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Description
We study an entropic regularization of optimal transport on spacetimes and directed acyclic graphs, where mass may only be transported along a causal relation, the cost is a power of the time separation, and one maximizes rather than minimizes the total cost (or benefit). Unlike classical entropic optimal transport, the causal constraint makes the cost take the value minus infinity, so standard tools break down and new phenomena appear. For finitely supported marginals, a causal Hall-type condition characterizes when a causal coupling exists. A strict version of it decides whether the Sinkhorn algorithm converges to an exact scaling or only approximately, with diverging dual potentials. In general we prove existence, uniqueness, and duality of entropic optimizers. Under a quantitative feasibility assumption we also show convergence to Lorentz optimal transport as the regularization vanishes. Without that assumption, explicit examples show that the entropic problem can collapse completely.