Bounds on the photon sphere radius for spherically symmetric black holes in n-dimensional Einstein gravity

The photon sphere, a hypersurface of null circular geodesics, plays a fundamental role in characterizing black hole spacetimes, influencing phenomena such as black hole shadows, gravitational lensing, and quasinormal modes. In this work, we derive both upper and lower bounds on the photon sphere radius for static, spherically symmetric, asymptotically flat black holes within n -dimensional Einstein gravity (n≥ 4) ( n ≥ 4 ) , assuming an anisotropic matter field satisfying the weak energy condition and a non-positive trace of the energy–momentum tensor. For the upper bound, we obtain r_γ ≤ [(n-1)M]1/n-3, r γ ≤ [ ( n - 1 ) M ] 1 n - 3 , where M is the ADM mass. In the four-dimensional case $$(n=4)$$ ( n = 4 ) , this reduces to r_γ ≤ 3M, r γ ≤ 3 M , in agreement with previous results. For the lower bound, under the additional assumption that |rⁿ⁻¹pᵣ(r)| | r n - 1 p r ( r ) | is monotonically decreasing, we prove r_γ ≥ (n-1/2)1/(n-3)rH, r γ ≥ ( n - 1 2 ) 1 / ( n - 3 ) r H , where rH r H is the radius of the outer event horizon; for $$n=4$$ n = 4 this gives r_γ ≥ 3/2rH,