Triple Path to the Exponential Metric

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.