Homeomorphism group
In relativistic physics, Supplee's paradox (also called the submarine paradox) arises when considering the buoyant force exerted on a relativistic bullet immersed in a fluid subject to an ambient gravitational field. If a bullet with neutral buoyancy passes through a fluid at a relativistic speed, will its increased mass in the fluid's frame of reference cause it to sink, or will the fluid's increased mass in the bullet's frame of reference cause it to rise? The paradox was apparently first discussed by James M. Supplee.
A bit about buoyancy
To simplify the analysis, it is customary to neglect drag and viscosity, and even to assume that the fluid has constant density.
Consider a small object immersed in a container of fluid subject to a uniform gravitational field. Then the object will be subject to a net downward gravitational force. Compare this with the net downward gravitational force on an equal volume of the fluid. If the object is less dense than the fluid, the difference between these two vectors is an upward pointing vector, the buoyant force, and the object will rise. If things are the other way around, it will sink. If the object and the fluid have equal density, we say that the object has neutral buoyancy and it will neither rise nor sink.
Statement of the paradox
It is simplest to assume that at rest, the bullet has neutral buoyancy.
In the rest frame of the fluid, if the bullet moves at speed , then according to the kinematic laws of special relativity, its density (as measured in the frame of the fluid) increases by the square of the Lorentz Factor . Therefore, a rapidly moving bullet should sink.
In the rest frame of the bullet, the density of fluid, as measured in the frame of the bullet, increases by the same factor, so the moving bullet should rise.
Resolution of the paradox
The resolution comes down to observing that the gravitational force lies outside the domain of kinematics; when it is treated properly, the paradox disappears.
Supplee himself concluded that the paradox can be resolved by noting that in the frame of the bullet, the shape of the container of fluid is altered (viz. the sea floor is curved upwards). Given certain assumptions about how to treat the gravitational force, he argued that the bullet sinks with acceleration , where g is the acceleration due to gravity (assumed to be uniform over the scale of the thought experiment) and is the factor mentioned above.
The paradox has also been studied by George Matsas, who used mathematical methods from general relativity to remove Supplee's assumptions. In particular, he modeled the situation using a Rindler chart. Matsas concluded that the paradox can be resolved by noting that in the frame of the fluid, the shape of the bullet is altered, and derived the same result which had been obtained by Supplee. Matsas has applied a similar analysis to shed light on certain questions involving the thermodynamics of black holes.
See also
References
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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External links
- Light Speed Submarine - article about the paradox in Physical Review Focus