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General Relativity | Lecture 4

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us begin begins tonight I want to just go through a very quick review some equations I wanna move church tonight they hope to get some physics and to discuss gravity if nothing else in EU uniform gravitational field with red-carpet costs about a uniform gravitational field that's just be a normal work that is just near the surface of the earth but the question is how would you describe that uniform gravitational field in general relativity OK we wanna get their understand what kind of magic it describes was describes it but let's just write down some of the 3 members some equations just throughout summer review 1st we talked about coal variably differentiating the idea called variant derivatives act on cancers in particular on vectors and we worked out the call Marion derivative as it applies to particular cancers in particular 1st of all the call variant derivative of course D & call very derivative of the metric itself is always 0 although December 1 more thing about the Korean the plight of the very derivative is to write a cancer such that given the strict such that gal in normal coordinates it is just a derivative in endow seemed normal coordinates the coordinates were truck closest the Cartesian coordinates at a particular point in the space we call variant derivative is just derivative but other coordinates it takes on a more complicated form our if we wanted to be a cancer I'm sorry Kobe derivative but could not just in awe of GM would balance the statements and show a basically special coupled with time you'll military days as best they said is that set in on maybe using may be using dousing normal court who were restricted or stand on knees by gaps normal coordinates that may be using the term to restrict way the duo troubles assume a tremor about wet call them the flat as possible court that have been most Cartesian coordinates at a point now is to communicate with somebody make a series of letters out of that the most Cartesian coordinates report which is close means it means exactly what we said before that the metric at that point at that point is a chronic at Delta and its 1st derivatives 0 0 the bats what I mean by the most Cartesian coordinates at port I may be list using the term gaps in normal coordinates my education was not turf was left something to be desired are sometimes use terms inappropriately but move were to score on the best coordinates they are the best coordinates a point the best score point and derivatives equals 0 in Gaza have the metric having derivatives equal to 0 and the problem of finding it cancer which in the best school at point reduced to nothing but the ordinary derivatives is the core variant derivative alright so here's a covariant derivative it acts on any component of the net and always give 0 White because in the best coordinates the derivative of the metric is a out OK now next we week considered they call variant derivative of a vector got hot hot Dr he S and I chose study of vector with coal variance components Kohleria derivative of the covariance and here's or we found it is that starts out as just a derivative of the S and then has a term which would be 0 in the best coordinates of a point or is it a plus gamma Pa S. 4th Court P. V. P. remember the summation convention was summing over Oh yes is plus or minus I believe it's minus the might and my ears is a convention and we worked out what these symbols or the gamma PRS they are not cancers then made up of derivatives of the threat that means in the best Kuerten it's 0 0 at a point in the best Courtnage because derivatives the metric equal 0 but what they were a cancer then they would have to be 0 in every coordinate and the lots of Uganda's what's right and then calm down P AS is equal and PRS is he called G P N reward GP In inverse metric
metric except with upper components times partial on G and are with respect to X S plus parcel of Jean guests with respect to rule X Miners partial of G of as a factor of 2 Excuse me a factor of 2 murders partial GRS with respect to x shouldn't take it at this this should be memorized it doesn't take too long to memorize it the GP and OK that's just erase some index but you look at it has G N R it has G In and it's as GRAS that I do this right arm 2 chorus with respect the P not or with respect can end there's not right it has all 3 combinations of blah blah with respect to x Schlafly parents becoming 3 key combinations 2 within positive 1 of the negative and a half year that's offal symbol was 1 of index in 1 Lauren decks and benches goes right in the year that ends the makeover derivatives rather complicated objects but what barometric for you it's a nuisance writerly things down because of what it means by next let's talk about call variant derivatives of vectors with contrary various components serve with scored D and beat end M and as always it starts out with a simple the river but I'm not going to work it out for you are not going to work it out you work out the same way you were about recovery derivatives of covariance answers but simple trick you could do it yourself doing Hallmark value with the trick is you begin by writing V it would operate acts as equal to GE and Danilo ever VP and LU say I want to construct the cold Mary derivative of Vienna and Sosa call them derivative is a derivative it's a derivative In the best frame it is a derivative it will satisfy here this is a product of product of 2 things and it will satisfy call variant derivative of G & P P Plus Jean and tee times call variant derivative Our VP of index down here and in bracket around here with this says is simply that a derivative of a product is a sum of terms the derivative of the 1st term as this 1 times VP plus a derivative of the 2nd 1 times G this holds for call derivatives that calls for just about every kind of derivative you could ever think of the mail what we know about