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In the Schwarzschild coordinates it is stated that it is clear that an object appears to slow down near the horizon form the perspective of an outside observer.

How do we do this calculation in in Eddington–Finkelstein coordinates? Where the path of the projectile has a continuous smooth path over the horizon?

To calculate its apparent position, would we have to calculate the path of a beam of light from the projectile to an observer? Wouldn't we also have to do that in Schwarzschild coordinates? This seems like a non-trivial calculation mathematically.

Thus to do a simulation of a moving object from the perspective of an observer would we have to continously calculate beams of light emitted by the projectile to see where it's apparent position was?

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To calculate its apparent position, would we have to calculate the path of a beam of light from the projectile to an observer? Wouldn't we also have to do that in Schwarzschild coordinates? This seems like a non-trivial calculation mathematically.

This is exactly what you have to do to show this. Fortunately, this is not a very hard calculation. Outward going radial null rays travel along lines of constant retarded time $u = t-r_*$. While the advanced time $v = t+r_*$ of an object falling through the horizon stays finite, the retarded time blows up. (This can be easily seen from the observation that $t\to\infty$ for geodesic approaching the horizon, while from it definition one also readily sees (from its definition) that the tortoise coordinate $r_*\to-\infty$.)

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