 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-311/3/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

For any nonzero

g

-causal vector

v^{a}

g_{ab} v^{a} v^{b} = g_{ab} v^{a} v^{b} - (t_{a} v^{a})^{2} .

For timelike

v

, the first term is negative. For null

v

, a nonzero null vector cannot be orthogonal to a timelike vector, so

(t \cdot v)^{2} > 0

. Thus

g (v, v) < 0

in both cases. Every

g

-causal direction is strictly timelike for

g

, so its light cone is strictly wider.

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