Transcript of gr-l11
==========


_0:09_: Hello everybody. Welcome to Lecture 11,

_0:12_: which is our last lecture of GRG one

_0:17_: preparing you for the likes of G2,

_0:20_: so our next semester.

_0:21_: So we are astonishing for

_0:24_: me in very good time.

_0:26_: So we've made excellent progress

_0:28_: and so I think we might well.

_0:31_: We'll have we're not under

_0:33_: time pressure this week.

_0:35_: It might even finish early.

_0:38_: Where I got to last time was I was

_0:42_: talking about and I hope justifying

_0:44_: the idea of the the the the physical

_0:48_: statement of Einstein's equations

_0:50_: relating the Einstein tensor which

_0:52_: is composed of the richer tensor

_0:55_: and the cover change and the metric

_0:58_: with and equating that for making

_1:00_: that proportional to the energy

_1:02_: momentum tensor which characterizes

_1:05_: the distribution of energy momentum.

_1:08_: In our particular volume,

_1:09_: that sounds rather abstract.

_1:11_: What it means is if you have a single

_1:13_: mass in the in the center of the universe,

_1:16_: that is a characterized by an

_1:18_: appropriate Environmental Center.

_1:19_: If you have an extended object

_1:23_: or you have a universe full of

_1:26_: certain distribution of argumentum,

_1:27_: that's characterized by the energy

_1:30_: momentum tensor and intuition

_1:32_: tells you what the consequent.

_1:35_: Distribution shape or space-time is.

_1:37_: And that is.

_1:40_: Basically it as far as the

_1:43_: the equations it goes.

_1:45_: However,

_1:45_: one of the first one of

_1:48_: the solutions to this was.

_1:51_: Which you will learn about N G2

_1:53_: I think all you which you've also

_1:55_: learned about in in in the cosmology

_1:57_: lectures you've had I'm sure is the

_1:59_: notion of the expanding universe.

_2:00_: There's a solution to Einstein's

_2:02_: equations which consists of a

_2:04_: universe which is expanding,

_2:05_: and this was felt.

_2:07_: And at that time in the 30s,

_2:10_: I think to be obviously unphysical,

_2:13_: obviously unreasonable.

_2:13_: That can't be, that can't be an

_2:15_: answer so thought to be illegitimate.

_2:18_: Solution, and that was therefore fair enough.

_2:21_: That's fine. That's that.

_2:22_: If physical solutions.

_2:23_: So I'm saying to you that he his

_2:26_: work was not done at this point

_2:29_: and he added another term to this.

_2:32_: And

_2:35_: ah.

_2:37_: Sorry, which I will write down.

_2:39_: In that case which was G?

_2:43_: You knew plus Lambda. Lambda G.

_2:49_: You you equals Kappa T mu where Lambda is.

_2:57_: Auto focus off where Lambda is just a scalar.

_3:02_: Multiplying the.

_3:05_: The metric and he said OK that's

_3:07_: the that that's clearly what I

_3:09_: should have said first time and

_3:10_: and when you add that term one of

_3:13_: the the the the the whole universe

_3:15_: solution you can get a a static

_3:18_: whole universe solution out of it.

_3:21_: And it was only later,

_3:22_: when the Hubble Metro expansion

_3:24_: was discovered that that you know,

_3:26_: and it became clear as a matter

_3:28_: of observation that the universe

_3:30_: was in fact expanding,

_3:31_: that he referred to this

_3:33_: as his greatest blunder.

_3:34_: That he had wimped out effectively,

_3:37_: effectively by believing experiment

_3:39_: to by believing experiment too much

_3:42_: and not going with the original plan

_3:44_: of just sticking with the original

_3:46_: version with Lambda equals 0.

_3:48_: Now in a further twist,

_3:50_: as you probably are aware.

_3:51_: Much later on, starting in the.

_3:55_: 2000s. It became.

_3:58_: Clear from supernova observations that the.

_4:03_: Aiming to discover whether

_4:04_: whether the universe was,

_4:06_: on the largest scales,

_4:08_: positively negatively curved or flat,

_4:10_: it became clear that that the most distant

_4:13_: supernovae were retreating from us.

_4:15_: Fair enough, but at an increasing rate.

_4:18_: So the universe seemed to be a inflating,

_4:21_: inflating, being driven by some sort

_4:23_: of of of of external pressure term,

_4:26_: and that was the motivation.

_4:29_: In the you know the 2000s for bringing

_4:32_: this version of Einstein equation back,

_4:34_: gain nonzero cosmological constant a

_4:37_: nonzero value of Lambda no no referred

_4:40_: to as the cosmological constant

_4:42_: which in has the effect was super

_4:46_: children values of of Lambda of.

_4:49_: Adding our.

_4:50_: A negative pressure to what is effectively

_4:52_: a negative pressure to the universe,

_4:55_: which means the universe ends up

_4:58_: expanding at a slightly accelerating rate.

_5:02_: That's all very far from the future.

_5:04_: The point of of saying this is to

_5:07_: think what I said last time really,

_5:09_: that this is, I guess,

_5:11_: which is corroborated by observation.

_5:13_: Einstein thought better of it.

_5:16_: Then she just been again and we've

_5:18_: seen change her mind again to

_5:19_: bring that that that term back.

_5:21_: But the point is that this is that

_5:23_: these steps here of this version

_5:25_: or that version of this version of

_5:27_: that version are by this point not

_5:29_: mathematically but physically and

_5:31_: observationally and astronomically motivated.

_5:35_: And, and and there's a lot more one could say

_5:37_: there about where does this term come from,

_5:39_: quantum gravity or something like that.

_5:41_: And and you know the there are a variety

_5:43_: of rabbit holes we could dive down at

_5:46_: this point but we shall forbear. And.

