘.(28) In (28), the 𝑋^𝑗are the coordinates of the ultimately flat spacetime,𝑋=𝑟sin𝜃cos𝜉,𝑌=𝑟sin𝜃sin𝜉,𝑍=𝑟cos𝜃,(29) and 𝑆^𝑘is the angular momentum vector of the source.

Comparing (28) with (27) (expanded in powers of 1/𝑟), we can confirm that the Kerr central mass is 𝑀=𝑚₀. Moreover, as was proved by Cohen [18], the angular momentum of the Kerr mass is 𝑛𝑚₀, so that the parameter 𝑛 in (25) is the angular momentum per unit mass.

If we now retain (25) except that we change 𝑃 to𝑃(𝑥,𝑦,𝑢)=1+𝑥𝑦2𝑢(30) and follow through (26), then we have a form of the radiating Kerr metric (cf. [4, 5]), namely,d𝑠²=𝑟²+𝑛²𝑡⁴cos(𝜃)²𝑡²+sin(𝜃)₂d𝜃²−4𝑛³𝑡sin(𝜃)³cos(𝜃)d𝜃d𝜉𝑛²𝑡²+𝑛⁴sin(𝜃)⁴+𝑟²𝑡²+2𝑛²𝑟sin(𝜃)𝑡²+2𝑛²𝑚₀𝑟sin(𝜃)²𝑟²+𝑛²𝑡⁴cos(𝜃)²d𝜉²−2𝑛sin(𝜃)²d𝜉d𝑟−4𝑛²𝑡sin(𝜃)cos(𝜃)d𝜃d𝑡+2sin(𝜃)²×𝑛³sin(𝜃)²+2𝑛𝑟+2𝑛𝑚₀𝑡𝑟−𝑡𝑟²+𝑛²𝑡⁴cos(𝜃)²d𝜉d𝑡−2d𝑟d𝑡₂−𝑛²sin(𝜃)²−2𝑟𝑡−2𝑚⁰𝑟𝑟²+𝑛²𝑡⁴cos(𝜃)²d𝑡².(31) It may be shown for this metric that the central mass (of parameter 𝑚₀) has at epoch 𝑢 the value 𝑚₀/𝑢³(see [14]), confirming its steady loss to radiation.

In the light of the foregoing, we see that our present solution is a nonaxisymmetric spacetime with a radiating rotating source. It has of course extra complication mathematically. But if we retain the expressions (26) for 𝑥 and 𝑦, then we find that the components of the Riemann tensor for our nonaxisymmetric case →0as either 𝑟 or 𝑢→∞. In particular, if we adopt the following values:𝑎=12,𝑐=1,𝑛¹=𝑛,𝑛₀=2𝑏𝑛𝑛₂=0,(32) and the assignment (26), then the components of the Riemann tensor →0as 𝑟→∞at rates varying from 𝑟⁻² to 𝑟⁻⁴and as 𝑡→∞at rates between 𝑡⁻⁴ and 𝑡⁻⁸.Hence the associated metric is flat at infinity. We give its metric coefficients in the Appendix. There we see that if we set 𝑏=0 in this nonaxisymmetric pure radiation solution, then we obtain the axisymmetric radiating Kerr metric given at (31).

Our general metric, specified by (19) and (20), is of Petrov Type II and without Killing vectors.

5. Conclusion

A solution of Einstein’s equations has been obtained, which features pure radiation from an isolated rotating source in a nonaxisymmetric environment. It contains the axisymmetric radiating Kerr metric as a special case.

An interesting aspect of the solution, therefore, is that its lack of symmetry suggests a possible theoretical model for the time-dependent configuration of the catastrophic collapse of a rotating star, during the approach to its final state of axial symmetry.

