The Gregory–Laflamme instability of the Schwarzschild black string exterior

In this paper, a direct rigorous mathematical proof of the Gregory–Laflamme instability for five-dimensional Schwarzschild black string is presented. Under choice ansatz perturbation and gauge choice, linearized vacuum Einstein equation reduces to an ordinary differential (ODE) problem single function. work, suitable rescaling change variables applied, which casts ODE into Schrödinger eigenvalue energy functional assigned. It then shown by variational methods that lowest eigenfunction gives rise exponentially growing mode solution, has admissible behavior at future event horizon spacelike infinity. After addition pure regular solution in harmonic/transverse-traceless gauge.