_5:51_: OK. I'll move you feel swiftly on

_5:55_: unless there any questions about that.

_5:59_: No. OK. So what I did the last last time

_6:04_: was this all leads us to the the point

_6:07_: where space and this is a formulation

_6:10_: attributable to I think it's John,

_6:13_: John Wheeler space tells matter how to move.

_6:15_: And match your speech to curve.

_6:17_: You be able to see that slogan before,

_6:18_: but now you have the,

_6:20_: the essentially all of the mathematical

_6:22_: background to understand what's

_6:24_: actually going on in that slogan,

_6:26_: what that slogan is actually summarizing.

_6:28_: And to some extent, at this point,

_6:30_: the job of G1 is done.

_6:33_: My job of G1 is to give you

_6:36_: the mathematical technology.

_6:38_: That will allow you to go in and

_6:40_: make sense of G2G2 next term Doctor

_6:43_: which is taking over this year.

_6:45_: It's actually about solutions to I,

_6:48_: I sense equation, it's partial solution,

_6:50_: the Freedman, Robertson Walker,

_6:53_: Friedman, Robertson Walker solution,

_6:55_: graphical waves and so on and so on.

_6:57_: And we can do that.

_6:59_: So that's the that's the payoff.

_7:02_: But I can't take you to the

_7:03_: threshold of of that and I'm not

_7:05_: give you any solutions at all.

_7:07_: So we're going to look at one solution

_7:10_: of Einstein's equation in outline.

_7:12_: I just know.

_7:15_: The pretty picture to show you

_7:17_: what an orbit in our in our

_7:20_: space-time is supposed to look like.

_7:24_: I said that. OK.

_7:27_: So you will recall that somewhere,

_7:30_: I think in part one,

_7:33_: I mentioned the idea of natural units,

_7:34_: units in which the speed

_7:36_: of light is equal to 1.

_7:38_: Which essentially is deciding to.

_7:42_: Use the light meter as our source,

_7:45_: as as our unit of our unit of time,

_7:48_: in units of which light light moves

_7:51_: at one metre per light meter or one.

_7:53_: It's not to mention this is merely unitless.

_7:56_: People doing worrying about GR

_7:58_: tend to work in a set of units in

_8:02_: which seat G big is equal to 1.

_8:05_: And that means that the.

_8:09_: Well, we can take the squared equal

_8:12_: to 0, equal to 1 or not, but the.

_8:18_: The big had the value 7.2 to the

_8:21_: minus 28 meters per kilogram,

_8:23_: which, just like CB equal to 1,

_8:25_: is a conversion between seconds and meters.

_8:28_: This is a conversion between

_8:30_: kilograms and meters,

_8:32_: so that masses in these units

_8:34_: are measured in meters.

_8:35_: Which makes sense when you do,

_8:37_: which seems weird, but makes sense when

_8:40_: you discover the functional solution.

_8:42_: Discover that the structural

_8:44_: solution has in it a parameter which.

_8:48_: Has the dimensions of meters,

_8:49_: which is essentially the the event the size

_8:51_: of the event horizon of a collapsed object.

_8:54_: So jumping ahead a bit in

_8:56_: the structural solution,

_8:57_: there is a size parameter inside which

_9:02_: the only time logistics are inwards.

_9:06_: So in other words,

_9:08_: even light cannot escape that there

_9:10_: is no time like there's no you're not

_9:13_: even a null path to greater radius.

_9:15_: What is the radius?

_9:16_: In other words,

_9:17_: the black hole.

_9:19_: So the.

_9:21_: What this far should solution tells

_9:22_: you is that there are such things

_9:24_: as black holes which have a radius

_9:26_: which is proportional to to G&M.

_9:28_: In which context it makes

_9:31_: sense that there is a natural?

_9:33_: And correspondence between mass and the

_9:37_: gravitational radius of of that mass.

_9:40_: So the the mass of the sun

_9:42_: for example is 3 kilometres.

_9:46_: That being the side of the black

_9:48_: hole into the the the the size,

_9:51_: which if you compress the sun

_9:53_: inside would make a black hole.

_9:55_: And.

_9:56_: And you you can divert yourself

_9:57_: by work by working out what your

_9:59_: gravity your own gravity should

_10:01_: radius is and survey small number

_10:03_: well it'll be 10 * 28 times whatever

_10:07_: your weight in kilograms in meters.

_10:10_: And as a curiosity there.

_10:13_: What this also?

_10:15_: A curiosity here is that even in

_10:19_: classical gravitation physics,

_10:21_: the thing that controls the orbit

_10:27_: Newton's gravity potential is g / R.

_10:34_: As you were just recall and it's GM,

_10:38_: the controls the orbits,

_10:40_: the behaviour of of particles in

_10:42_: the source in the solar system

_10:44_: and not G or M separately.

_10:46_: They never appear separately.

_10:47_: And what that means is that it's

_10:51_: relatively easy to find what GM is and

_10:53_: and and and if you're doing classical

_10:56_: mechanics of the in the solar system

_10:59_: what you the parameter of interest

_11:01_: is GM and it's quite easy to to

_11:03_: determine that from looking at the FMD.

_11:05_: Of planets going around the sun

_11:06_: and you can get, you can get,

_11:08_: you can estimate GM to I think

_11:10_: one part in 10 to the ten.

_11:12_: It's extremely accurate.

_11:13_: Do you to an accuracy where general

_11:17_: statistic corrections matter.

_11:20_: But the only way you can find what G

_11:22_: is is using terrestrial experiments,

_11:25_: such as you're looking at plumbs

_11:26_: next to mountains and so on.

_11:28_: As you will be aware,

_11:29_: we can only do that to about

_11:31_: one part in 10 to the five.

