The problem of reconstruction for static spherically-symmetric 4D metrics in scalar-Einstein–Gauss–Bonnet model
Abstract We consider the 4D gravitational model with a scalar field $$\varphi $$ φ , Einstein and Gauss–Bonnet terms. The action of the model contains a potential term $$U(\varphi )$$ U ( φ ) , Gauss–Bonnet coupling function $$f(\varphi )$$ f ( φ ) and a parameter $$\varepsilon = \pm 1 $$ ε = ± 1 ,...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2025-07-01
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| Series: | European Physical Journal C: Particles and Fields |
| Online Access: | https://doi.org/10.1140/epjc/s10052-025-14481-7 |
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| Summary: | Abstract We consider the 4D gravitational model with a scalar field $$\varphi $$ φ , Einstein and Gauss–Bonnet terms. The action of the model contains a potential term $$U(\varphi )$$ U ( φ ) , Gauss–Bonnet coupling function $$f(\varphi )$$ f ( φ ) and a parameter $$\varepsilon = \pm 1 $$ ε = ± 1 , where $$\varepsilon = 1$$ ε = 1 corresponds to ordinary scalar field and $$\varepsilon = -1 $$ ε = - 1 - to phantom one. Inspired by the recent works of Nojiri and Nashed, we explore a reconstruction procedure for a generic static spherically symmetric metric written in the Buchdal parametrization: $$ds^2 = \left( A(u)\right) ^{-1}du^2 - A(u)dt^2 + C(u)d\Omega ^2$$ d s 2 = A ( u ) - 1 d u 2 - A ( u ) d t 2 + C ( u ) d Ω 2 , with given $$A(u) > 0$$ A ( u ) > 0 and $$C(u) > 0$$ C ( u ) > 0 . The procedure gives the relations for $$U(\varphi (u))$$ U ( φ ( u ) ) , $$f(\varphi (u))$$ f ( φ ( u ) ) and $$d\varphi /du$$ d φ / d u , which lead to exact solutions to equations of motion with a given metric. A key role in this approach is played by the solutions to a second order linear differential equation for the function $$f(\varphi (u))$$ f ( φ ( u ) ) . The formalism is illustrated by two examples when: a) the Schwarzschild metric and b) the Ellis wormhole metric, are chosen as a starting point. For the first case a) the black hole solution with a “trapped ghost” is found which describes an ordinary scalar field outside the photon sphere and phantom scalar field inside the photon sphere. For the second case b) the sEGB-extension of the Ellis wormhole solution is found when the coupling function reads: $$f(\varphi ) = c_1 + c_0 ( \tan ( \varphi ) + \frac{1}{3} (\tan ( \varphi ))^3)$$ f ( φ ) = c 1 + c 0 ( tan ( φ ) + 1 3 ( tan ( φ ) ) 3 ) , where $$c_1$$ c 1 and $$c_0$$ c 0 are constants. |
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| ISSN: | 1434-6052 |