Well, it's always possible that we'll come up with something better in the future, (ie relativity replacing Newtonian mechanics) but for the time being both special relativity and general relativity are very well accepted, and have been demonstrated for decades.
One of the most commonly mentioned modern examples is the GPS system. To determine your precision requires knowing extremely precise times and time differences. GPS positions would diverge from accuracy by something like 10KM/day if they were building the system they hadn't corrected for both the motion impact (special relativity) and our location deeper in a gravity well than the satellites (general relativity).
One of the most commonly mentioned modern examples is the GPS system. To determine your precision requires knowing extremely precise times and time differences. GPS positions would diverge from accuracy by something like 10KM/day if they were building the system they hadn't corrected for both the motion impact (special relativity) and our location deeper in a gravity well than the satellites (general relativity).

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