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We discuss non-relativistic limits of general relativity. In particular, we define a special fine-tuned non-relativistic limit, inspired by string theory, where the Einstein-Hilbert action has been supplemented by the kinetic term of a one-form gauge field. Taking the limit, a crucial cancellation takes place, in an expansion of the action in terms of powers of the velocity of light, between a leading divergence coming from the spin-connection squared term and another infinity that originates from the kinetic term of the one-form gauge field such that the finite invariant non-relativistic gravity action is given by the next subleading term. This non-relativistic action allows an underlying torsional Newton-Cartan geometry as opposed to the zero torsion Newton-Cartan geometry that follows from a more standard limit of General Relativity but it lacks the Poisson equation for the Newton potential. We will mention extensions of the model to include this Poisson equation.

This chapter discusses inverse gravimetry as an ill-posed problem in Newtonian potential theory. It is shown that all criteria of Hadamard’s classification are violated for terrestrial data. Consequently, inverse gravimetry is considered “too ill-posed” to use regularization techniques other than mollifier regularization.

The Newtonian potential, or the potential of volume, associated to the Laplace equation \(\Delta u=0\) is, by definition, the following improper integral: $$\begin{aligned} U(\xi )=-\frac{1}{(n-2)\omega _n}\int _{\Omega }\frac{f(x)}{r^{n-2}}\mathrm {d}x, \end{aligned}$$ where \(r=r_{\xi x}=\left| \overline{\xi x}\right| =\sqrt{ \sum \limits _{i=1}^n\left( x_i-\xi _i\right) ^2}\), and \(\omega _n\) is the area of the unit sphere from the n-dimensional space \(\mathrm{I}\!\mathrm{R}^n\).

Particle trajectories in different Spherically Symmetric static space-time metrics are studied employing Pseudo Newtonian potential (PNP).

The gravitational potential of (a part of) the Earth is assumed to be available, the inverse gravimetry problem is to determine the density contrast function inside (the specified part of) the Earth from known potential values. This paper deals with the characteristic ill-posed features of transferring input gravitational information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined dependent on the data source. Three cases of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as Haar-type solutions are proposed in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently available features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Finally, reproducing kernel Hilbert space solutions are formulated for use of gravimeter data, independent of a specifically chosen input area, i.e., in whole Euclidean space \(\mathbb{R}^3\)

We present a new geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center of mass of geometrostatic systems. We then explain the pseudo-Newtonian character of these formulas and show that they converge to Newtonian mass and center of mass in the Newtonian limit, respectively, using Ehlers’ frame theory. Moreover, we present a novel physical interpretation of the level sets of the canonical lapse function and apply it to prove uniqueness results. Finally, we suggest a notion of force on test particles in geometrostatic space-times.

The Inflationary account for the emerging of the seeds of cosmic structure from quantum fluctuations is a central part of our current views of cosmology. It is, on the one hand, extremely successful at the phenomenological level, and yet, it retains an aspect that is generally regarded as controversial: The exact mechanism by which quantum fluctuations transmute into actual inhomogeneities. We will review the considerations that lead us to conclude that the fully satisfactory resolution of the issue requires novel physics, and we will discuss an option we have been considering in this regard.

The so-called f(R)-gravity addresses issues like dark energy and dark matter from the point of view of gravitational field instead of requiring new material ingredients. In particular, several self-gravitating structures, like spiral and elliptical galaxies, can be consistently described, in the weak-field limit regime, due to the presence of Yukawa-like corrections in the Newtonian potential. We give here some examples related to the observations.