We study strong gravitational lensing due to a Schwarzschild black hole. Apart from the primary and the secondary images we find a sequence of images on both sides of the optic axis; we call them relativistic images. These images are formed due to large bending of light near $r=3M$ (the closest distance of approach $r_o$ is greater than $3M).$ The sources of the entire universe are mapped in the vicinity of the black hole by these images. For the case of the Galactic supermassive “black hole” they are formed at about $17$ microarcseconds from the optic axis. The relativistic images are not resolved among themselves, but they are resolved from the primary and secondary images. However the relativistic images are very much demagnified unless the observer, lens and source are very highly aligned. Because of this and some other difficulties the observation of these images does not seem to be feasible in the near future. However, it would be a great success of the general theory of relativity in a strong gravitational field if they ever were observed and it would also give an upper bound, $r_o=3.21M,$ to the compactness of the lens, which would support the black hole interpretation of the lensing object.

We model massive dark objects in galactic nuclei as spherically symmetric static naked singularities in the Einstein massless scalar field theory and study the resulting gravitational lensing in detail. Based on whether or not a naked singularity is covered within a photon sphere we classify naked singularities into two kinds: weakly naked (those contained within at least one photon sphere) and strongly naked (those not contained within any photon sphere). The qualitative features of gravitational lensing due to a weakly naked singularity are similar to those due to a Schwarzschild black hole (these give rise to one Einstein ring but no radial critical curve). However, the gravitational lensing due to a strongly naked singularity is qualitatively different from that due to a Schwarzschild black hole; a strongly naked singularity gives rise to either two or nil Einstein ring(s) and one radial critical curve. A light ray passing close to a photon sphere of a black hole or a weakly naked singularity goes around its photon sphere once, twice, or many times (before reaching an observer) depending upon the impact parameter and thus gives rise to a sequence of theoretically infinite number of relativistic images, which are highly demagnified. A strongly naked singularity produces no relativistic images.

We present the time drift of the cosmological redshift in a general spherically symmetric spacetime. We demonstrate that its observation would allow us to test the Copernican principle and so determine if our Universe is radially inhomogeneous, an important issue in our understanding of dark energy. In particular, when combined with distance data, this extra observable allows one to fully reconstruct the geometry of a spacetime describing a spherically symmetric underdense region around us, purely from background observations.

Both bottom-up and top-down causation occur in the hierarchy of structure and causation. A key feature is multiple realizability of higher level functions, and consequent existence of equivalence classes of lower level variables that correspond to the same higher level state. Five essentially different classes of top-down influence can be identified, and their existence demonstrated by many real-world examples. They are: algorithmic top-down causation; top-down causation via non-adaptive information control, top-down causation via adaptive selection, top-down causation via adaptive information control and intelligent top-down causation (the effect of the human mind on the physical world). Through the mind, abstract entities such as mathematical structures have causal power. The causal slack enabling top-down action to take place lies in the structuring of the system so as to attain higher level functions; in the way the nature of lower level elements is changed by context, and in micro-indeterminism combined with adaptive selection. Understanding top-down causation can have important effects on society. Two cases will be mentioned: medical/healthcare issues, and education—in particular, teaching reading and writing. In both cases, an ongoing battle between bottom-up and top-down approaches has important consequences for society.

We reconsider the issue of proving large-scale spatial homogeneity of the Universe, given isotropic observations about us and the possibility of source evolution in both numbers and luminosities. Two theorems make precise the freedom available in constructing cosmological models that will fit the observations. They make it quite clear that homogeneity cannot be proven without either a fully determinate theory of source evolution, or availability of distance measures that are independent of source evolution. We contrast this goal with the standard approach that assumes spatial homogeneity a priori, and determines source evolution functions on the basis of this assumption.