_11:33_: And the way you find what the

_11:35_: mass of the sun is is by finding

_11:37_: what GM is and dividing it by G.

_11:39_: Through the mass of the sun in

_11:42_: kilograms is obtained by GM over G

_11:44_: and has the error of G which 10 to

_11:47_: minus 12:50 and 10:00 to the five.

_11:49_: But the error uncertainty of the sun's

_11:52_: mass in meters is essentially the

_11:54_: error of of the of the gravitation power GM.

_11:57_: The mass of the sun in meters

_11:58_: is known to about 10 * 110 to

_12:01_: 10 kilograms .10 to the five,

_12:04_: so there's a nice inversion of

_12:05_: what you might expect there.

_12:09_: Sorry, that's all in a

_12:10_: big parenthesis really.

_12:13_: Moving on the. Point here is that

_12:17_: we're now going to look at the.

_12:20_: Solution of intense equations

_12:22_: in a particular approximation,

_12:24_: namely the weak field approximation,

_12:25_: the approximation of.

_12:29_: Small masses, so small central mass,

_12:31_: something planet size or star size.

_12:33_: Milk that just is isolated

_12:35_: in in the in the universe.

_12:37_: Or or equivalently the approximation

_12:39_: where you're looking at the solution

_12:41_: for a mass but you're you're quite

_12:43_: far away from the mass of the masses,

_12:45_: so 2nd order terms disappear.

_12:48_: And then we do that and I'm going

_12:50_: to go through this in in outline

_12:52_: rather than in in line by line

_12:53_: detail is we approximate the metric.

_13:00_: By the Minkowski metric.

_13:03_: Which is minus plus plus plus

_13:07_: diagonal plus. A perturbation.

_13:11_: And the point here that.

_13:16_: This is a perturbation.

_13:17_: H is small in the sense that

_13:20_: the magnitude of all the values

_13:22_: of H is much less than one.

_13:24_: So each squad is is is ignorant.

_13:28_: Now that's a matrix equation

_13:30_: and not a tensor equation.

_13:32_: But it turns out that.

_13:35_: For reasons which we could expand on,

_13:37_: but might make expander if we

_13:38_: have more time at the end,

_13:39_: this can be treated as if it were a tensor.

_13:44_: And what we can then do is. Reexpress.

_13:50_: The point I want to make here are that.

_13:53_: What you're doing here is essentially

_13:56_: changing into coordinates in which the.

_13:60_: In which each can be regarded as

_14:02_: a tensor now, and there's a couple

_14:03_: of ways of thinking of that.

_14:04_: One is that you are making a particular.

_14:08_: Particularly our particular coordinate

_14:09_: transformation which is constrained

_14:11_: by the the constraint that this be

_14:13_: small or you can regard this and

_14:14_: this is quite quite a productive way

_14:16_: of regarding of thinking about it.

_14:18_: You could regard this as being as is

_14:22_: it asking. About the behaviour of.

_14:26_: Tensor each in a Minkowski background

_14:30_: as fluctuations. Unlucky background.

_14:33_: The point is that this is.

_14:35_: Using that you can then calculate

_14:39_: what expression is for the.

_14:45_: A connection for the Riemann tensor

_14:48_: in terms of H as opposed to G.

_14:53_: And then express Einstein's equations,

_14:57_: which are of course obtained from

_14:60_: the room tense contractions in terms

_15:01_: of each and and because you then

_15:03_: at that at that point are dealing

_15:05_: with something which is small,

_15:07_: where second order terms can be neglected,

_15:10_: that becomes easier to solve.

_15:13_: And the solution is look at this

_15:16_: just to get the terms right where

_15:19_: each the as a quasi tensor is.

_15:24_: Diagonal. Each nought nought

_15:32_: H11H22H33. Yeah. All of these.

_15:41_: Are equal to. The same.

_15:47_: 5 Phi.

_15:51_: I'm plugging this back into the minkovski.

_15:54_: Metric using the copy metric again

_15:56_: the that means that our solution. Is.

_16:04_: Diagonal and minus 1 + 2 Phi. 1 -,

_16:10_: 2 Phi, 1 -, 2 Phi, 1 -, 2 Phi, or.

_16:17_: With an interval of.

_16:23_: That's good. Plus one minus.

_16:26_: Just checking up to make sure

_16:29_: you get the signs right.

_16:37_: Where the Sigma there is the.

_16:47_: Is the the the spatial sector.

_16:51_: And.

_16:55_: So that that's our,

_16:57_: so that's the the metric in that

_16:60_: low mass weak field approximation.

_17:03_: And I've. Mr Bit here which is

_17:07_: fiddly rather than hard and and

_17:09_: and and shoots for example goes

_17:12_: does go through it step by step.

_17:14_: It's not terrifically edifying but

_17:16_: it's sort of reassuring that it's

_17:18_: actually quite a short calculation.

_17:20_: Roughly I think even they're part

_17:22_: of what he says is if you then go

_17:25_: through this and and and work out

_17:27_: what are the components of our are

_17:30_: in in details several pages of

_17:32_: algebra but it's not hard algebra.

_17:34_: Just turning the handle.

_17:37_: Umm.

_17:41_: So that's all very nice that's that looks

_17:43_: like pretty and jumping ahead because we

_17:46_: have time and it's quite interesting you

_17:49_: will discover that when you look at the.

_17:52_: We discovered this partial

_17:54_: solution next semester use.

_17:56_: You discover that this ends

_17:58_: up being the, which is the.

_18:02_: Exact solution for this same

_18:04_: problem of our single central mass.

_18:07_: You get an expression for the for

_18:10_: the for the metric for which which

_18:12_: is is equal to this to 1st order.

