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Exploring Relativity: Subjective Time and Curved Space in Cosmology

This introductory lecture by Paul Stankus from ORNL provides a foundational understanding of cosmology, emphasizing the concept of subjective or proper time as a fundamental observable in physics. The lecture delves into the implications of Newtonian mechanics, the significance of the metric tensor, and introduces Minkowski's insights on the intertwining of space and time. Key topics include the Twin Paradox, geodesics of free-falling objects, and the relationship between gravity and time. This course aims to illuminate the complex yet fascinating interactions of time and space through a scientific lens.

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Exploring Relativity: Subjective Time and Curved Space in Cosmology

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  1. Prelude: GR for the Common Man Intro Cosmology Short Course Lecture 1 Paul Stankus, ORNL

  2. Same path through space-time Same subjective elapsed time What is Calvin & Hobbes’ primary misconception? Subjective time -- “proper time” -- is a fundamental physical observable, independent of coordinates

  3. Newtonian: Most general: (dt, dx) x t gmn Metric Tensor B. RiemannGerman Formalized non-Euclidean geometry (1854)

  4. Assuming Newton found parking…. t for all paths Car You Me x Bank News Stand Galilean/ Newtonian

  5. t Down-going muon decays Down-going photon Lab Stopped muon decays 2.2 msec Down Up Basement x Basement detector Cosmic ray lab Down-going muons v < c Lifetime of down-going muons > 2.2msec We observe:

  6. Rest frame of lab-stopped muon Rest frame of down-going muon t t’ Curve of constant Dt=2.2 msec x x’ 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)

  7. The Twin “Paradox” made easy t Castor inbound DT Pollux at rest Castor outbound It’s just that simple! x DX

  8. New, generalized laws of motion B t 1. In getting from A to B, all free-falling objects will follow the path of maximal proper time (“geodesic”). A x 2. Photons follow “null” paths of zero proper time.

  9. Recovering Newton’s First Law t t B B T T A A x x e X X An object at rest tending to remain at rest

  10. Gravitational Red Shift t Photon period DtUpstairs DtUpstairs Photon period DtDownstairs DtDownstairs Up Down x Detector Upstairs Source Downstairs Non-inertial frame -- curved space!

  11. Motion in curved 1+1D space t B T T A x h Conservative (Newtonian) Potential!

  12. Albert Einstein German General Theory of Relativity (1915) Force law: action at a distance Isaac Newton British Universal Theory of Gravitation (1687) Metric: a local property

  13. 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 Newton’s 1st law • Slightly curved space reproduces Newtonian gravitation

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