Rindler's Just Flat Out Pretty Derivation of the Special Relativistic Time Dilation Factor Gamma

Rindler demonstrates[2] the best way ever to derive the relativistic Lorentz contraction/time dilation factor, gamma, (of twin paradox fame), from four velocity! I've been working through Rindler's paper on hyperbolic motion as a result of constant acceleration[1] lately.

Starting from the constancy of the speed of light, Einstein's theory of special relativity posited, and Minkowski refined the idea that the universe is actually four dimensional with space and time sitting on an equal footing. Starting from here, we can write an expression for the distance squared along an infinitesimal line element in four dimensions, (think Pythagorean theorem)...

Brian Greene popularized the idea of four velocity[3] in one of his books and although it isn't mentioned as much as some of the other aspects of special rel, it's a simple idea. Everything is moving at the speed of light. Something might have more of it's velocity pointed into either the space dimensions, or the time dimension, but at the end of the day, its four dimensional speed is equal to the speed of light, c. So, let's write down the infinitesimal distance along a four dimensional line divided by an infinitesimal amount of time as a velocity

which can be rearranged to give

From here on out, it's all Rindler. Factor the increment of time out of the above expression

At this point, the velocity, (distance over time), squared is sitting in the second term, so we can write

Leaving us with

Which can be rearranged once again to give

Which is the formula for time dilation, (the time in the moving frame is equal to gamma times the time in the rest frame), so

and we're done!

References:
1. http://dx.doi.org/10.1103%2FPhysRev.119.2082
Rindler W. (1960). Hyperbolic Motion in Curved Space Time, Physical Review, 119 (6) 2082-2089. DOI: 10.1103/PhysRev.119.2082

2. Special Relativity by Wolfgang Rindler
http://books.google.com/books?id=8kqBAAAAIAAJ

3. On four velocity

http://copaseticflow.blogspot.com/2013/02/spheres-special-relativity-and-rotations.html

Comments

The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1. While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero. This is typically explained in the following way. While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare. That doesn't stop us from looking for them though! Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2]. Cabrera was utilizing a loop type of magnetic monopole detector. Its operation is in concept very sim

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems , there seem to be fewer explanations of the procedure for deriving the conversion, so here goes! What do we actually want? To convert the Cartesian nabla to the nabla for another coordinate system, say… cylindrical coordinates. What we’ll need: 1. The Cartesian Nabla: 2. A set of equations relating the Cartesian coordinates to cylindrical coordinates: 3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system: How to do it: Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables. The chain

More Cowbell! Record Production using Google Forms and Charts

First, the what : This article shows how to embed a new Google Form into any web page. To demonstrate ths, a chart and form that allow blog readers to control the recording levels of each instrument in Blue Oyster Cult's "(Don't Fear) The Reaper" is used. HTML code from the Google version of the form included on this page is shown and the parts that need to be modified are highlighted. Next, the why : Google recently released an e-mail form feature that allows users of Google Documents to create an e-mail a form that automatically places each user's input into an associated spreadsheet. As it turns out, with a little bit of work, the forms that are created by Google Docs can be embedded into any web page. Now, The Goods: Click on the instrument you want turned up, click the submit button and then refresh the page. Through the magic of Google Forms as soon as you click on submit and refresh this web page, the data chart will update immediately. Turn up the:

Copasetic Flow

Search This Blog