_18:14_: So in this case we have obtained

_18:17_: this by demanding that well,

_18:19_: but by building on the fact

_18:22_: that each is small.

_18:23_: We can recover this as as the low mass

_18:27_: limit of the Schwarzschild solution by.

_18:31_: Depending on Phi being small

_18:33_: at that hand, yes.

_18:36_: I second. What these fees?

_18:40_: Physical significance, good point,

_18:43_: good point. This ends up being.

_18:54_: Remarkably enough, what comes out of

_18:56_: this is that if I just the numerically

_18:60_: the same as Newton's gravitational

_19:03_: potential. So using gravitational

_19:05_: potential pops out of this.

_19:07_: And exactly the place you'd expect.

_19:10_: And or or. And it it turns out that this.

_19:19_: Readius.

_19:24_: It's 2.

_19:28_: The the radius two GM.

_19:31_: Turns out to be the that that

_19:34_: that this, which is basically.

_19:35_: You can see the two coming from there.

_19:38_: That radius is the radius which

_19:40_: I mentioned, which is the.

_19:44_: Size where the structural

_19:46_: solution gets interesting and

_19:48_: where the black hole appears.

_19:50_: So the size of a black hole is.

_19:54_: Dependently it directly linked to

_19:56_: this GM parameter which comes up

_19:59_: which appears just as this potential

_20:01_: factor in in in inside the metric.

_20:09_: So that's very nice.

_20:11_: But what can we do with that well?

_20:13_: We've been half the thing what we we

_20:16_: have worked out at this point that.

_20:19_: Are are.

_20:25_: A solution for instance equations.

_20:27_: The next thing we have to do is workout.

_20:29_: How do things move into that space-time.

_20:32_: And we can do that by using

_20:35_: the geodesic equation. Uh.

_20:39_: Right.

_20:45_: So. Point.

_20:51_: So the geodesic equation

_20:53_: we've seen versions of of it.

_20:56_: But if you look back one of the versions

_20:59_: that the sort of prime Prime primal

_21:01_: version of the judges equation is,

_21:04_: is that one which is the one saying

_21:07_: that as as you parallel transport the

_21:10_: tangent to a geodesic along the geodesic,

_21:13_: it stays tangent to the geodesic.

_21:15_: So that's the mathematical version

_21:17_: of walking in a straight line.

_21:19_: And the the part you you, you,

_21:21_: you draw out by walking straight

_21:23_: line is as you desire.

_21:25_: That's not particularly convenient,

_21:27_: but let's instead recall that the.

_21:33_: We can talk about the full momentum

_21:35_: of an object as just being the

_21:38_: mass times the that that this.

_21:40_: The the the full velocity.

_21:41_: In this case, we're taking the

_21:43_: full velocity to be the full

_21:45_: velocity along a judic, so this.

_21:53_: Judy equation turns into an expression

_21:57_: involving the. Momentum of an object.

_22:03_: And then the. But that's the

_22:06_: geometrical version of it.

_22:09_: You would recall that the

_22:10_: component version of that.

_22:14_: And get everything in the right place.

_22:18_: And choose their indexes

_22:19_: that I am consistent with.

_22:24_: Who's that? It will evolve in

_22:29_: the covariant derivative of the.

_22:32_: Of the vector P so that's just

_22:34_: the component version of this,

_22:35_: which is is the A scaling

_22:38_: of the duties equation.

_22:39_: So asking what are the what?

_22:41_: What are the are the is the the field

_22:44_: of P vectors of momentum vectors which

_22:49_: satisfies this equation and thus which.

_22:53_: Indicate.

_22:53_: The judaics in this space

_22:55_: tech in the space-time.

_22:60_: And that in this space dangers and

_23:02_: because this covad derivative is

_23:04_: picking up curve the the the way

_23:06_: that the the components change as

_23:08_: you move around the the space.

_23:10_: So the curvature if you like is in

_23:12_: the coverage of the of the space

_23:15_: we're looking at here is in that.

_23:17_: You could be derivative.

_23:20_: And now so that's exact.

_23:22_: This this far, no.

_23:24_: Because we're interested in

_23:25_: the weak field solution,

_23:27_: we're going to take the another

_23:29_: weak field approximation,

_23:30_: which is to say that things

_23:31_: are going to be moving slowly.

_23:35_: And what that means is that for

_23:38_: the momentum 4 vector of the

_23:40_: geodesic we're interested in.

_23:42_: The. Time component.

_23:47_: It could be much larger than

_23:49_: the spatial components.

_23:51_: So things are going to be

_23:52_: moving through time faster than

_23:53_: they're moving through space.

_23:54_: They are moving slowly, in other words.

_23:59_: So what this? Implies is and again

_24:05_: keeping things neat P alpha P mu

_24:10_: comma alpha plus gamma mu. And.

_24:17_: Alpha Beta P Alpha P beta. Equals zero.

_24:21_: All I'm doing there is simply.

_24:25_: Breaking that out in a slightly

_24:27_: longer version, but if.

_24:30_: The 0 component of these vectors

_24:33_: are much larger than the.

_24:36_: And spatial components,

_24:37_: then we can then discard all the

_24:40_: spatial components in that sum.

_24:43_: So the only terms that

_24:45_: will survive in that sum.

_24:48_: Are going to be.

_24:54_: The 00 times. And similarly if the

_24:59_: only term that survives in this sum

_25:02_: here over alpha is the zero term then.

_25:06_: The. And and given that.

_25:11_: The.

_25:14_: P is equal to gamma M.

_25:20_: When in M visa.

_25:24_: No comma M1. The. The momentum for

_25:33_: vector is proportional to the gamma the.

