Does the Alcubierre drive/negative energy manipulation imply "antigravity"

So assume your setting has the Alcubierre drive - it requires mastery over negative energy to some extent (at least, some form of exotic matter you can excite to generate that energy).

Is negative energy like this usable for creating antigravity, hypothetically speaking, i.e. if you have FTL based on Alcubierre warp in a setting, should you have antigravity in it, too? (I am writing a setting where you have FTL based on the warp drive but no antigravity, wondering if this is internally consistent)

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Does the Alcubierre drive/negative energy manipulation imply "antigravity"

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