General Relativity

Table of Contents

Notation

  • IFs is inertial frames

Newtonian spacetime

Notation

Stuff

Reformulation of Poisson laws:

general_relativity_8b943fb8afc8b3fdca735fbd34d0c38c05e84d0f.png

in terms of the Newtonian spacetime curvature as

general_relativity_6748429e0530f04af9926ff2ac5f8514b9674167.png

where general_relativity_dd108deec4cb36c3123789a922337b3b7976cb1c.png is the 00-component of the Ricci tensor.

Then Newtonian spacetime is tuple general_relativity_2853e32efd47b2fa3ac1e9266d81c388168d7722.png where

Absolute space

Absolute space at time general_relativity_c6bef10eba2db5e87e10b36de4a645e2b0a4a5d7.png is defined

general_relativity_ba6c64b6dcf123bae58637cdd2b5a082cbdca455.png

and so (Newtonian) spacetime can be written as the disjoint union of the absolute spaces:

general_relativity_686dce8825ff7088f6f7503f9e287f82a49182e5.png

One way to visualize absolute time is as follows:

newtonian_spacetime_absolute_space_at_times.png

If absolute time didn't "flow uniformly", then we would allow something like the following, where we can have time standing completely still:

newtonian_spacetime_time_stands_still.png

Motivation

Newton's laws

  • Motion: general_relativity_7095bfe1a32d5701862bc42ce72bbaef93980ecf.png and general_relativity_04fc17220ea78f779a7eea89c8aae5e9b5aa399f.png where
    • general_relativity_8c741f70c592c9dca214b174db75cb1143645a8a.png is the inertial mass
  • Gravity: general_relativity_e2b0bd857bff1bd95a337f6f56d5e3da4861f780.png where general_relativity_d649fbe4073fb36dd33e6c91540761f2fa63039b.png, which satisfies the (gravitational) Poisson equation

    general_relativity_e84630f6571578b696f815864c910474b88486d2.png

    where

    • general_relativity_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is Newton's gravitational constant
    • general_relativity_ccb973f0cf1a9e40c184ec65f392888f18a6471e.png is density
    • general_relativity_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png is the mass of the "source"
  • Experiment: inertial mass is equivalent to gravitational mass, i.e.

    general_relativity_9f47cee8432463cafe3431f2042626090fe943c3.png

    which is called the weak equivalence principle, thus

    general_relativity_0d48a17977cdd8091d5fa65ab0d76482e3000dad.png

    Therefore we get the fact that locally, gravity is uniform acceleration

    • Non-uniform general_relativity_cfc2328de87e5f3d2b2f44278e9cd777a14272aa.png gives rise to tidal forces. Follows from the deviation equation

      general_relativity_ecc1065518925034495a015d93d97add39741ea4.png

Special relativity

  • Spacetime general_relativity_765537b15d3a74b2e9d4f671198b8c4238181046.png where points are events, with inertial frames

    general_relativity_ad94361d516da50594c2161e34241c00ed020250.png

    and we use the Minkowski metric

    general_relativity_f978c7c37f956dc370d98ba36a65bfd82dce08ed.png

    (a quadratic form), which is invariant in all IFs. Notice general_relativity_3631c9e7c6b3c11be6cdb81417b3d1afe8220ecd.png

  • Motion is described by wordlines general_relativity_9cd7ccd0f575b397e3041ee2372e1dbc4719deee.png
    • Free motion follow straight lines
  • IFs related by Poincare group (which leaves general_relativity_cea2d867bfa7965f21798e314045c1bf799c365d.png invariant:
    • Rotations
    • Translations
    • Lorentz boosts (rotation between space and time)
  • Maxwell's eqns have PoincarĂ© symmetry, but NOT Galilean symmetry (which is what Newton's gravity follows)

gr-motivation.png

General relativity

  1. Laws of physics take same form in any coordinate system, i.e. invariant under general coord changes
    • Will obtain by formuating laws with tensor fields
  2. There exists lcoal IF with no gravity, where laws are as in special relativity (called the /strong equivalence principle)
    • Spacetime is (differentiable) manifold with Lorentzian metric

Postulates

Spacetime is a four-dimensional Lorentzian manifold general_relativity_0daa3711b0e8e42bbf80bb9b42536c0c6c068d4c.png.