_25:36_: Matter of the object being moving

_25:41_: plus this one V XYZ vector here.

_25:47_: The that that means that the.

_25:50_: And at low speed, regardless small,

_25:54_: gamma is 1, the zero the the 0

_25:58_: component is just M so it's M.

_26:00_: DP mu by D Tau that survives.

_26:10_: So what we're doing here is this current

_26:14_: derivative in this approximation.

_26:18_: I should probably see.

_26:25_: OK, now what we can then do?

_26:29_: It go back to the metric.

_26:34_: Here. And do the things that were were,

_26:38_: you know, I hope, fairly well rehearsed

_26:40_: that calculating the Christoffel

_26:42_: symbols corresponding to this metric.

_26:46_: And we find most of them are zero.

_26:49_: I just saw from the case and there's a

_26:52_: hydrogen symmetry and the ones that are not.

_26:55_: Are gamma 000. Which is equal to.

_27:04_: 5 comma 0 plus terms of order Phi

_27:08_: squared. Which number is small?

_27:10_: And gamma I-00. Which is.

_27:20_: Comma, G.

_27:25_: Both genes evolving fine before.

_27:32_: OK. And what if we then look at

_27:36_: this component by component,

_27:38_: we find that therefore M.

_27:45_: DP naughty by detour. Yeah,

_27:49_: I can't, right? Plus gamma.

_27:52_: Comma 005 comma 0. P nought squared.

_27:60_: Which is just. MDP nought by D Tour plus.

_28:12_: I'm expecting to see our.

_28:16_: The term here. No, it's good.

_28:22_: Ohh yes plus m ^2. Have. Yeah, squared.

_28:30_: Phi comma not equals 0 or.

_28:36_: DP nought by D Tau is equal

_28:40_: to minus MD Phi by DT. Toll.

_28:46_: And what that is saying is that.

_28:49_: The. Reach the the the.

_28:53_: Change in the energy.

_28:55_: Of this particle.

_28:57_: Is proportional to the change in time.

_29:01_: Of the potential,

_29:02_: and we given that there isn't more

_29:06_: mass certainly appearing here,

_29:08_: that's going to be 0.

_29:09_: In other words,

_29:10_: that's saying the energy is

_29:12_: conserved as the particle

_29:13_: moves along the geodesic.

_29:15_: Through phoned one of the relevant.

_29:21_: Descriptions of the motion.

_29:28_: Now looking at the special one.

_29:30_: Uh. And looking at. This one here.

_29:37_: What we then discovered

_29:39_: there is that DPI by D.

_29:42_: Tall. Is equal to minus M.

_29:48_: If I. Comma I which is the I

_29:53_: spatial derivative of the.

_29:59_: Of this potential, which is

_30:01_: just a funny way of writing.

_30:04_: The Richard change of momentum.

_30:06_: The force. Is equal to minus. Gradifi.

_30:14_: Which you will recognize as the equations

_30:18_: of motion in Newton's gravitational theory.

_30:22_: That the the particle moves in such

_30:24_: a way that the rate of change of its

_30:28_: momentum is directed along the gradient

_30:30_: of the gravitational potential.

_30:35_: Which is very gratifying

_30:37_: because this means that that

_30:39_: the low energy approximation,

_30:41_: that low energy approximation

_30:43_: of of of Einstein's theory

_30:45_: recovers the manifest successful.

_30:51_: Theory of gravity that Newton

_30:53_: developed for from starting from

_30:55_: a completely different place.

_31:00_: So.

_31:04_: I think I I've I've missed

_31:06_: these quick questions.

_31:10_: Those are in, in, in, in, in the notes.

_31:13_: So I think that's that's basically budget

_31:16_: and and I think that's a a remarkable thing.

_31:19_: I I do know what know what I may appear

_31:21_: to the world but to myself I seem to be

_31:23_: more like a a boy a boy playing on the

_31:25_: seashore and diverting myself now and now

_31:27_: and then finding a smoother Pebble or a

_31:30_: prettier shell than ordinary with great

_31:32_: ocean of truth lay all undiscovered before

_31:34_: me there's a certain rejection to that.

_31:37_: I think on Newton's part he

_31:39_: knew that he had done great.

_31:40_: Things and found beautiful,

_31:43_: mathematically beautiful

_31:44_: explanations of what happened.

_31:45_: But there was much more to find.

_31:47_: It took 300 years to find

_31:49_: something that was better.

_31:50_: But as we've discovered,

_31:52_: what he did is contained

_31:54_: within a later theory.

_31:59_: And that is essentially it.

_32:03_: I we were ahead of time.

_32:04_: I've slightly spun out by talking about

_32:07_: spatial solution and G2 and and and so on,

_32:09_: but I think it's actually a first that

_32:13_: I've managed to get to the end without

_32:17_: galloping through the last lecture

_32:19_: in a something of a mild panic but.

_32:22_: That we've got here,

_32:23_: we've got technology,

_32:24_: you've got one solution and you have

_32:26_: the browser uplands of G2 to find all

_32:29_: sorts of other solutions next semester.

_32:31_: And so we we might as well stop

_32:34_: there or we'll get questions.

_32:40_: Questions. But we can do.

_32:43_: If we want to run off, that's fine.

_32:45_: If if one turns into an impromptu

_32:48_: supervision question session,

_32:49_: question session or chat session,

_32:51_: then that's fine too.

_32:54_: How about it? Question over there.

_33:02_: Can you say where's tricked

_33:04_: ourselves to the motion by N

_33:05_: relativistic particle we have this,

_33:07_: where does this come from like.

_33:10_: So uhm. The question there is um.

_33:17_: This approximation.

_33:18_: Why that drove that approximation?

_33:21_: Is basically comes from.

