Novel ansatzes and scalar quantities in gravito-electromagnetism

Abstract

In this work, we focus on the theory of gravito-electromagnetism (GEM)—the theory that describes the dynamics of the gravitational field in terms of quantities met in electromagnetism—and we propose two novel forms of metric perturbations. The first one is a generalisation of the traditional GEM ansatz, and succeeds in reproducing the whole set of Maxwell’s equations even for a dynamical vector potential $\mathbf {A}$. The second form, the so-called alternative ansatz, goes beyond that leading to an expression for the Lorentz force that matches the one of electromagnetism and is free of additional terms even for a dynamical scalar potential $\varPhi$. In the context of the linearised theory, we then search for scalar invariant quantities in analogy to electromagnetism. We define three novel, 3rd-rank gravitational tensors, and demonstrate that the last two can be employed to construct scalar quantities that succeed in giving results very similar to those found in electromagnetism. Finally, the gauge invariance of the linearised gravitational theory is studied, and shown to lead to the gauge invariance of the GEM fields $\mathbf {E}$ and $\mathbf {B}$ for a general configuration of the arbitrary vector involved in the coordinate transformations.

Appendices

Appendix A: Components of the Christoffel symbols

Here, we present the components of the Christoffel symbols for both metric perturbations ansatzes that are necessary for the derivation of the Lorentz equation in each case.

Employing the expression of the Christoffel symbols (3.7) in the linear approximation and of the initial perturbations $h_{\mu \nu }$ from the line-element (3.10) for the case of the generalised traditional ansatz, we find the following components

$$\begin{aligned} \varGamma ^i_{00}= & {} \frac{1}{c^2}\,\partial _i \varPhi - \frac{3}{2c^4}\,\partial _i \lambda +\frac{2}{c^3}\,\partial _t A^i\,,\nonumber \\ \varGamma ^i_{0j}= & {} \frac{1}{c^2}\,F_{ij} - \frac{1}{c^3}\,\delta ^i_j \left( \partial _t \varPhi +\frac{1}{2c^2}\,\partial _t \lambda \right) + \frac{1}{c^5}\,\partial _t d^i_{\,\, j}\,, \nonumber \\ \varGamma ^i_{kj}= & {} -\frac{1}{c^2}\left( \delta ^i_j \,\partial _k \varPhi + \delta ^i_k \,\partial _j \varPhi -\delta _{kj} \,\partial _i \varPhi \right) \nonumber \\&-\frac{1}{2c^4}\left( \delta ^i_j \,\partial _k \lambda + \delta ^i_k \,\partial _j \lambda -\delta _{kj} \,\partial _i \lambda \right) + \frac{1}{c^4}\left( \partial _k d^i_{\,\, j} + \partial _j d^i_{\,\, k}- \partial ^i d_{kj} \right) . \nonumber \\ \end{aligned}$$

(A.1)

Note that above we have used the definition $F_{ij} \equiv \partial _i A_j -\partial _j A_i$.

In the case of the alternative ansatz for the metric perturbations, the original perturbations $h_{\mu \nu }$ may be deduced from the line-element (3.15). Then, the Christoffel symbols are found to be

$$\begin{aligned} \varGamma ^i_{00}= & {} \frac{(1+3\gamma )}{4c^2}\,\partial _i \varPhi +\frac{1}{c^3}\,\partial _t A^i\,, \qquad \varGamma ^i_{0j} = \frac{1}{2c^2}\,F_{ij} - \frac{(1-3\gamma )}{4c^3}\,\delta ^i_j \,\partial _t \varPhi + \frac{1}{2c}\,\partial _t {\tilde{h}}^i_{\,\, j}\,, \nonumber \\ \varGamma ^i_{kj}= & {} -\frac{(1-3\gamma )}{4c^2}\left( \delta ^i_j \,\partial _k \varPhi + \delta ^i_k \,\partial _j \varPhi -\delta _{kj} \,\partial _i \varPhi \right) + \frac{1}{2}\left( \partial _k {\tilde{h}}^i_{\,\, j} + \partial _j {\tilde{h}}^i_{\,\, k}- \partial ^i {\tilde{h}}_{kj} \right) . \end{aligned}$$

(A.2)

