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Gravitational plane waves in Einstein-aether theory

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Abstract

In this paper, we systematically study spacetimes of gravitational plane waves in Einstein-aether theory. Due to the presence of the timelike aether vector field, now the problem in general becomes overdetermined. In particular, for the linearly polarized plane waves, there are five independent vacuum Einstein-aether field equations for three unknown functions. Therefore, solutions exist only for particular choices of the four free parameters \(c_{i}\)’s of the theory. We find that there exist eight cases, in two of which any form of gravitational plane waves can exist, similar to that in general relativity, while in the other six cases, gravitational plane waves exist only in particular forms. Beyond these eight cases, solutions either do not exist or are trivial (simply representing a Minkowski spacetime with a constant or dynamical aether field).

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Notes

  1. By rescaling the null coordinate \(u \rightarrow u' = \int { e^{-M(u)} du}\), without loss of the generality, one can always set \(M = 0\).

  2. Polarizations of GWs in weak-field approximations were also studied in [60] in the framework of Einstein-aether theory.

  3. It is interesting to note that in Einstein’s theory the field equations \(G_{\mu \nu } = 0\) yields only a single equation [47, 51],

    figure a

    for the two unknown functions U(u) and V(u). In this sense, the problem is underdetermined in Einstein’s theory. Thus, for any given gravitational wave V(u), we can always integrate the above equation to find U(u).

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Acknowledgements

We would like to thank G. Cleaver, Bao-Fei Li and Bahram Shakerin for valuable discussions. The work of A.W. was supported in part by the National Natural Science Foundation of China (NNSFC), Grant Nos. 11375153 and 11675145.

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Authors and Affiliations

  1. GCAP-CASPER, Physics Department, Baylor University, Waco, TX, 76798-7316, USA

    Jacob Oost, Madhurima Bhattacharjee & Anzhong Wang

  2. Institute for Advanced Physics & Mathematics, Zhejiang University of Technology, Hangzhou, 310032, China

    Anzhong Wang

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  1. Jacob Oost
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Correspondence to Anzhong Wang.

Appendix A: The Einstein and aether tensors \(G_{\mu \nu }\) and \(T^{\ae }_{\mu \nu }\) and the aether vector

Appendix A: The Einstein and aether tensors \(G_{\mu \nu }\) and \(T^{\ae }_{\mu \nu }\) and the aether vector

For the spacetime of Eq. (3.1), the non-vanishing components of the Einstein tensor \(G_{\mu \nu }\) and \(T^{\ae }_{\mu \nu }\) are given by,

$$\begin{aligned} G_{00}= & {} \frac{1}{2}\Big (2U_{uu} - U_u^2 - V_u^2\Big ), \nonumber \\ T^{\ae }_{00}= & {} -\frac{1}{8}\Big [2c_2 U_{uu} + c_{13}\Big (V_u^2 + U_u^2\Big ) \nonumber \\&+ \,2\big (c_{13} + c_2 + 3c_{14}\big )\Big (h_{uu} - h_u U_u - h_u^2\Big )\Big ],\nonumber \\ T^{\ae }_{01}= & {} \frac{e^{-2h}}{4}\Big [c_2 \Big (U_{uu} - 2h_u U_u - U_u^2\Big ) \nonumber \\&+\, \big (c_2 + c_{13} - c_{14}\big )\Big (h_{uu} - h_u U_u - 2h_u^2\Big )\Big ],\nonumber \\ T^{\ae }_{11}= & {} -\frac{e^{-4h}}{8}\Big [2c_2 U_{uu} + c_{13}\Big (U_u^2 +V_u^2\Big ) \nonumber \\&+\, 2\big (c_2 + c_{13} - c_{14}\big )\Big (h_{uu} - h_u U_u - h_u^2\Big )\Big ],\nonumber \\ T^{\ae }_{22}= & {} \frac{e^{V-U-2h}}{8}\Big [c_{13}\Big (2V_{uu} - V_u^2 - 2 U_uV_u - 4 h_uV_u\Big ) \nonumber \\&-\, \big (c_{13} + 2c_2\big )\Big (2U_{uu} - U_u^2 - 4h_uU_u\Big )\nonumber \\&-\, 4 c_2h_{uu} + 2\big (3c_2 - c_{13} + c_{14}\big )h_u^2\Big ],\nonumber \\ T^{\ae }_{33}= & {} - \frac{e^{-(V+U+2h)}}{8}\Big [c_{13}\Big (2V_{uu} + V_u^2 - 2 U_uV_u - 4 h_uV_u\Big ) \nonumber \\&+\, \big (c_{13} + 2c_2\big )\Big (2U_{uu} - U_u^2 - 4h_uU_u\Big )\nonumber \\&+\, 4 c_2h_{uu} - 2\big (3c_2 - c_{13} + c_{14}\big )h_u^2\Big ], \end{aligned}$$
(A.1)

and , where

(A.2)

In the vacuum case, we have \(T^{m}_{\mu \nu } = 0\), and the Einstein-aether equations (2.7) reduce to

$$\begin{aligned} G_{\mu \nu } = T^{\ae }_{\mu \nu }, \end{aligned}$$
(A.3)

which yield five independent equations,

$$\begin{aligned}&2U_{uu} - \Big (V_u^2 + U_u^2\Big )+ 2c_{14}\Big (h_{uu} - h_u U_u - h_u^2\Big ) = 0, ~~~~~~~ \end{aligned}$$
(A.4)
$$\begin{aligned}&c_2 \Big (U_{uu} - 2h_u U_u - U_u^2\Big ) \nonumber \\&\quad +\, \big (c_2 + c_{13} - c_{14}\big )\Big (h_{uu} - h_u U_u - 2h_u^2\Big ) = 0, ~~~~~ \end{aligned}$$
(A.5)
$$\begin{aligned}&2c_2 U_{uu} + c_{13}\Big (U_u^2 +V_u^2\Big ) \nonumber \\&\quad +\, 2\big (c_2 + c_{13} - c_{14}\big )\Big (h_{uu} - h_u U_u - h_u^2\Big ) = 0, ~~~~~ \end{aligned}$$
(A.6)
$$\begin{aligned}&c_{13}\Big (2V_{uu} - V_u^2 - 2 U_uV_u - 4 h_uV_u\Big ) \nonumber \\&\quad -\, \big (c_{13} + 2c_2\big )\Big (2U_{uu} - U_u^2 - 4h_uU_u\Big )\nonumber \\&\quad -\, 4 c_2h_{uu} + 2\big (3c_2 - c_{13} + c_{14}\big )h_u^2 = 0, ~~~~~ \end{aligned}$$
(A.7)
$$\begin{aligned}&c_{13}\Big (2V_{uu} + V_u^2 - 2 U_uV_u - 4 h_uV_u\Big ) \nonumber \\&\quad +\, \big (c_{13} + 2c_2\big )\Big (2U_{uu} - U_u^2 - 4h_uU_u\Big )\nonumber \\&\quad +\, 4 c_2h_{uu} - 2\big (3c_2 - c_{13} + c_{14}\big )h_u^2 = 0 ~~~~ \end{aligned}$$
(A.8)

where in Eq. (A.4) we have used the fact that \(T^{\ae }_{00}\) can be expressed in terms of \(T^{\ae }_{11}\) which is equal to zero.

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Oost, J., Bhattacharjee, M. & Wang, A. Gravitational plane waves in Einstein-aether theory. Gen Relativ Gravit 50, 124 (2018). https://doi.org/10.1007/s10714-018-2453-6

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