More specifically, (relativistic) spacetime is defined by the tuple general_relativity_f71d89d14b1fae50334d04547cf1d5c14a35b55f.png where

Let general_relativity_a44bee37bb9b498a40b663ab30e6c7ce09f4ea71.png be a Lorentzian manifold.

Then a time-orientation is given by a smooth vector field general_relativity_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png that

  1. do not vanish anywhere, i.e. general_relativity_415c651c80dfc0eeea8a61fd259c3066ef5d2ee8.png for all general_relativity_d0bdf5211cf0345283a43cb4796f0a43f63fa7bf.png
  2. general_relativity_d4aa2a219f0bb38cbaf769aab3352da38d9fd100.png

Here general_relativity_e5019fbc8244ea31595481a7cfa6087e28e9c3d0.png denotes an oriented atlas.

  • Comparing with Newtonian spacetime we have gone from a "time" general_relativity_7858a550c696df9ba92761068d9d9803e3bb69e4.png to a metric general_relativity_c4f480233088a134e88f2426541b2f00ca318b55.png and time-orientation general_relativity_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png.
  • Reason for introducing a metric general_relativity_c4f480233088a134e88f2426541b2f00ca318b55.png now is that we want some way of enforcing the fact that a particle cannot move faster than the speed of light
  • If we instead considered the metric general_relativity_461fe29c8986a6726bb24e9cbcdffaae4a785827.png then we observe that

    general_relativity_ded5c43af4799dfdcc63c8b7d88e06b0b57e608f.png

    so if general_relativity_d4aa2a219f0bb38cbaf769aab3352da38d9fd100.png, we are basically saying that the flow along general_relativity_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png has a "speed" which less than the speed of light (as we want).

    • Observe then that (with our convention), that general_relativity_666e8dea6dff2df87eb63a64877ba199c8da11b2.png is "illegal"
  • The above produces the well-known light-cone, where if we make a choice of time-orientation to enforce the fact that a particle cannot move into the past

light-cone.png

Connections and curvature

Notation

  • general_relativity_93868c7080638936e809863f0c473a2dfd0a4353.png denotes a basis of a vector fields on general_relativity_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png
  • general_relativity_b636b54c70fd36efee191cd32609ddc5b375138b.png denotes the dual basis of covector fields on $M4
  • In GR, the term connection will often be used to refer to the tensor general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png; this is because of how it relates to the connection 1-form
  • Covariant derivative as a (1, 1)-tensor field

    general_relativity_d5a919f6c4d0bf3efab5c4a6d2c3a9472cf70ebf.png

    with components

    general_relativity_c5fe61820d7c784ddd77eb3cedc4294197779dbd.png

  • Components of connection

    general_relativity_7409010e892f725f52c2aff50c971ba781346189.png

  • Torsion is denoted general_relativity_1b50c4edc464f08e6f7a654b36da460854e7d3b8.png or in components general_relativity_21d6cbaee47314e5c70d4e5016384da4af72c8bb.png and is given by

    general_relativity_ffe58b5cca41e344bd89d96a6cd0a81bbc10e28e.png

    and in components

    general_relativity_2eabf4aa1870ed17510bdfe3227ff657466bce28.png

Covariant derivative

Let general_relativity_93868c7080638936e809863f0c473a2dfd0a4353.png be a basis of vector fields on general_relativity_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png. The components of the connection general_relativity_8b2fa9eb6afba6571a5696ad238cf6ef4a33c001.png in a basis is defined by

general_relativity_7ba472fc3782758573c75c909669f88c26f17029.png

Letting

general_relativity_94f9650c596c1433e179492aec09c90a48e490df.png

the covariant derivative can be written

general_relativity_ae9731295be3d284aa97bf24a55feb636e9ebc50.png

and so

general_relativity_52159af013e9c06e13fcdad94e7ce5ddaef82b87.png

where general_relativity_b636b54c70fd36efee191cd32609ddc5b375138b.png is the dual basis of general_relativity_93868c7080638936e809863f0c473a2dfd0a4353.png. In a particular coordinate basis general_relativity_06d72864d170d54c14176e581ff857704f17b3cd.png

