quantum gravity
Physics rests on two towering theories that refuse to speak to each other. General relativity describes gravity as smooth, curved spacetime and governs the very large: stars, galaxies, the cosmos. Quantum mechanics describes matter as probabilistic and granular and governs the very small: atoms, particles, fields. Each is superbly tested in its own domain, yet they are built on incompatible foundations. Quantum gravity is the sought-after theory that would unite them, describing spacetime itself as a quantum object.
The clash becomes unavoidable where both theories are needed at once: at extreme density and tiny scales, such as the centre of a black hole or the first instant of the Big Bang, near the Planck scale of about 10^-35 metres. There, the smooth geometry of general relativity should itself fluctuate quantum-mechanically, but applying the standard recipe of quantum field theory to gravity fails: the resulting theory is non-renormalizable, meaning it produces uncontrollable infinities that cannot be absorbed by redefining a finite set of parameters, unlike quantum electrodynamics. A working theory of quantum gravity must tame those infinities and reduce to both general relativity and quantum field theory in their respective limits.
There is as yet no complete, experimentally confirmed theory of quantum gravity. The two leading research programmes are string theory, which replaces point particles with tiny vibrating strings and naturally contains a graviton, and loop quantum gravity, which quantizes the geometry of spacetime directly into discrete chunks of area and volume. Both make sense mathematically in various ways, but Planck-scale energies are so far beyond any conceivable accelerator that direct tests are extraordinarily hard, which is why this remains the deepest open frontier of theoretical physics.
At the singularity inside a black hole, general relativity predicts infinite curvature and simply breaks down; only a quantum theory of gravity could say what really happens there. Similarly, running the Big Bang back toward t = 0 hits Planck-scale conditions where quantum gravity, not classical relativity, must take over.
Where relativity and quantum theory both apply, only quantum gravity can speak.
Gravity being 'quantum' does not mean it is simply general relativity with hbar sprinkled in; naively quantizing it gives a non-renormalizable theory. No candidate theory has yet been confirmed, because Planck-scale energies lie far beyond any experiment we can currently build.