_33:24_: From this. So you may recall,

_33:28_: you may not recall that when you

_33:31_: talk when a special activity,

_33:33_: you talk about the relativistic.

_33:38_: Velocity.

_33:39_: It's it's a form momentum

_33:41_: which includes the the.

_33:43_: Let's not talk to the

_33:45_: momentum rather than velocity.

_33:47_: The full momentum involves the spatial

_33:50_: momentum and the time component

_33:52_: of the four momentum which is the

_33:55_: energy of the particle and you

_33:57_: discover that the low speed limit

_33:59_: of that what that in in in the

_34:01_: frame in which the word is that.

_34:05_: Does that look sort of familiar?

_34:07_: Have you seen something like that?

_34:09_: Before or vaguely enough that you

_34:11_: believe me that that's the key, right?

_34:13_: So that's that's the key and and

_34:16_: the low speed in the frame of

_34:17_: which the particle is not moving,

_34:19_: you discover,

_34:20_: good heavens,

_34:20_: that the the 0 component has is

_34:23_: gamma M which doesn't go to zero as.

_34:29_: As the speed goes to 0,

_34:31_: the the 0 component energy is gamma M

_34:33_: or in physical units gamma Mt squared?

_34:36_: Or will be SU equals MC squared.

_34:38_: That's where equals MC squared comes from.

_34:40_: But in this case. The.

_34:47_: This time component is always gamma

_34:49_: M but in the case where you're

_34:52_: looking at particles which are moving

_34:55_: only slowly and by slowly meaning.

_34:58_: Much less than the speed of light.

_34:60_: Then. Each of these spatial components

_35:05_: VVZ will be much less than one,

_35:07_: much less than C.

_35:09_: And so these people,

_35:11_: and that will be therefore true of the.

_35:15_: Overall momentum component,

_35:16_: so the spatial components.

_35:18_: Will be small more than the energy

_35:20_: components simply because the energy

_35:21_: component is primarily the particles mass.

_35:23_: So in the case where essentially

_35:25_: all of the particles energy is in

_35:28_: the form of its mass as opposed to

_35:30_: its mass and its kinetic energy.

_35:32_: Then we can solve this in in that

_35:37_: limit to get Newton's theory.

_35:39_: So that's telling us that Newton's

_35:42_: theory goes wrong.

_35:43_: When things move at rustic speeds,

_35:46_: which is terribly surprising.

_35:48_: And what that means is things go

_35:50_: wrong when things were rustic speeds,

_35:52_: a because they're moving rapidly

_35:54_: and B because there is a component,

_35:56_: there's an element of of energy. In the.

_36:01_: Simply by virtue of the particles motion.

_36:05_: Which, which is which will grab,

_36:07_: which gravitates.

_36:09_: So the the the the.

_36:13_: The energy that's in our particles motion.

_36:17_: The instruments in particle motion,

_36:18_: it was Einstein's theory.

_36:21_: It's Andrew Mentum, the gravity.

_36:22_: It's not mass.

_36:24_: And that is a thing which does not.

_36:28_: It was something moves faster

_36:30_: than it gravitates more.

_36:32_: And that's that's completely

_36:34_: alien to Newton's theory.

_36:35_: I just you cannot be there and your theory.

_36:37_: That's why in a sense this this

_36:40_: has to be the the limit in which

_36:43_: Newton's theory will pop out,

_36:45_: the case where we're

_36:47_: ignoring the gravitation,

_36:48_: the gravitating influence of kinetic energy.

_36:51_: That's the physical interpretation of.

_36:57_: So this is the. As I mentioned,

_37:02_: this links to this partial

_37:03_: solution in the sense that.

_37:08_: There. This metric here is derivable

_37:13_: as the as the little file limit

_37:16_: of the partial solution. The.

_37:21_: As far as your solution is the exact

_37:23_: solution to an approximate problem.

_37:26_: In the sense that it is the

_37:28_: solution to the approximation where

_37:29_: the universe has one mass in it.

_37:32_: And that's not actually true there.

_37:33_: There's more than one star in the universe.

_37:35_: But in certainly in our environment there

_37:39_: it is a very, very good approximation.

_37:42_: And it's the solution that is used

_37:45_: for essentially all of the relativity

_37:47_: corrections to things that GPS,

_37:49_: to things like precise timing,

_37:51_: to things like the deflection of radio

_37:54_: waves by going near the start that,

_37:57_: that, that the, the, the,

_37:59_: the sun and the Eddington Dyson.

_38:02_: Observations of the deflections of the.

_38:06_: Ohh of um.

_38:09_: Dilate as it comes past the sun in Eclipse,

_38:12_: which you've heard of.

_38:13_: Yes, eddington.

_38:16_: And perhaps I didn't mention that

_38:17_: and well I didn't mention though,

_38:19_: but I thought yeah it might be

_38:21_: that that that that sort of

_38:23_: normally comes later than the,

_38:24_: the what I'm seeing but earlier

_38:28_: than G2 because amongst the the

_38:31_: the effects of this of this.

_38:34_: Solution to.

_38:37_: GR Well,

_38:37_: if you remember back in in

_38:40_: lecture one beginning in part,

_38:42_: one of the things we discovered

_38:44_: was the coolest principle tells

_38:46_: you that that that light must

_38:47_: bend in the gravitational field,

_38:49_: that the wholeness of things falling down,

_38:51_: down lift shafts.

_38:52_: But we didn't calculate how

_38:55_: much that that deflection was.

_38:57_: Now you can calculate it.

_38:59_: Based on the. Gravitational.

_39:06_: Red shift of our particle.

_39:09_: So one of the other things we

_39:11_: mentioned in part one was the

_39:13_: idea that as our photon.

