What is Calvin & Hobbes primary misconception? Subjective time -- proper time -- is a fundamental physical observable, independent of coordinates Same path through space-time Same subjective elapsed time
x t (dt, dx) B. Riemann German Formalized non- Euclidean geometry (1854) g Metric Tensor Newtonian: Most general:
x t BankNews Stand You Me Car for all paths Assuming Newton found parking…. Galilean/ Newtonian
x t UpDown Basement detector Down-going muon decays Stopped muon decays 2.2 sec Cosmic ray lab Lab Basement Down-going photon 1.Down-going muons v < c 2.Lifetime of down-going muons > 2.2 sec We observe:
x t Curve of constant =2.2 sec H. Minkowski German Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" (1907) x t Rest frame of lab-stopped muon Rest frame of down-going muon
The Twin Paradox made easy x t Castor inbound T X Castor outbound Pollux at rest Its just that simple!
New, generalized laws of motion 1. In getting from A to B, all free- falling objects will follow the path of maximal proper time (geodesic). 2. Photons follow null paths of zero proper time. x t A B
Recovering Newtons First Law x t A B x t A B T T XX An object at rest tending to remain at rest
Gravitational Red Shift x t Up Down Source Downstairs Detector Upstairs Photon period t Downstairs Downstairs Photon period t Upstairs Upstairs Non-inertial frame -- curved space!
Motion in curved 1+1D space x t A B h T T Conservative (Newtonian) Potential!
Albert Einstein German General Theory of Relativity (1915) Isaac Newton British Universal Theory of Gravitation (1687) Metric: a local property Force law: action at a distance
Points to take home Subjective/proper time as the fundamental observable Central role of the metric Free-fall paths maximize subjective time Minkowski metric for empty space recovers Newtons 1 st law Slightly curved space reproduces Newtonian gravitation