general_relativity_db9d988bb4ebfbd1a1f402a28b516fd5a3701e4d.png

Under a basis change general_relativity_34589d5d4f4a98457dbea6aec610738c937bf5bd.png and general_relativity_2b4052a311d187b5a7335c529b7f984cafd43413.png, we have

general_relativity_55e9b2309eacddda13f69caca5e1f5fed6f17250.png

whre the basis change coefficients general_relativity_93c0d6f83781c8f5b8e8620b4196f8547a073689.png.

In particular, under a change of coordinate basis, this becomes

general_relativity_2f15d346883ca21a8f600b769a391e5cb51d5796.png

We use general_relativity_cec7bc07628b47897d64e92f9982ef83c7a0a8f5.png to denote fixed indices, and general_relativity_bb9caa7586bcb90194f3e6b9e0824c7247ada8f2.png for contractible ones.

general_relativity_0373b7c352f9e759a8ce00eba973bf1ab85bfb2e.png

If we then let general_relativity_d9218e4a3e729add0e60e45d77e2cbc6304e7466.png and general_relativity_146807c348d75dd207c184b9cd5c41bcabde3ce3.png we obtain our proof (up to commutation between general_relativity_8f83b870f7d94292644407b5c7e32b9fab164b3e.png functions, so we're fine).

Moreover, in a specific chart general_relativity_fc520ded29354a737ae8a918fdc413c524dde37e.png and changed basis general_relativity_77aa8328c68bfe998583846a725a5056ced6f9f9.png we simply have

general_relativity_7535fa9651307089c7884974702da81ab3b80633.png

Substituting back into the above, we obtain our result.

Let general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png be a connection on a smooth manifold general_relativity_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png.

A vector field general_relativity_a4242ffc7d6e6fc3298db18c548b41b6309e0a68.png is said to be parallely transported along a curve with tangent vector field general_relativity_f40ad5f32532ae52dd17a4315b7711042277a778.png if

general_relativity_87bdaf3bf9b7e451d605ad9a3a9718b8274880de.png

The parallel transport condition in components reads

general_relativity_91cc3da7cfba29033fb4f83ce788c504175b50e6.png

Along an integral curve general_relativity_f40ad5f32532ae52dd17a4315b7711042277a778.png we have general_relativity_b1e5b465a928693ce29468649fd96e8a46a3890c.png, and so we can write

general_relativity_23c9512e42f72f95a8d2d061bc844ea64b2b4006.png

A geodesic is an integral curve of a vector field general_relativity_f40ad5f32532ae52dd17a4315b7711042277a778.png that satisfies

general_relativity_3e5c6ae507a717b6e04c8fb083d266a2a207e0cd.png

In other words, a geodesic vector field is parallely transported wrt. itself.

In a coordinate basis, the integral curves satisfy general_relativity_d487dc5ea021abc9ae07e77403fa06b3ae8e1820.png, and so the geodesic eqation is

general_relativity_d724eabcd8f93fc7ea3077b1e9927b4126c698cf.png

Let general_relativity_0cd642656f0c33750baa9dd02c90fdcb10acff6d.png open for which general_relativity_46b172c9fc826b59528a62e9f98fcb46a8287bd7.png is a bijection.

Normal coordinates at general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png, of a point general_relativity_e350d0a7f0d5744f4f015dd5d8270bc46af374e2.png, are given by general_relativity_46144df8d2b23dfbf5ce581b849145993912b4b7.png where general_relativity_a925b1d65ef0f19ce6dab184da92ee0c5d0a3c24.png are the components of

general_relativity_ee7c9110fde43a14df1703fc07d82f80cf6227cc.png

in a basis general_relativity_9f853586cdb2fb4a6b602bd4dc0da59bfb31907f.png.

