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BUG: Index.get_indexer will change behaviour for nulls with arrow strings #55833

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@phofl

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@phofl

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Reproducible Example

idx = pd.Index(["a", "b", None], dtype="string[pyarrow_numpy]")
idx.get_indexer([None])

Issue Description

This returns -1 for the new arrow string dtype because we cast None to np.nan when creating the array. We wanted to be as close to object dtype as possible, but patching this does not seem ideal. Thoughts?

cc @jorisvandenbossche

Expected Behavior

see above

Installed Versions

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    API DesignArrowpyarrow functionalityStringsString extension data type and string data

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