Attention conservation notice:<\/b> This post requires a little general relativity to understand the early bits, and quite a bit of general relativity later on. Later in the post I’ve also used some of the munged LaTeX beloved of theorists collaborating by email; hopefully this won’t obscure the main point.<\/p>\n
One of the most exciting developments in physics over the past twenty years was the discovery that the cosmological constant (\u00e2\u20ac\u0153Einstein\u00e2\u20ac\u2122s greatest mistake\u00e2\u20ac\u009d) was not, in fact, a mistake, but appears to be very real.<\/p>\n
In standard general relativity \u00e2\u20ac\u201c the type where the cosmological constant is zero \u00e2\u20ac\u201c it can be shown that test particles follow geodesics of spacetime, i.e., locally, they simply fall freely, with relative acceleration between different particles due to spacetime curvature.<\/p>\n
(Note that this geodesic behaviour is sometimes presented as though it\u00e2\u20ac\u2122s a fundamental assumption of Einstein\u00e2\u20ac\u2122s theory. Actually, it\u00e2\u20ac\u2122s not \u00e2\u20ac\u201c it follows as a consequence of the Einstein field equations, as we\u00e2\u20ac\u2122ll see below.)<\/p>\n
I got to wondering recently what paths test particles take when the cosmological constant is non-zero. Do they still follow geodesics, or is their behaviour modified?<\/p>\n
The answer, which I\u00e2\u20ac\u2122m sure is well-known to relativists, is that they still follow geodesics. So even when the cosmological constant is non-zero, particles still fall freely. What does change is the geometry, since the modification of the field equations means that the same distribution of matter will give rise to a different geometry, and thus to different geodesics.<\/p>\n
Here\u00e2\u20ac\u2122s an outline of the argument deriving geodesic paths, which is a simple variation on the argument given in Dirac\u00e2\u20ac\u2122s text on general relativity for the case where the cosmological constant \\Lambda = 0.<\/p>\n
Start with the field equations, G+ \\Lambda g = 8 \\pi T. Suppose we evaluate the divergence of both sides. In index notation we obtain:<\/p>\n
G^{\\mu \\nu}_{; \\nu} + \\Lambda g^{\\mu \\nu}_{;\\nu} = 8 \\pi T^{\\mu \\nu}_{;\\nu}<\/p>\n
The first term on the left vanishes because of the Bianchi identies, while the second term vanishes because of the metric compatability condition imposed on the connection. This tells us that the divergence of the stress-energy tensor vanishes:<\/p>\n
T^{\\mu \\nu}_{;\\nu} = 0<\/p>\n
Now, suppose the stress-energy tensor has the form T^{\\mu \\nu} = \\rho v^{\\mu} v^{\\nu}, where \\rho is matter density, and v^\\mu is the four-velocity. Using the product rule to evaluate the divergence, and conservation of mass-energy (\\rho v^{\\mu})_{;\\mu} = 0) we end up with the equation v^{\\nu} v ^{\\mu}_{;\\,nu} = 0, which is the geodesic equation.<\/p>\n","protected":false},"excerpt":{"rendered":"
Attention conservation notice: This post requires a little general relativity to understand the early bits, and quite a bit of general relativity later on. Later in the post I’ve also used some of the munged LaTeX beloved of theorists collaborating by email; hopefully this won’t obscure the main point. One of the most exciting developments… Continue reading Geodesics and the cosmological constant<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-194","post","type-post","status-publish","format-standard","hentry","category-3","entry"],"_links":{"self":[{"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/posts\/194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/comments?post=194"}],"version-history":[{"count":0,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/posts\/194\/revisions"}],"wp:attachment":[{"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/media?parent=194"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/categories?post=194"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/tags?post=194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}