Consider the geodesic through general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png and general_relativity_b69731d35b07e541a22e4ba109da945693880aad.png.

In normal coords. the geodesic takes the form

general_relativity_cfd3f8ebc1d28ed0e43f3b52e611829ccb80dafb.png

Inserting this into the geodesic equation and evaluating at general_relativity_1521f78eb9a3ed17a5379fd7b8ebcfa5f1733e50.png, we deduce that

general_relativity_50a739bf01e4b0f858de489fd313b869c62b5625.png

Hence

general_relativity_69d9b5e36eb24cf5b3eed4d4a9b989292a4c645f.png

Since this is true for all general_relativity_c834963a8b62d5c5f439573b918bb377633a1ce3.png in an open set general_relativity_3edc3a696095d8013a488370920e789f5134e468.png, we have our proof.

Torsion and curvature

Torsion

Let general_relativity_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png be a smooth manifold with a connection general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png.

The torsion general_relativity_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png is a general_relativity_2436bc257ed61f5bf0d6a08a00cebea2d799f1c9.png tensor defined by

general_relativity_ffe58b5cca41e344bd89d96a6cd0a81bbc10e28e.png

where general_relativity_80ddfe12ba242336d843ae285476c8dfe0c8351e.png are smooth vector fields.

One can easily check that this is indeed a tensor.

Componentwise,

general_relativity_6c8616bec4d98afd83f23f600ee9fa5a90f6a05b.png

Geodesics are determined by the symmetric part of the connection components general_relativity_eefc620b6202742ddb3e0be51ca7d03503f81469.png, thus the torsion does not affect the geodesics!

Let general_relativity_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png be a manifold with a covariant derivative general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png.

For any function general_relativity_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png,

general_relativity_8c716341ff9ea12ed6c1f298429880adcdb19723.png

We prove this by working in a basis general_relativity_9f853586cdb2fb4a6b602bd4dc0da59bfb31907f.png.

Since general_relativity_e48deef168535fb894e740d2915f915552fb51fc.png, the covariant derivative of the covector general_relativity_c7eb9493749b626a34346c7b0b5999e619536fff.png is

general_relativity_4eb0d8992e4dd50278f36ffc23044c79ff94ebb3.png

Therefore, antisymmetrising, we get

general_relativity_875a45fa936b4921fc0467d3d1c58b229b659b13.png

Since this is a relation between tensors, it follows that it must hold in any arbitrary basis.

Curvature

The Riemann curvature of a connection general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png is a general_relativity_476508defcfd65e538bf589d49e8dffe4c72f9e4.png defined by

general_relativity_3e1af124e5ca59a64e1104c322afec3b0e0416bc.png

where general_relativity_eacd876977da071531503fdbc3fc20278cc5a51e.png are smooth vector fields.

One can verify that this is indeed a tensor by checking that it's linear in general_relativity_a4242ffc7d6e6fc3298db18c548b41b6309e0a68.png and general_relativity_819394bc33557d54ecc2b0d62dfae8541b0d6a97.png.

The Riemann tensor is the (1, 3) tensor defined by general_relativity_122adaf19ebf2d24c722494306801da04b15c174.png, where the ordering of the arguments is standard notation. The components are then defined

general_relativity_ce178d7510d3ad3d9871365cac27702dd532b0cf.png

The component-wise expression for the Riemann tensor can be seen as follows:

general_relativity_d451f8085965de6069d6a3059dcf98fc81a4942e.png

where we first observe that

general_relativity_fb56878d5d4fad9fef917ff98dac282c0fc4e4e4.png

from which the Riemann tensor general_relativity_5adc86fd3e99c0e70dc0297e300f0fdf73bdecaa.png immediately follows.

Let general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png be a torsionless connection.

For any vector field general_relativity_819394bc33557d54ecc2b0d62dfae8541b0d6a97.png,

general_relativity_11a33910ea37a19f71a35664e9275a65606d51d3.png

This is called the Ricci identity.

