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You're not far off, but there are a few key points to clarify. Hawking radiation is a result of quantum mechanics in curved spacetime and isn't predicted by general relativity alone. GR doesn't account for radiation from black holes; that's where quantum field theory comes in.

Regarding gravitational waves, if we assume a quantum theory of gravity, black holes might emit gravitons, resulting in gravitational waves. However, these emissions would be minuscule compared to electromagnetic radiation due to the weaker gravitational force. Thus, while theoretically possible, gravitational waves from such black holes would be negligible in practice.