_39:15_: Claims through graphical field it's

_39:17_: frequency changes and to the extent

_39:19_: the frequency is are a proxy for a clock,

_39:23_: a photon oscillation is proxy for

_39:26_: clock that is telling us that time

_39:30_: moved differently at different

_39:32_: at different gravitational

_39:34_: potentials and from that.

_39:36_: You can deduce.

_39:40_: Through a few a few steps,

_39:42_: but not too many,

_39:43_: that there will be a particular

_39:44_: deflection of Starlight as it

_39:46_: comes a past a gravitating body.

_39:48_: You can work out what the

_39:49_: selection will be purely from that.

_39:52_: And you can also use this solution

_39:56_: this approximate solution.

_39:58_: The this week full solution

_39:60_: of Einstein's equations.

_40:01_: To work out what the deflection what what,

_40:04_: what, what the geodesic or

_40:06_: photon is as it goes near a mass,

_40:09_: whatever near counts as and

_40:11_: you discover it's deflected.

_40:12_: Of course you know very way

_40:14_: better that it didn't happen,

_40:15_: so there's a deflection of

_40:17_: light as it goes near mass.

_40:19_: And you can calculate what the

_40:20_: angle of that deflection is.

_40:22_: You discover is twice the angle

_40:24_: that you got when you use only

_40:27_: the gravitational redshift.

_40:30_: Which and. And so Einstein and Eddington.

_40:34_: Got to the first answer first.

_40:37_: I think Einstein got the I think

_40:39_: it would be representing 13 or

_40:41_: something that he worked out how

_40:42_: much the deflection would be based

_40:44_: purely on on on the gravitational

_40:46_: redshift and that was the prediction

_40:48_: for how much that affection would be

_40:51_: and it was only after in about 19.

_40:55_: 16 or 17 I think when the when this

_40:57_: solution was was available to work

_40:60_: out what the deflection would be

_41:02_: based on the field equations and so

_41:05_: there's going to be a reflection.

_41:07_: The OR the prediction and how

_41:08_: do you find that deflection?

_41:10_: You what you look at star at

_41:12_: stars as the as the lake from the

_41:15_: moves near a battered body.

_41:17_: There could be massive

_41:18_: body in the neighborhood.

_41:19_: Yes, there is the sun.

_41:20_: Unfortunately you can't see the stars in

_41:22_: the daylight because the sun's very bright.

_41:24_: So you wait for an eclipse.

_41:27_: And conveniently,

_41:28_: there was a total solar eclipse visible from

_41:31_: some parts of the of the planet in 1919,

_41:34_: so just after the First World War.

_41:38_: And.

_41:40_: You know, I could go on the

_41:43_: story for quite a long time.

_41:45_: With all sorts of layers of of interest,

_41:48_: but the short version,

_41:49_: the focusing on the on the physics version.

_41:52_: Is that it was a fairly prediction

_41:55_: at this point of GR that there

_41:57_: would be deflection.

_41:59_: And through the next edition

_42:00_: mounted by Edison,

_42:02_: who is the head of the

_42:04_: Cambridge Observatory and.

_42:08_: Herbert Dyson. Some Frank Frank Dyson,

_42:12_: who was the director of the grand jury,

_42:15_: and they put together the equipment you've

_42:17_: scattered because of the First World War,

_42:19_: but the equipment to make an

_42:21_: expedition to Brazil and the.

_42:27_: I don't want to keep Verde islands anyway,

_42:29_: somewhere in the southern Atlantic where

_42:32_: where there was it was going to seasonality

_42:35_: and long observational story later the the,

_42:37_: the, the three three possible outcomes of

_42:39_: that of the observation were no deflection,

_42:41_: which is what in the sense that the the

_42:44_: the intoning, they would see the what

_42:46_: was called the Newtonian deflection,

_42:48_: which was the the one that that

_42:50_: Einstein and Co had produced based on

_42:52_: purely graphical redshift and the full

_42:55_: Einsteinian deflection which was the.

_42:57_: And they obtained from this,

_42:59_: which is twice the Newtonian 1. And the.

_43:02_: Observational but opposition nightmare things

_43:05_: which were supposed to work didn't work.

_43:08_: There was rain though, you know,

_43:11_: in the field, literally covered in

_43:13_: mud in the 10 minutes of totality.

_43:17_: But they did manage to exclude the the,

_43:21_: the, the reflection case and.

_43:24_: Arguably and correctly exclude

_43:26_: the Newtonian version,

_43:27_: and thus confirmed by direct observation

_43:30_: that the deflection was what Einstein,

_43:32_: Einstein Field equation set,

_43:34_: and Einstein became a worldwide

_43:36_: celebrity and so on.

_43:37_: And there's also a footnote to

_43:39_: that story which are fascinating,

_43:40_: which I might put something and pass

_43:43_: on to you because it's interesting.

_43:46_: Before we start off with this,

_43:47_: ohh yes,

_43:48_: but the the point is that that's

_43:50_: an approximate solution,

_43:53_: but the smart solution is the exact

_43:55_: solution to the same problem which.

_43:58_: Can be approximated and the

_43:59_: smart solution because the metric

_44:01_: would be approximated by this.

_44:07_: So that was a very long answer

_44:08_: to a question which I've slightly

_44:10_: lost track of, but I other more.

_44:19_: You have a box?

_44:20_: Yeah, we know that photo.

_44:21_: That will be the gravitational well,

_44:23_: yeah. Four, yes, so, so.

_44:29_: Right, so the question is why do

_44:31_: why are photons deflected by this?

_44:33_: And the answer to that is.

_44:38_: That this is the. What we have here is the.

_44:45_: Well. That's the equation which solving.

_44:49_: Uh, what? Asking what is the?

