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

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216 CHAPTER 8. GENERAL RELATIVITY AND CURVED SPACETIME When an object is perfectly round (spherical), the high symmetry of the situation simplifies the mathematics. The point is that if the object is round, and if the object completely determines the gravitational field, then the gravitational field must be round as well. So, the first simplification we will perform is to assume that our gravitational field (i.e., our spacetime) is spherically symmetric. The second simplification we will impose is to assume that there is no matter (just empty spacetime), at least in the region of spacetime that we are studying. In particular, the energy, momentum, etc., of matter are equal to zero in this region. As a result, we will be describing the gravitational field of an object (the earth, a star, etc.) only in the region outside of the object. This would describe the gravitational field well above the earth’s surface, but not down in the interior. For this case, the Einstein equations were solved by a young German mathematician named Schwarzschild. There is an interesting story here, as Schwarzschild solved these equations during his spare time while he was in the trenches fighting (on the German side) in World War I. I believe the story is that Schwarzschild got his calculations published but, by the time this happened, he had been killed in the war. Because of the spherical symmetry, it was simplest for Schwarzschild to use what are called spherical coordinates (r, θ, φ) as opposed to Cartesian Coordinates (x, y, z). Here, r tells us how far out we are, and θ, φ are latitude and longitude coordinates on the sphere at any value of r. Object r=10 r = 20 Schwarzschild found that, for any spherically symmetric spacetime and outside of the matter, the metric takes the form: ( ds 2 = − 1 − R ) s dt 2 + dr2 r 1 − Rs r + r 2 (dθ 2 + sin 2 θdφ 2 ). (8.17) Here, the parameter R s depends on the total mass of the matter inside. In particular, it turns out that R s = 2MG/c 2 . The last part of the metric, r 2 (dθ 2 +sin 2 θdφ 2 ), is just the metric on a standard sphere of radius r. This part follows just from the spherical symmetry itself. Recall that θ is a latitude coordinate and φ is a longitude coordinate. The factor

8.5. EXPERIMENTAL VERIFICATION OF GR 217 of sin 2 θ encodes the fact that circles at constant θ (i.e., with dθ = 0) are smaller near the poles (θ = 0, π) than at the equator (θ = π/2). North Pole p θ φ 8.5 The experimental verification of General Relativity Now that the Schwarzschild metric (8.17) is in hand, we know what is the spacetime geometry around any round object. Now, what can we do with it? Well, in principle, one can do just about anything. The metric encodes all of the information about the geometry, and thus all of the information about geodesics. Recall that any freely falling worldline (like, say, that of an orbiting planet) is a geodesic. So, one thing that can be done is to compute the orbits of the planets. Another would be to compute various gravitational time dilation effects. Having arrived at the Schwarzschild solution, we are finally at the point where Einstein’s ideas have a lot of power. They now predict the curvature around any massive object (the sun, the earth, the moon, etc.). So, Einstein started looking for predictions that could be directly tested by experiment to check that he was actually right. This makes an interesting contrast with special relativity, in which quite a bit of experimental data was already available before Einstein constructed the theory. In the case of GR, Einstein was guided for a long time by a lot of intuition (i.e., guesswork) and, for the most part, the experiments would only be done later, after he had constructed the theory. Recall that although we have mentioned a few pieces of experimental evidence already (such as the Pound-Rebke and GPS experiments) these occurred only in 1959 and in the 1990’s! Einstein finished developing General Relativity in 1916 and certainly wanted to find an experiment that could be done soon after. 8.5.1 The planet Mercury We have seen that Einstein’s theory of gravity agrees with Newton’s when the gravitational fields are weak (i.e., far away from any massive object). But, the discrepancy increases as the field gets stronger. So, the best place (around

8.5. EXPERIMENTAL VERIFICATION OF GR 217<br />

of sin 2 θ encodes the fact that circles at c<strong>on</strong>stant θ (i.e., with dθ = 0) are smaller<br />

near the poles (θ = 0, π) than at the equator (θ = π/2).<br />

North Pole<br />

p<br />

θ<br />

φ<br />

8.5 The experimental verificati<strong>on</strong> of General <strong>Relativity</strong><br />

Now that the Schwarzschild metric (8.17) is in h<strong>and</strong>, we know what is the<br />

spacetime geometry around any round object. Now, what can we do with it?<br />

Well, in principle, <strong>on</strong>e can do just about anything. The metric encodes all<br />

of the informati<strong>on</strong> about the geometry, <strong>and</strong> thus all of the informati<strong>on</strong> about<br />

geodesics. Recall that any freely falling worldline (like, say, that of an orbiting<br />

planet) is a geodesic. So, <strong>on</strong>e thing that can be d<strong>on</strong>e is to compute the orbits of<br />

the planets. Another would be to compute various gravitati<strong>on</strong>al time dilati<strong>on</strong><br />

effects.<br />

Having arrived at the Schwarzschild soluti<strong>on</strong>, we are finally at the point where<br />

Einstein’s ideas have a lot of power. They now predict the curvature around<br />

any massive object (the sun, the earth, the mo<strong>on</strong>, etc.). So, Einstein started<br />

looking for predicti<strong>on</strong>s that could be directly tested by experiment to check that<br />

he was actually right. This makes an interesting c<strong>on</strong>trast with special relativity,<br />

in which quite a bit of experimental data was already available before Einstein<br />

c<strong>on</strong>structed the theory. In the case of GR, Einstein was guided for a l<strong>on</strong>g time by<br />

a lot of intuiti<strong>on</strong> (i.e., guesswork) <strong>and</strong>, for the most part, the experiments would<br />

<strong>on</strong>ly be d<strong>on</strong>e later, after he had c<strong>on</strong>structed the theory. Recall that although<br />

we have menti<strong>on</strong>ed a few pieces of experimental evidence already (such as the<br />

Pound-Rebke <strong>and</strong> GPS experiments) these occurred <strong>on</strong>ly in 1959 <strong>and</strong> in the<br />

1990’s! Einstein finished developing General <strong>Relativity</strong> in 1916 <strong>and</strong> certainly<br />

wanted to find an experiment that could be d<strong>on</strong>e so<strong>on</strong> after.<br />

8.5.1 The planet Mercury<br />

We have seen that Einstein’s theory of gravity agrees with Newt<strong>on</strong>’s when the<br />

gravitati<strong>on</strong>al fields are weak (i.e., far away from any massive object). But,<br />

the discrepancy increases as the field gets str<strong>on</strong>ger. So, the best place (around

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