Triple Path to the Exponential Metric
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Triple Path to the Exponential Metric Maxim Makukov1 · Eduard Mychelkin1 Received: 16 April 2020 / Accepted: 16 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The exponential Papapetrou metric induced by scalar field conforms to observational data not worse than the vacuum Schwarzschild solution. Here, we analyze the origin of this metric as a peculiar space-time within a wide class of scalar and antiscalar solutions of the Einstein equations parameterized by scalar charge. Generalizing the three families of static solutions obtained by Fisher (Zhurnal Experimental’noj i Teoreticheskoj Fiziki 18:636, 1948), Janis et al. (Phys Rev Lett 20(16):878. https:// doi.org/10.1103/PhysRevLett.20.878, 1968), and Xanthopoulos and Zannias (Phys Rev D 40(8):2564, 1989), we prove that all three reduce to the same exponential metric provided that scalar charge is equal to central mass, thereby suggesting the universal character of such background scalar field. Keywords Exponential metric · Scalar field · Janis–Newman–Winicour solution · Scalar charge
1 Introduction Understanding the universe evolution depends on the correct answer to the question—what kind of physical vacuum do we live in? In general relativity there is a mainstream in the study of various effects and processes related to known vacuum solutions of the Einstein equations, such as black holes and gravitational waves. On the other hand, similar phenomena may arise due to (or be influenced by) some background scalar field 𝜙(x𝛼 ) suggested, in particular, by the dark side of the universe. E.g., the presence of such background scalar field affects whether one will observe regular black holes or, rather, compact objects without horizons [1–3]. In this regard, the exponential spherically symmetric Papapetrou metric [4]:
* Maxim Makukov [email protected] Eduard Mychelkin [email protected] 1
Fesenkov Astrophysical Institute, 050020 Almaty, Republic of Kazakhstan
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( ) ds2 = e−2𝜙(r) dt2 − e2𝜙(r) dr2 + r2 d𝛺2 ,
(1)
with 𝜙(r) = M∕r , represents a viable alternative to pure vacuum approach [3] (we adopt 𝜙 > 0 ). The fact that the source for the exponential metric represents a minimal scalar field in antiscalar regime (see below) was first pointed out by Yilmaz [5]. Note also that the exponential metric satisfies the “Papapetrou ansatz” g𝜇𝜈 = g𝜇𝜈 (𝜙(x𝛼 )) , which picks out the class of metrics whose coefficients depend on coordinates solely through the background scalar field. As evident from (1), this metric is horizon-free. In the isotropic form (1) of the metric, both finite and infinitesimal spatial intervals have the same scale factor. Besides, this spatially-conformal factor is reciprocal to that of the time interval, so that in the weak field approximation this automatically leads to the Newtonian gauge widely used, in particular, in cosmological applications. The potential in exponents in (1) plays a key role in that it allows to make quantitative estimations of ef
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