First, observe that for a torsionless connection general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png, we have

general_relativity_bca89fcec977f90ac2f8e910de58ee7a24ae0884.png

Now

general_relativity_3b5b4b3572d0287a9645490129c4904fb4658fd8.png

Since this holds for all general_relativity_80ddfe12ba242336d843ae285476c8dfe0c8351e.png, we conclude our proof.

Riemann tensor has an important geometrical interpretation.

It can be shown that general_relativity_d53e871087d7bdea453e988dc90f065e1e262b06.png is the change in general_relativity_819394bc33557d54ecc2b0d62dfae8541b0d6a97.png upon parallel transport around a small quadrilateral whose opposite sides are integral curves of vector fields general_relativity_f40ad5f32532ae52dd17a4315b7711042277a778.png and general_relativity_a4242ffc7d6e6fc3298db18c548b41b6309e0a68.png. Hence, if general_relativity_8afc12ff81de044ba941589e53a38ff8563b6f0f.png parallel transport is locally path-independent.

The torsion and curvature of a connection general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png vanish if and only if for any general_relativity_5c0d2bbc9e56c3c4bdf0edbb52455078685fb8a9.png, there exists a chart general_relativity_fc520ded29354a737ae8a918fdc413c524dde37e.png such that general_relativity_73ba794404dd6494e74951a912853ac15584b546.png and

general_relativity_bda316906961d33a68119d58e9090561ee9d5a34.png

In short,

general_relativity_8b82f9b60bb2bfafe8cd438c5949607b608f0b5a.png

The Ricci tensor is the general_relativity_99fbcf96d404e8ca7311e7e5f090d8430bd689d8.png tensor defined by contraction of the Riemann tensor

general_relativity_726e8fff7be1a2f9246f016c64c5517238097935.png

In components,

general_relativity_4d0201f1f206a6f6aff176490a0f0eeb9bb3f4b8.png

Suppose general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png is a torsionless connection. Then we have

general_relativity_ddfd9df0cc6fb1c1f7082d7b090c365fd3265a7c.png

We will be working in normal coordinates.

Recall that in normal coordinates at general_relativity_5c0d2bbc9e56c3c4bdf0edbb52455078685fb8a9.png, we have general_relativity_1652ebb9d5d47640b7701dba66323c331ad39b0c.png.

Hence, for a torsionless connection, this reduces to general_relativity_9965442dc0a6240f5d21401b4f82c29655c9d59e.png.

Therefore, the Riemann tensor in normal coordinates at general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png is simply

general_relativity_526ca6658ac8071af8633fb32a2747186d237975.png

  1. Due to anti-symmetry in the last two indices,

    general_relativity_d7304903c2066ea1d0f346109ce0a3070c17f416.png

    Substituting in the above identity (since we are in normal coordinates),

    general_relativity_9bbd1f743b363b837a5981efcf5c239ba89d2d43.png

    since general_relativity_88773308e180ddf85abfec2a0bd5577928e4d889.png.

  2. We have

    general_relativity_ae7ea0ac8ddf6b2883e6e6bfe5122a2a1aa83985.png

    So we need the last term to equal general_relativity_25773875a2509134ab0f26c565b85f85bda7955a.png. To see this we write the expressions out

    general_relativity_f1cfa6a549e5516e83037344bdafcab0cc457664.png

    as we wanted. Substituting into the expression above,

    general_relativity_79e2e63ac312ca89fabb5396610537649adbee08.png

    as claimed.

Suppose general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png is a torsionless connection. Then we have

general_relativity_a843895d906763f3a87dfd08e5ed59835ec8c305.png

This is called the Bianchi identity.

general_relativity_6dd2367a5aad5e1dc9b762fdff8453725ceb9a7c.png

which follows directly from

general_relativity_e736ffd73d684267e44c910a3089b4a5a190e033.png

Levi-Civita connection

Let general_relativity_e0c52d45e43977e22a9d861fa1e5f34b791e6226.png be a psuedo-Riemannian manifold. There exist a unique connection general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png with vanishing torsion and satisfying general_relativity_5969c63444130c5e91981b8c26017c3636b7d5e0.png.