Appendix B: Components of the novel gravitational tensors

By using the components of the perturbations ${\tilde{h}}_{\mu \nu }$, as these appear in the generalised traditional ansatz (3.3), we find the following explicit forms for the components of the new tensor $F_{\alpha \mu \nu }$, defined in Eq. (4.5):

$$\begin{aligned} F_{000}= & {} -\frac{2}{c^2}\,\partial _i A^i, \,\, F_{i00}=\frac{4 E^i}{c^2} - \frac{2}{c^4} \left( \partial _i \lambda + \partial ^j d_{ij} \right) , \,\,\nonumber \\ F_{ij0}= & {} - \frac{2}{c^2} F_{ij} + \frac{2}{c^5}\partial _t \left( \lambda \,\eta _{ij} +d_{ij} \right) ,\; F_{00i}=\frac{4}{c^2} \partial _i \varPhi , \nonumber \\ F_{0ij}= & {} \frac{2}{c^2} \left[ (\partial _i A_j + \partial _j A_i) - \eta _{ij} \left( \frac{2}{c}\,\partial _t \varPhi + \partial _k A^k \right) \right] - \frac{2}{c^5} \partial _t (\lambda \, \eta _{ij} +d_{ij}), \nonumber \\ F_{ijk}= & {} \frac{2}{c^4} \bigl [ \partial _j (\lambda \eta _{ik} + d_{ik}) - (j \leftrightarrow k) - (j \leftrightarrow i) \bigr ]\nonumber \\&-\frac{2 \eta _{jk}}{c^3} \left[ \partial _t A_i +\frac{1}{c} \,\partial ^l (\lambda \eta _{il} + d_{il}) \right] . \end{aligned}$$

(B.1)

Similarly, employing the components of ${\tilde{h}}_{\mu \nu }$ of the alternative ansatz (3.13), we find the results

$$\begin{aligned} F_{000}= & {} -\frac{1}{c^2} \partial _i A^i, \qquad F_{i00}= \frac{1}{c^3} \partial _t A_i - \frac{1}{c^4} \partial ^j d_{ij}, \qquad F_{00i} =\frac{1}{c^2}\partial _i \varPhi , \nonumber \\ F_{ij0}= & {} -\frac{F_{ij}}{c^2} - \frac{\partial _t \varPhi }{c^3} \eta _{ij} +\frac{1}{c^5} \partial _t d_{ij}, \nonumber \\ F_{0ij}= & {} \frac{1}{c^2} \left[ ( \partial _iA_j +\partial _jA_i) - \eta _{ij} \partial _k A^k\right] -\frac{\partial _t d_{ij}}{c^5}, \nonumber \\ F_{ijk}= & {} -\frac{1}{c^2} \left( \eta _{ij}\,\partial _k \varPhi +\eta _{ik}\,\partial _j \varPhi -2\eta _{jk}\,\partial _i \varPhi \right) - \frac{1}{c^3}\,\eta _{jk}\,\partial _t A_i -\frac{1}{c^4}\,\eta _{jk}\,\partial ^l d_{il} \nonumber \\&+ \frac{1}{c^4}\,(\partial _j d_{ik} + \partial _k d_{ij} -\partial _i d_{jk}). \end{aligned}$$

(B.2)

We now move to the components of the gravitational tensors $Q_{\alpha \mu \nu }$ and $H_{\alpha \mu \nu }$ defined in Eqs. (4.11). For the generalised ansatz (3.3) (where we have set for simplicity $\lambda =0$), we find the following components of the $Q_{\alpha \mu \nu }$

$$\begin{aligned} Q_{000}= & {} \frac{6}{c^3}\,\partial _t \varPhi , \qquad Q_{i00}= \frac{4}{c^3}\,\partial _t A_i - \frac{2}{c^2}\,\partial _i \varPhi , \qquad Q_{00i} =\frac{2}{c^2}\,\partial _i \varPhi , \nonumber \\ Q_{0i0}= & {} \frac{6}{c^2}\,\partial _i \varPhi , \qquad Q_{ij0} = -\frac{2}{c^2}F_{ij} - \frac{2}{c^3}\,\partial _t \varPhi \,\eta _{ij} +\frac{2}{c^5}\,\partial _t d_{ij}, \nonumber \\ Q_{0ij}= & {} \frac{2}{c^2} ( \partial _iA_j +\partial _jA_i) + \frac{2 \partial _t \varPhi }{c^3} \eta _{ij} -\frac{2 \partial _t d_{ij}}{c^5}, \nonumber \\ Q_{i0j}= & {} -\frac{2 F_{ij}}{c^2} + \frac{2 \partial _t \varPhi \,\eta _{ij}}{c^3} +\frac{2 \partial _t d_{ij}}{c^5}, \nonumber \\ Q_{ijk}= & {} -\frac{2}{c^2} \left( \eta _{ij}\,\partial _k \varPhi -\eta _{ik}\,\partial _j \varPhi -\eta _{jk}\,\partial _i \varPhi \right) + \frac{2}{c^4}\,(\partial _j d_{ik} + \partial _k d_{ij} -\partial _i d_{jk}).\nonumber \\ \end{aligned}$$

