Notes on Relativity and Cosmology - Physics Department, UCSB
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Notes on Relativity and Cosmology - Physics Department, UCSB

Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB

web.physics.ucsb.edu
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27.09.2014 Views

108 CHAPTER 4. MINKOWSKIAN GEOMETRY Because boost parameters are part of the native Minkowskian geometry of spacetime, they allow us to see the rule for combining boosts in a simple form. In particular, they allow us to avoid the confusion created by first splitting things into space and time and introducing the notion of “velocity.” 4.4 2+1 and higher dimensional effects: A return to Stellar Aberration So, we are beginning to understand how this relativity stuff works, and how it can be self-consistent. In the last section we even saw that relativity looks pretty when viewed it is viewed in the right way! However, we are still missing something.... Although we now ‘understand’ the fact that the speed of light is the same in all inertial reference frames (and thus the Michelson-Morely experiment), recall that it was not just the Michelson-Morely experiment that compelled us to abandon the ether and to move to this new point of view. Another very important set of experiments involved stellar aberration (the tilting telescopes) – a subject to which we need to return. One might think that assuming the speed of light to be constant in all reference frames would remove all effects of relative motion on light, in which case the stellar aberration experiments would contradict relativity. However, we will now see that this is not so. 4.4.1 Stellar Aberration in Relativity Recall the basic setup of the aberration experiments. Starlight hits the earth from the side, but the earth is “moving forward” so this somehow means that astronomers can’t point their telescopes straight toward the star if they actually want to see it. This is shown in the diagram below. Light Ray hits side instead of reaching bottom Telescope moves through ether Must tilt telescope to see star To reanalyze the situation using our new understanding of relativity we will have to deal the fact that the star light comes in from the side while the earth

4.4. 2+1 DIMENSIONS: ABERRATION 109 travels forward (relative to the star). Thus, we will need to use a spacetime diagram having three dimensions – two space, and one time. One often calls such diagrams “2+1 dimensional.” These are harder to draw than the 1+1 dimensional diagrams that we have been using so far, but are really not so much different. After all, we have already talked a little bit about the fact that, under a boost, things behave reasonably simply in the direction perpendicular to the action of the boost: neither simultaneity nor lengths are affected in that direction. We’ll try to draw 2+1 dimensional spacetime diagrams using our standard conventions: all light rays move at 45 degrees to the vertical. Thus, a light cone looks like this: We can also draw an observer and their plane of simultaneity.

4.4. 2+1 DIMENSIONS: ABERRATION 109<br />

travels forward (relative to the star). Thus, we will need to use a spacetime<br />

diagram having three dimensi<strong>on</strong>s – two space, <strong>and</strong> <strong>on</strong>e time. One often calls<br />

such diagrams “2+1 dimensi<strong>on</strong>al.” These are harder to draw than the 1+1<br />

dimensi<strong>on</strong>al diagrams that we have been using so far, but are really not so<br />

much different. After all, we have already talked a little bit about the fact that,<br />

under a boost, things behave reas<strong>on</strong>ably simply in the directi<strong>on</strong> perpendicular<br />

to the acti<strong>on</strong> of the boost: neither simultaneity nor lengths are affected in that<br />

directi<strong>on</strong>.<br />

We’ll try to draw 2+1 dimensi<strong>on</strong>al spacetime diagrams using our st<strong>and</strong>ard c<strong>on</strong>venti<strong>on</strong>s:<br />

all light rays move at 45 degrees to the vertical. Thus, a light c<strong>on</strong>e<br />

looks like this:<br />

We can also draw an observer <strong>and</strong> their plane of simultaneity.

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