What is the evanescent evidence, and hot pursuit exception to the warrant requirement?

Sometimes police do not have time to go get a warrant before performing a search or seizure because the nature of the item they wish to search or seize is likely to—or, at least, at risk of—change. For example, in Raiders of the Lost Ark, Indiana Jones was in search of the Ark of the Covenant and is able to determine its location by placing a staff into a specific location of a specific room so that sunlight at a specific time would shine through the headpiece of the staff and generate a red beam of light that would point to a specific place on a map. The evidence of the Ark's location was extremely evanescent, in that it is only available to be observed for a short moment in time before disappearing. This doesn't necessarily forgive Indiana Jones' trespass, but this is the kind of situation that can sometimes justify a police officer's decision to act rather than first get a warrant.

A more practical example of evanescent evidence comes from Cupp v. Murphy: Daniel Murphy's wife was murdered by strangulation in her home, leaving bruises and wounds on her throat. Police asked Daniel to come in for questioning. When he did, officers immediately noticed he had some dark spots underneath his fingernails, which they believed may be dried blood. Officers asked Daniel for permission to take a sample of the substance, and Daniel denied their request. Over Daniel's protests, officers took a sample anyway. The substance turned out to be his wife's dried blood and helped convict Daniel of the murder. Daniel sued, claiming that this amounted to an unconstitutional search since the officers did not first get a warrant. The SCOTUS defended the officer's actions due to the evanescent nature of the evidence. In other words, Daniel clearly didn't realize he was carrying around his dead wife's blood under his fingernails and once he was alerted to this fact, all it would take to destroy the evidence would be washing his hands or even placing them in his mouth. This risk of destroying the evidence, paired with the other surrounding probable cause, means that the officers were justified in forcibly taking samples without a warrant or Daniel's consent.

Similar to evanescent evidence, the hot pursuit exception recognizes that police may be unable to secure a warrant before entering certain private locations if they are in the middle of chasing someone who has committed a serious crime, like a felony. In other words, if police are chasing a person who just committed an armed robbery, there is a chance that the suspect may flee and escape into someone's private residence (like running through a backyard, or into an apartment complex). It would be silly to require police to stop chasing the suspect once they committed their trespass. Thus, while police are in hot pursuit of such a suspect, they are allowed to enter and pass through private places that would otherwise require a warrant and while they are there, the plain view exception applies, meaning they can seize any evidence of any other crimes they may notice while in hot pursuit.