Appendix

Here we give the metric coefficients for the (nonaxisymmetric) subcase:𝑎=12,𝑐=1,𝑛¹=𝑛,𝑛₀=2𝑏𝑛𝑛₂=0,(A.1) having made the transformation𝑥=√2tan𝜃2𝑒^𝑖𝜉,𝑦=√2tan𝜃2𝑒^−𝑖𝜉𝑔𝑟=𝑟,𝑢=𝑡,_𝜃𝜃,=512𝑏²𝑛²sin²𝜉𝐴⁴(1+cos𝜃)²21−2𝑏²𝑡²−𝐾−2𝑡𝑟−2𝑚⁰𝑟+1024√𝑟²+Σ²2𝑏𝑛³𝐵𝑡sin𝜉𝐴⁵(1+cos𝜃)²tan𝜃2+√+16𝑟₂2𝑏cos𝜉𝐴²+𝑛²𝐵²𝑡⁴𝐴⁴𝑡²(1+cos𝜃)²𝑔_𝜃𝜉,=−128√2𝑏𝑛²tan(𝜃/2)sin𝜉𝐴⁴(1+cos𝜃)tan𝜃2+√×𝐾−2𝑟2𝑏cos𝜉𝑡−2𝑚⁰𝑟+512√𝑟²+Σ²2𝑏1−2𝑏²𝑛²𝑡²tan(𝜃/2)sin𝜉×tan𝜃𝐴⁴(1+cos𝜃)2+√+512𝑛₃2𝑏cos𝜉𝐵𝑡tan(𝜃/2)×2𝐴𝑏⁵₂(1+cos𝜃)𝑔2cos²𝜉−1+2√2𝑏tan𝜃2cos𝜉+tan²𝜃2_𝜉𝜉,=−256𝑛²tan²(𝜃/2)𝐴⁴tan𝜃2+√2𝑏cos𝜉²×𝐾−2𝑟𝑡−2𝑚⁰𝑟+5121−2𝑏₂𝑟²+Σ²𝑛²𝑡²tan²(𝜃/2)𝐴⁴tan𝜃2+√2𝑏cos𝜉²−1024√2𝑏𝑛³𝐵𝑡tan²(𝜃/2)sin𝜉𝐴⁵tan𝜃2+√+16tan₂2𝑏cos𝜉(𝜃/2)𝐴⁴𝑡²𝑔_𝜃𝑡𝑟²𝐴²+𝑛²𝑡⁴𝐵²,=16√2𝑏𝑛sin𝜉𝐴²(1+cos𝜃)1−2𝑏²𝑡²−𝐾−2𝑡𝑟−2𝑚⁰𝑟+32𝑛₂𝑟²+Σ²𝐵𝑡𝐴³(1+cos𝜃)tan𝜃2+√𝑔2𝑏cos𝜉_𝜉𝑡,=16𝑛tan(𝜃/2)tan(𝜃/2)+√2𝑏cos𝜉𝐴²×1−2𝑏²𝑡²−𝐾−2𝑟𝑡−2𝑚⁰𝑟−32√𝑟²+Σ²2𝑏𝑛²𝐵𝑡𝐴³tan𝜃𝑔2sin𝜉,_𝜃𝑟=−16√2𝑏𝑛sin𝜉𝑔𝐴²(1+cos𝜃)_𝜉𝑟,=−16𝑛tan(𝜃/2)𝐴²tan𝜃2+√𝑔2𝑏cos𝜉_𝑡𝑟,𝑔=−1,_𝑡𝑡=−𝐾−2𝑟𝑡−2𝑚⁰𝑟𝐾=𝑛₂𝑟²+Σ²,(1+cos𝜃)1−2𝑏²cos𝜃−2√2𝑏sin𝜃cos𝜉−2𝑏²−1+1−2𝑏₂1+2√2𝑏sin𝜃cos𝜉+2𝑏²sin²𝜃cos²𝜉𝐴=4𝑡²,1+cos𝜃1+√𝐵=42𝑏sin𝜃cos𝜉,1+cos𝜃2𝑏²Σ=𝐵𝑛𝑡₂(1+cos𝜃)+√2𝑏sin𝜃cos𝜉−cos𝜃,𝐴.(A.2) Setting 𝑏=0 in the above nonaxisymmetric coefficients gives the axisymmetric radiating Kerr metric of (31).