_44:56_: Geodesic traced out by a Momentum 4 vector.

_44:59_: Now we motivated here by by describing the

_45:03_: momentum of a massive massive particle, but.

_45:09_: I don't think we covered that here.

_45:12_: You can also talk about

_45:12_: the momentum of a photon.

_45:13_: Even classically you could talk

_45:15_: about the momentum of photon.

_45:16_: How much I as all the momentum

_45:18_: of an electric field.

_45:20_: As an electric field interacts

_45:22_: with with matter, it will transmit.

_45:25_: Momentum to it, in some cases through

_45:28_: the Lorentz force law and so on.

_45:32_: And that you so you can talk about

_45:35_: the momentum of our classical field

_45:36_: and if you think of of the quantum

_45:39_: mechanics you know you will know

_45:41_: that the photons have have have 4

_45:44_: vectors they they have, they have,

_45:46_: they have energy and momentum and

_45:49_: So what we're solving here is the.

_45:53_: Judic. Of the momentum 4 vector.

_45:58_: Something irrespective of what

_45:59_: the momentum 4 vector of.

_46:01_: So for a massive particle it'll be the

_46:04_: mass times the four four velocity of that.

_46:09_: Of that object for a photon that the

_46:11_: the previous didn't really mean much

_46:12_: in the matter 0 so we have a different

_46:15_: definition of what the full mentum is.

_46:17_: But it's still that we're,

_46:19_: we're we're solving however in.

_46:23_: In this expression for the.

_46:27_: The potential Phi.

_46:28_: The mass here is the mass

_46:30_: of the central object.

_46:32_: That's the mass of your star or

_46:34_: your planet or or or whatever

_46:37_: you're talking about.

_46:37_: So does that sound evasive or

_46:39_: is that the does that cover?

_46:48_: Because of the.

_46:51_: And this this mindset here.

_46:56_: Of the final final. Well,

_47:02_: I think that's that comes just because.

_47:08_: This is our a rewrite of.

_47:15_: MPIBYD. Tall plus.

_47:34_: And I think the whoops,

_47:38_: we're just rearranging that equation.

_47:40_: That's the geodesic, that's the

_47:42_: space part of the geodesic equation.

_47:44_: And so just rearranging that it's where the.

_47:48_: It's where that when you're saying. Appears.

_47:57_: It had, but it worked. A question there.

_48:03_: Lucky.

_48:07_: Because we go back to.

_48:10_: Yeah. And. Each. Here in detention,

_48:16_: because there's no reason why it should be.

_48:21_: All we've done here is, is, is.

_48:24_: Write down the. Component of the metric

_48:29_: in a particular frame. Being the.

_48:33_: Components of the of the Minkowski metric

_48:37_: plus A+ some other other components.

_48:40_: So this is just a matrix equation.

_48:45_: So there's nothing.

_48:47_: We're right in that and not care and not

_48:49_: make any constraints of what each is.

_48:51_: Each could be as big as because we're like.

_48:54_: We are, however, choosing.

_48:55_: The the point of doing this is that we

_48:59_: want to see these are perturbations,

_49:02_: so we want to say these are are small.

_49:07_: But that's not attention thing to say.

_49:09_: You can't really talk about attention

_49:11_: in that context, attention being small.

_49:14_: So we couldn't write G equals ETA

_49:19_: plus H and say each is small.

_49:23_: Because that doesn't really mean that

_49:25_: that that's not a sensory thing to say.

_49:28_: If you're like,

_49:30_: there's a better way of expressing that,

_49:32_: but we.

_49:33_: Are saying we're making this constraint,

_49:35_: this constraint as a matrix constraint.

_49:37_: So it's true, it's this approximation is

_49:41_: meaningful only in a particular frame.

_49:44_: So yeah so that's basically this

_49:46_: is a frame dependent approximation.

_49:49_: It's only in one frame in basically

_49:52_: the local inertial frame that we

_49:54_: that that this we can talk about

_49:56_: these components being small.

_49:58_: And it's then involves a bit of

_50:01_: stepping back and I think about it to

_50:04_: discover that when you turn the handle,

_50:06_: you can review this as a tensor

_50:09_: on a on a on a flat background.

_50:13_: So I think that, that, that,

_50:14_: that,

_50:15_: that the basic AHA is that that

_50:17_: approximation,

_50:17_: that those those pair of things

_50:19_: is meaningful only in one in in,

_50:20_: in in a small set of coordinate choices.

_50:25_: Namely,

_50:25_: those which are which are almost minkowsky.

_50:31_: And and and the there's a dangerous

_50:33_: bend as a section there which I think

_50:36_: was added after veteran requestion

_50:38_: what one of the year which talks of

_50:40_: which goes into more detail about

_50:42_: that which talks about what how what

_50:45_: you're doing here is either talking

_50:48_: about tensor on a flat background or

_50:50_: you're talking about age choice here.

_50:53_: And if you've done a quantum field

_50:55_: theory as some of you will have

_50:57_: done you and certainly if you've

_50:60_: done classical electromagnetic.

_51:01_: 80 and the notion of the wrench gauge.

_51:04_: You will discover that there is a

_51:06_: the notion of gauge fixing being

_51:08_: engaged choice essentially the

_51:09_: the exotic mathematical version

_51:11_: of choosing the right coordinates.

_51:13_: If you choose the right coordinates

_51:14_: then you can do all your calories

_51:16_: in a particular gauge.

_51:17_: Where things make are simple and

_51:19_: this essentially therefore engage

_51:21_: choice in those terms.

_51:24_: And that is time up, I think.

_51:27_: So I have the usual, usual second

_51:32_: officer tomorrow and I think we

_51:35_: have a supervision a week on Friday.

_51:37_: So I meet you some of you