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Does the spacetime curvature in the vicinity of a massive body such as the sun increase, decrease or remain unchanged with respect to an observer's increasing velocity relative to that massive body?

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The curvature of the spacetime geometry most often is expressed in terms of curvature invariant, examples are the Kretschmann scalar, the Chern–Pontryagin scalar and the Euler scalar. Look at the discussion in Wikipedia.

The scalar invariants are the same for all observers, regardless of their velocity.

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