This choice of connection is often called the Levi-Civita connection.

general_relativity_1811330fb9b0d4a8787c8cbd11dce56d84e40905.png

Curvature of Levi-Civita connection

Let general_relativity_c5680e5fed5c8fe3c23edf6089761ab4d873db71.png be the Levi-Civita connection. We define the (0, 4)-tensor

general_relativity_180f91eb65758c8e1cea38a8702092da888240ad.png

general_relativity_19e8c13fe939e1cbe505023bd9d63dde88f5a671.png

We work in normal coordinates at general_relativity_5c0d2bbc9e56c3c4bdf0edbb52455078685fb8a9.png, so

general_relativity_48fc69de7a881c66c4dd95eb275e0bd6ccdd1b69.png

general_relativity_82726d191effe60d5f9bb021b61d9fda9e01b6ee.png

and so

general_relativity_564ca37dbf15fe8897ac96cecdbc80d267e6b2ab.png

(at general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png) differentiate wrt. general_relativity_eebec262dbcb5cc268135d5be1d170740f7bbfe9.png and evaluate at general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png:

general_relativity_8b4d8f0ebfc1de34f26abdaaf2db36c1333e0b0b.png

at general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png.

AND MORE.

general_relativity_21b5aaf0373fb651c9e88b202b84a021dc10f722.png

for Levi-Civita connection.

general_relativity_825c902870e29f0112e59133f40a0057dd9017fa.png

using Proposition proposition:4.61-lectures.

The Ricci scalar of general_relativity_21fcbcf5589c5f9f13449cbf9390b17a8aa30b06.png is

general_relativity_0b8fce31ce23b73a4688daca2aef6272c8730796.png

The Einstein tensor

general_relativity_fa38ee1b32d8e2ea74af9d4649bfc58be85b01f3.png

general_relativity_f4deb710f52e0f60bf72ad12e074052b63fbe5c5.png

From the Bianchi identity

general_relativity_be31e9dc6cdff99df53b9796c121b272ac38149f.png

We contract by general_relativity_b0f69f745066a1c03af69e00112342a59dd1f94f.png

general_relativity_a5fd8114779ca6adfb69a88ce26f2729433d77e0.png

Contract with general_relativity_3c76e5defd184a4aa18c5d801b45dbbd0c28f0ed.png, noting that general_relativity_31ec4ed02db21f2d2608fce4552800aa7f448815.png by general_relativity_5969c63444130c5e91981b8c26017c3636b7d5e0.png

general_relativity_791d8b503d4f2ca579c45dc0649246c5bc8d6dc4.png

Riemann tensor of general_relativity_0daa3711b0e8e42bbf80bb9b42536c0c6c068d4c.png vanishes if and only if for every general_relativity_5c0d2bbc9e56c3c4bdf0edbb52455078685fb8a9.png, there exists a chart containing general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png such that

general_relativity_1a649e077297fe4fed9306a13c915329490923e8.png

Recall if torsion vanishes, then Riemann tensor vanishes if and only if there exists charts where general_relativity_3a4528ccc7e20087dfd68c7b4157da96b90a6d06.png. Then

general_relativity_14b0cea0c3c0e2ade941148c3c942d3e1a33b7c7.png

in such charts.

This implies that general_relativity_049b9126e54c34a258c7b80e2e03ab55695a5534.png is constant in chart, if we choose basis at general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png to be orthonormal, which implies

general_relativity_1a649e077297fe4fed9306a13c915329490923e8.png

Theorem thm:Riemann-tensor-vanish-iff-Euclidean-or-Minkowski-metric shows us that the Riemann tensor sort of measure the deviation from Euclidean or Minkowski metric, locally.

Special relativity

Spacetime is just a Lorentizan manifold general_relativity_29e57b9addf9649cd29f1cea7e6134b9d60898bb.png (Minkowski spacetime),

general_relativity_1b2f408604a3b853a9a4d6dabf4f50dc59c8c851.png

where

general_relativity_70f4877f2aac6daa67931a238e120b9e16db3360.png

and general_relativity_2bc372ebe57df35586c41067896f9ee5fded32f1.png are the inertial coordinates, i.e. general_relativity_b6338a2129047c05b739f038c9131a92437c3712.png.

