Abstract
These lecture notes were prepared for a 25-hour course for advanced undergraduate students participating in Perimeter Institute’s Undergraduate Summer Program. The lectures cover some of what is currently known about the possibility of superluminal travel and time travel within the context of established science, that is, general relativity and quantum field theory. Previous knowledge of general relativity at the level of a standard undergraduate-level introductory course is recommended, but all the relevant material is included for completion and reference. No previous knowledge of quantum field theory, or anything else beyond the standard undergraduate curriculum, is required. Advanced topics in relativity, such as causal structures, the Raychaudhuri equation, and the energy conditions are presented in detail. Once the required background is covered, concepts related to faster-than-light travel and time travel are discussed. After introducing tachyons in special relativity as a warm-up, exotic spacetime geometries in general relativity such as warp drives and wormholes are discussed and analyzed, including their limitations. Time travel paradoxes are also discussed in detail, including some of their proposed resolutions.
Highlights
3.2.2 Geodesic Deviation3.2.3 The Raychaudhuri Equation3.2.4 The Strong Energy Condition3.2.5 The Null, Weak and Dominant Energy Conditions3.2.6 Energy Density and Pressure3.2.7 The Averaged Energy Conditions3.3 Violations of the Energy Conditions3.3.1 Basic Facts about Scalar Fields3.3.2 The Scalar Field and the Energy Conditions3.3.3 The Jordan and Einstein Frames
The requirement for the curve to be inextendible is necessary since we could always have short extendible curves which pass through p and do not intersect S, but would intersect it if we extended them
As we will see later, the strategy taken with the exotic metrics – which allow faster-thanlight travel and/or time travel – is the one we described in the beginning; one first writes down the desired metric, and calculates the form that Tμν must take in order to solve the Einstein equation
Summary
Relativistic time dilation means that, while for an observer on Earth it would seem that the spaceship takes at least 4.4 years to complete the trip to Alpha Centauri, an observer on the spaceship will measure an arbitrarily short proper time on their own clock, which will get shorter the closer they get to the speed of light3 Both scientists and science fiction authors have even imagined journeys lasting hundreds or even thousands of years, while the passengers are in suspended animation in order for them to survive the long journey. In a typical science-fiction scenario, the captain of a spaceship near Earth might instantaneously receives news of an alien attack on a human colony on Proxima Centauri b and, after a quick 4.4-light-year journey using the ship’s warp drive, will arrive at the exoplanet just in time to stop the aliens5 Such scenarios require faster-than-light communication and travel, both of which are considered by most to be disallowed by the known laws of physics.
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