(B.3)

and $H_{\alpha \mu \nu }$ tensors

$$\begin{aligned} H_{000}= & {} \frac{6}{c^3}\,\partial _t \varPhi , \qquad H_{i00}= \frac{2}{c^2}\,\partial _i \varPhi , \qquad H_{00i} =\frac{6}{c^2}\,\partial _i \varPhi , \nonumber \\ H_{0i0}= & {} \frac{4}{c^3}\,\partial _t A_i -\frac{2}{c^2}\,\partial _i \varPhi , \qquad H_{ij0} = \frac{2}{c^2}F_{ij} + \frac{2}{c^3}\,\partial _t \varPhi \,\eta _{ij} +\frac{2}{c^5}\,\partial _t d_{ij}, \nonumber \\ H_{0ij}= & {} -\frac{2 F_{ij}}{c^2} - \frac{2 \partial _t \varPhi }{c^3} \eta _{ij} +\frac{2 \partial _t d_{ij}}{c^5}, \nonumber \\ H_{i0j}= & {} \frac{2}{c^2} ( \partial _iA_j +\partial _jA_i) + \frac{2 \partial _t \varPhi }{c^3} \eta _{ij} -\frac{2 \partial _t d_{ij}}{c^5} \nonumber \\ H_{ijk}= & {} \frac{2}{c^2} \left( \eta _{ij}\,\partial _k \varPhi +\eta _{ik}\,\partial _j \varPhi -\eta _{jk}\,\partial _i \varPhi \right) + \frac{2}{c^4}\,(\partial _j d_{ik} + \partial _k d_{ij} -\partial _i d_{jk}).\nonumber \\ \end{aligned}$$

(B.4)

We follow a similar analysis for the case of the alternative ansatz (3.13) (with $\gamma =1$). Then, we find the following components for the $Q_{\alpha \mu \nu }$

$$\begin{aligned} Q_{000}= & {} 0=Q_{0i0}, \qquad Q_{i00}= \frac{2}{c^3}\,\partial _t A_i - \frac{2}{c^2}\,\partial _i \varPhi , \qquad Q_{00i} =\frac{2}{c^2}\,\partial _i \varPhi , \nonumber \\ Q_{ij0}= & {} -\frac{1}{c^2}F_{ij} +\frac{1}{c^5}\,\partial _t d_{ij} \qquad Q_{0ij} = \frac{1}{c^2}\left( \partial _iA_j +\partial _jA_i \right) -\frac{1}{c^5}\,\partial _t d_{ij}, \nonumber \\ Q_{i0j}= & {} -\frac{F_{ij}}{c^2} - \frac{2 \partial _t \varPhi }{c^3} \eta _{ij} +\frac{\partial _t d_{ij}}{c^5}, \nonumber \\ Q_{ijk}= & {} -\frac{2 \partial _j \varPhi }{c^2}\eta _{ik} + \frac{1}{c^4} (\partial _j d_{ik} + \partial _k d_{ij} -\partial _i d_{jk}). \end{aligned}$$

(B.5)

and $H_{\alpha \mu \nu }$ tensors

$$\begin{aligned} H_{000}= & {} 0=H_{00i}, \qquad H_{i00}=\frac{2}{c^2}\,\partial _i \varPhi , \qquad H_{0i0} = \frac{2}{c^3}\,\partial _t A_i -\frac{2}{c^2}\,\partial _i \varPhi , \nonumber \\ H_{ij0}= & {} \frac{1}{c^2}\,F_{ij} - \frac{2}{c^3}\,\partial _t \varPhi \,\eta _{ij} +\frac{1}{c^5}\,\partial _t d_{ij}, \qquad H_{0ij} =-\frac{1}{c^2}\,F_{ij} +\frac{1}{c^5}\,\partial _t d_{ij}, \nonumber \\ H_{i0j}= & {} \frac{1}{c^2} ( \partial _iA_j +\partial _jA_i) -\frac{\partial _t d_{ij}}{c^5}\,,\nonumber \\ H_{ijk}= & {} -\frac{2 \partial _k \varPhi }{c^2} \eta _{ij} + \frac{1}{c^4}\,(\partial _k d_{ij} + \partial _i d_{jk} -\partial _j d_{ik}). \end{aligned}$$

(B.6)