lec24 - Scott Hughes 12 May 2005 Massachusetts Institute of...

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Unformatted text preview: Scott Hughes 12 May 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 24: A (very) brief introduction to general relativity. 24.1 Gravity? The Coulomb interaction between two point charges looks essentially identical to the grav- itational interaction between two masses. Does this mean that everything we have done so far can also be used to describe gravitational interactions (perhaps with some slight modifi- cations)? The answer to this turns out to be almost — but not quite. To understand what this means in detail, we need to do a little bit of background work. The key point to be made here is that Maxwell’s equations and electricity and magnetism are relativistically correct: all of E&M “knows about” special relativity, in the sense that the equations make sense in any frame of reference. (When we transform reference frames, we modify the ~ E and ~ B fields according to those relativistic transformation laws we all learned to love ages ago. However, the new fields still satisfy Maxwell equations.) This is NOT the case for gravity! Significant modifications need to be made to make gravity relativistic. What we end up with is Einstein’s theory of general relativity . In this lecture we will look at some basic properties of general relativity. To begin, we need to introduce a bit of notation. 24.2 4-vectors Since space and time are unified in relativity, it doesn’t make much sense to treat them separately. Accordingly, many concepts are described using 4-vectors , quantities that are essentially just like the vectors we’ve known and loved since kindergarten, but with an extra “timelike” component. For example, the position 4-vector is written x μ = ( ct,x,y,z ) . The way to read this is that μ is an index , ranging from 0 to 3: x = x μ =0 = ct ; x 1 = x μ =1 = x ; x 2 = x μ =2 = y ; x 3 = x μ =3 = z . Whenever you see an x with a number superscript, it means the index — it does NOT mean take it to a higher power! This system can be a little confusing when you first encounter it. That the “timelike component of position” is time makes perfect sense. What do we get when we try to make something like momentum into a 4-vector? It turns out that the “timelike component of momentum” is the energy : p μ = ( E/c,p x ,p y ,p z ) . We’ll introduce a few other 4-vectors before we’re done here. 220 24.2.1 Invariant interval Back on Pset 6, we showed that the quantity Δ s 2 =- c 2 Δ t + Δ x 2 + Δ y 2 + Δ z 2 is invariant : all inertial observers agree on its value. This can be written this in terms of a quantity called the metric : Δ s 2 = 3 X μ =0 3 X ν =0 g μν Δ x μ Δ x ν where the metric tensor g μν can be represented as a 4 × 4 matrix: g μν = - 1 1 1 1 ....
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This note was uploaded on 07/19/2010 for the course 8 8.022 taught by Professor Scotthughes during the Spring '10 term at MIT.

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lec24 - Scott Hughes 12 May 2005 Massachusetts Institute of...

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