Free motion: are timelike geodesics, ligthrays null geodesics

Physics: described by tensor fields on general_relativity_29e57b9addf9649cd29f1cea7e6134b9d60898bb.png which obey evolution equations.

Examples

Scalar field

Let general_relativity_5dbb4ffc8f3e55c261a67680486057fdaedcf30f.png be a scalar field, then

general_relativity_0464c9f8a4fb7ea4b174a4dc374b527d87195c95.png

which is just the wave-equation:

general_relativity_c10d236ff19675811796cfc7731ab3200b72ff15.png

Energy-momentum tensor

general_relativity_643fafe6a404cad978d5e91bd84590c2b66cf512.png

satisfies

general_relativity_3de03681effe6615197fe4a162099728ad59dd11.png

which is conservation of energy / momentum.

Maxwell's theory of E. M.

  • Electromagnetic field strength general_relativity_e4c92a35afb55a0466f95c27ca0310ec872b5585.png which obeys the Maxwell's equations:

    general_relativity_11a284fb4eb525277eb0132bd155e2e974662812.png

  • The corresponding energy-momentum tensor

    general_relativity_ce17f7943b2ebb851188e49b4c00f6e26f22a2a1.png

Any matter distribution is described by energy-momentum tensor

general_relativity_bc353e93fa81f7f48a98a52cd5f941dbca8e8aec.png

Fluids

  • Described by vector field general_relativity_25509b68ba0d89052ffbe80aaccd9fbe57a21853.png, and typically normalized such that general_relativity_95e493930a8f5f96865fa01ecfdd877df128ca10.png
  • Perfect fluids:

    general_relativity_5309e68fe3c0556b271ee88a10d38b90274cf318.png

    where

    • general_relativity_ccb973f0cf1a9e40c184ec65f392888f18a6471e.png is the energy density
    • general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png is the pressure
  • Then general_relativity_58eb7edc52245b74d6f505e3963b626ca5c6e33a.png is the relativitic eqns. of fluid dynamics

General relativity

Main idea: there exist local freely flowing frames with no gravity.

This is achieved by the following postulates.

Postulates:

  1. Space-time is a 4D Lorentzian manifold general_relativity_0daa3711b0e8e42bbf80bb9b42536c0c6c068d4c.png
  2. Free particles follow timelike / null-geodesics wrt. Levi-Civita of general_relativity_0daa3711b0e8e42bbf80bb9b42536c0c6c068d4c.png
  3. Energy-momentum distribution of matter fields described by general_relativity_61d60344c84a658d5d223f0292c0f54b70596209.png symmetric tensor field general_relativity_ded0bb6434cb4484efe9aad49c61a7865c20d543.png which is conserved

    general_relativity_c46960e675787ad4a9601d91887991fa744c60ab.png

  4. The curvature of general_relativity_e0c52d45e43977e22a9d861fa1e5f34b791e6226.png is related to energy-momentum tensor of matter by the Einstein's equations:

    general_relativity_df48f63c74ef797f4008f3536b8e455a9043d0ec.png

    where general_relativity_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is the Newton gravitational constant.

    • Note that general_relativity_ded0bb6434cb4484efe9aad49c61a7865c20d543.png might also depend on general_relativity_2c1aa1d89a5f7e8a94f2bd2467fdd442542f8bd3.png, so we cannot simply fix general_relativity_ded0bb6434cb4484efe9aad49c61a7865c20d543.png and solve

Laws of physics governed by:

  1. General covariance: laws indep. of basis / coord system
  2. Equivalence principle: in any local inertial frame (normal coordinate system) laws reduce to the laws in Minkowski spacetime (Minkowski space)

Do not fix laws uniquely! But suggests simple rule (called minimal coupling):

  • given any equation in general_relativity_29e57b9addf9649cd29f1cea7e6134b9d60898bb.png (Minkowski space), we replace

    general_relativity_6e1ca259f058c5c07f4240294bcd9e9832208977.png

    to get laws on curved spacetime general_relativity_0daa3711b0e8e42bbf80bb9b42536c0c6c068d4c.png

