What happens if someone gets away with a serious crime and never gets caught in their entire rest of their life?

If they authorities ever solve the case, it is closed. Even if you can convict a dead person, actual punishment isn't possible.

The criminal would answer to God, but that is outside of earthly government authority.

No, nothing happens to the "law people" if the person who committed a crime is never caught.

They will still have to answer to God for their crime.

Business as usual, there are cold cases going back to the days of prohibition.

That is MOST of what happens.

Contrary to "crime stories" and movies there are no supercops who know the guilty party as soon as they arrive on scene.

The vast majority of criminals who actually get caught are turned in by a relative or accomplice or just shoot off their mouth too much.

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Look up how the Manson family was finally identified for the Tate / LaBianca murders. Instills no faith in your government at all.

An investigation happens but never gets solved.

They still have to live with what they did.