It has been proven that in standard Einstein gravity, exotic matter (i.e., matter violating the pointwise and averaged Weak and Null Energy Conditions) is required to stabilize traversable wormholes. Quantum field theory permits these violations due to the quantum coherent effects found in any quantum field. Even reasonable classical scalar fields violate the energy conditions. In the case of the Casimir effect and squeezed vacuum states, these violations have been experimentally proven. It is advantageous to investigate methods to minimize the use of exotic matter. One such area of interest is extended theories of Einstein gravity. It has been claimed that in some extended theories, stable traversable wormholes solutions can be found without the use of exotic matter. There are many extended theories of gravity, and in this review paper, we first explore $f\left(R\right)$ theories and then explore some wormhole solutions in $f\left(R\right)$ theories, including Lovelock gravity and Einstein Dilaton Gauss–Bonnet (EdGB) gravity. For completeness, we have also reviewed ‘Other wormholes’ such as Casimir wormholes, dark matter halo wormholes, thin-shell wormholes, and Nonlocal Gravity (NLG) wormholes, where alternative techniques are used to either avoid or reduce the amount of exotic matter that is required.

Motivated by recently explored examples, we undertake a systematic study of conformal invariance in one-dimensional sigma models where an isometry group has been gauged. Perhaps surprisingly, we uncover classes of sigma models which are only scale invariant in their ungauged form and become fully conformally invariant only after gauging. In these cases the target space of the gauged sigma model satisfies a deformation of the well-known conformal geometry constraints. We consider bosonic models as well as their $$ \mathcal{N} $$N = 1, 2, 4 supersymmetric extensions. We solve the quantum ordering ambiguities in implementing (super-) conformal symmetry on the physical Hilbert space. Examples of our general results are furnished by the D(2, 1; 0)-invariant Coulomb branch quiver models relevant for black hole physics.

Superconformal mechanics describes superparticle dynamics in near-horizon geometries of supersymmetric black holes. We systematically study the minimal compatible set of constraints required for a gauged superconformal symmetry. Our study uncovers classes of sigma models, which are only scale invariant in their ungauged form and become fully conformal invariant only after gauging.

Wedevelop a fully gauge-invariant analysis of gravitational-wave polarizations in metric f(R) gravity with a particular focus on the modified Starobinsky model f(R) = R +αR² −2Λ, whose constant curvature solution R_d = 4Λ provides a natural de Sitter background for both early- and late-time cosmology. Linearizing the field equations around this background, we derive the Klein–Gordon equation for the curvature perturbation δR and show that the scalar propagating mode acquires a mass m_ψ² = 1/(6α), highlighting how the same scalar degree of freedom governs inflationary dynamics at high curvature and the ropagation of gravitational waves in the current accelerating Universe. Using the scalar–vector–tensor decomposition and a decomposition of the perturbed Ricci tensor, we obtain a set of fully gauge-invariant propagation equations that isolate the contributions of the scalar, vector, and tensor modes in the presence of matter. We find that the tensor sector retains the two transverse–traceless polarizations of General Relativity, while the scalar sector supports a massive breathing/longitudinal mode determined by the massive scalar propagating mode. Through the geodesic deviation equation—computed both in a local Minkowski patch and in fully covariant de Sitter form—we independently recover the same polarization content and identify its tidal signatures. The resulting framework connects the extra scalar polarization to cosmological observables, providing a unified, gauge-invariant link between gravitational-wave phenomenology and the cosmological implications of metric f(R) gravity.

In this work we will investigate the structures of invariants within the class of plane waves in linearized General Relativity, to introduce a new approach to determine the linearized gauge invariants encoding the polarization of the plane waves. The Newman–Penrose (NP) formalism is traditionally used to analyze the polarization content of gravitational waves, while the gauge-invariant Bardeen formalism provides a complementary description based on the irreducible scalar, vector, and tensor perturbations of the metric. We will apply the Bardeen formalism to compute all scalar, vector, and tensor gauge-invariant variables explicitly and verify that only the two transverse–traceless tensor modes survive. We then compare these Bardeen variables with curvature-based invariants constructed using the linearized Cartan–Karlhede (CK) algorithm. Although one might anticipate a correspondence, our analysis shows that the CK invariants do not capture the polarization modes: the Weyl tensor possesses only a single non-zero Newman–Penrose scalar and the CK algorithm terminates without producing invariants that distinguish the $$\oplus $$ ⊕ and $$\otimes $$ ⊗ states. However, by computing invariant quantities obtained from the translational Killing vector fields of the Minkowski background that are retained under linear perturbation, we provide an algorithmic approach that reproduces the same physical tensor degrees of freedom captured by the Bardeen variables. The algorithm is a new result which is expected to be applicable to determine linearized gauge invariants for any linearized gravity solution admitting symmetries.

It has been proven that in standard Einstein gravity, exotic matter (i.e. matter violating the point-wise and averaged weak and null energy conditions) is required to stabilize traversable wormholes. Quantum field theory permits these violations due to the quantum coherent effects found in any quantum field . Even reasonable classical scalar fields violate the energy conditions. In the case of the Casimir effect and squeezed vacuum states, these violations have been experimentally proven. It is advantageous to investigate methods of minimizing the use of exotic matter. One such area of interest is extended theories of Einstein gravity. It has been claimed that in some extended theories, stable traversable wormholes solutions can be found without the use of exotic matter. There are many extended theories of gravity and in this review paper we first explore f(R) theories and then explore some wormhole solutions in f(R) theories including Lovelock gravity.