  • Rules ensure general covariance as they output tensor equations.
  • Local intertial frames are normal coords at general_relativity_5c0d2bbc9e56c3c4bdf0edbb52455078685fb8a9.png, i.e.

    general_relativity_4e3b164d0818e9b22011be5db2df968a175b82da.png

Examples

Wave equation

Applying these rules to wave-equation of S.R. we get

general_relativity_cbc014cd463f69852ad1c58839cce5f2504d8f87.png

Consider the wave equation

general_relativity_a9c12cd9693422812100fb0b33de47b6f5c538a8.png

also reduces to the wave-equation on spacetime general_relativity_d32c913882c470d8774a9e8196fc9b1249f2b46d.png since general_relativity_6ffe4b76846a2e4e332c8d7a230f28760148b98b.png.

In local inertial frame, observe that we have

general_relativity_ddf7f0e0cd127992c12252e3ebce88a9af9e338c.png

where the RHS is the wave-equation in general_relativity_29e57b9addf9649cd29f1cea7e6134b9d60898bb.png.

Postulate 3: we have

general_relativity_35b27499e1386ea51e021693b66a63760ffd3fc9.png

and by wave-equation

general_relativity_c46960e675787ad4a9601d91887991fa744c60ab.png

Maxwell's equations

Applying these rules to Maxwell's equations in spacetime, we get

general_relativity_5ddfa38457d98fe10fb3cc604fae0c57a42d1a97.png

Postulate 3: we have

general_relativity_21a6686037a7cc3cbfc8abf367c9a1e52cad15d1.png

combined with Maxwell's equations we get

general_relativity_c46960e675787ad4a9601d91887991fa744c60ab.png

Fluids

  • Described by vector field general_relativity_25509b68ba0d89052ffbe80aaccd9fbe57a21853.png, and typically normalized such that general_relativity_95e493930a8f5f96865fa01ecfdd877df128ca10.png
  • Perfect fluids:

    general_relativity_adccd784db7ecc7bdf4cc3e86ca63dfea7ce9c16.png

    where

    • general_relativity_ccb973f0cf1a9e40c184ec65f392888f18a6471e.png is the energy density
    • general_relativity_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png is the pressure

Then

general_relativity_fc34cf3a33286385e64a76679c1fc0721846de66.png

Motion of fluid given by

general_relativity_27707a86871c771a3711c89db65f9c7419fe03cc.png

Observe that if the pressure vanish, i.e. general_relativity_38407d3674fe196c96befcb0c41f03c8d7e53318.png, then we're left with

general_relativity_86fde0f56615febce8510e880236e2f2adbcb9e8.png

which is the equation describing geodesic, fluid moves as "free particles".

Einstein's equations

  • Motivation:
    • Newtonian
      • Graviational field described by scalar potential general_relativity_1827136eeff990e82cb2b9950aa049a32cc2f910.png
      • Equation of motion

        general_relativity_4bbb263d2ddf9a53290510b3cca29e32fcb54359.png

        where general_relativity_f7267dc63abc18da4ed9c6e8b04a62a5986eafcd.png in cartesian coordinates

      • Relative acceleration of nearby particles ("tildal force") governed by deviation equation

        general_relativity_0a9982529e5aad08d27b48faa1d7d7f57757055d.png

        where general_relativity_ebc1684855eeada7e358d484fcf6c36d2acfb61a.png is the separation vector of the particles

    • GR:
      • Relative acceleration of two nearby particles following timelike geodesics is given by the geodesic deviation equation

        general_relativity_2d582bced14d1f1fd992f70cd282ad1a6835a5ff.png

        where general_relativity_335bb451b439bbef5529eceb0d3a9c1ca3ac72dd.png is the tangent to the geodesics and general_relativity_9b92e2dd91835582d01cbe3baaf51261c0e62073.png is now the deviation vector.

    • Comparison suggests

      general_relativity_76e9afd4c3b3c2a238186f8556c80bc2ef4c94ee.png