A polar bear's guard hairs are hollow, colorless keratin tubes — not white and not pigmented. Fur looks white because light partially reflects off thousands of hair surfaces and scatters in random directions, the same reason snow or ground glass looks white. A single hair, isolated, is clear.
This diagram traces one ray through a hair's wall using real Snell's-law refraction at the tip and total internal reflection at the wall (guided whenever the wall angle exceeds the critical angle asin(1/n)), riding down to the bear's actual black skin, where it is absorbed as heat. How much that mechanism really warms a live bear — versus simple insulation doing most of the work — is a genuine, still-debated question in the literature; this sim shows the optical mechanism, not a final verdict.