the derivative of the metric now this is the co-vary interim remember the metric In the best coordinates has no derivatives so in the best coordinates the metric itself of Laura disease is constant at least the 1st order released a tour the 1st derivatives that means In 1st cancer the inverse matrix Is also can't the 1st derivatives and so would also argue that the call variant derivative Of these metric tensor with 2 contrary indices is also 0 we know differentiator think with lower indices Rice's formula Over here just plug that in here and manipulated a little bit you'll find out how to differentiate covalently differentiate a cancer with upper indices alright formula for you areas it's the same kind of thing as appear to live with respect the emir of these and the river with respect to X M V and they always start out that way and then burial plots this to work out but doing this little exercise Here simple little exercise fix a few minutes now plus gamma in end call P I guess what the accord to keep my notation consistent in em are know your are br OK let's look at it 4 minutes as before it begins with just a simple derivative that has turned which would be Zero in the best coordinates because the best coordinates began as a row 0 0 but about 0 in general coordinates this has 1 or index 1 next here it has won on a book upper index has same upper index here he has a low index contracted with an
upper index so the formula makes sense it satisfies the roles upper indices contracted with sees summed over and indices generally in place if plus as what's that plus that's the Luo peculiarity that get by doing this exercise its blasts that's all you really have to remember that when you go from cholera in the country during the reform the way the indices fit together you can mistake a lawyer and that another index that Saint Laurent access to appear on the side of of square there's nothing else economy but the law index of because Stossel symbol the upper and access to appear here and there has to be a contraction OK that that's called variant derivative of that contrary Asharian components of a cancer now we can't an idea called parallel transport River ready moreover spoken about it but let's however detail pools what did supposing you have occurred space tough all I sit right down 1 more than just To down from Merrill the curvature tensor figure walked we know we will you too much that I will be back and now we have some vector field let that be a vector field our living on the space and I'm interested in knowing whether I want a long here with the field parallel To itself were vector fields stays parallel To itself I have some vector reached point here at each point along the curve and I want to know what the field studies parallel to itself power will to itself means that the call variant derivatives your along the direction that's what it means call very derivative it is the difference between the vectors except Britain In the best court if a derivative of the vector from 1 point to others bad sector is maintaining a relationship of being parallel to itself as you move along OK so that's actually a definition but would do we have written down a right down again the Cauvery river of course use it D. M. V. ended a strike found for more physical parcel of me and with respect to X M plus emirate in em aura the ah that's that's the Ecole derivatives now I want to consider we derivative along the trajectories euro along curve how the vector changes from point to point that simply corresponds to taking the call derivative the and multiplying at between these 2 points by D X be acts of I differentiate with respect to X M and then multiply like the X M that could just be called the small change in the vector going from point the pouring the VX except keeping track are the fact that water niche themselves may banned as you go from point to point and that's for you have to use a core variant derivative This is a change in the vector V I'm going from 1 point to its neighbor was equal to watch just courses given name D V it's a small change in the back going from 1 point to a neighboring point more precisely it's a small change in the sector In the best quarter of about 6 hours equal to its equal to DVD and by DD X and times DX and I'm simply multiplying this equation by DX em With plus gamma In In d X and I did nothing here except multiplied this bike and some over M so let's erase the new expression he misses the call variant changed in a vector going from 1 point to another now this thing here that simple but derivative of something with respect to X X back in just be called differential change and that's differential change in 3 voters equation we call variant differential change in very busy ordinary change in B plus the purse powerful symbol multiplied by DX This is the formula which tells you how any vector changes from point to point now supposing I'm interested in the finding of vector which is parallel to itself as move along risk Kerr parallel to itself means it doesn't changes you along here at each point your erect some best coordinates in those coordinates to test whether the vector is changing not say I vectors constant along a little segment they go to the next segment direct best coordinates that pouring through the same thing if it every point the change in the vector is 0 equals 0 vector is said to be parallel to itself alone that occur as you move along at specific curve the vectors maintains a relationship of being parallel to itself were always worked as well known unnecessary once you find the best set of coordinates you could rotate it now they don't have to be parallel to the curve but they have to be fixed and well-defined but they could be parallel to occur the best admits not unique just like an ordinary flat space be Cartesian coordinates not unique because you go always rotate
them you could also rotate the best caught out point that's that's some report that won't change worthy of vector stays Parra of to itself Moody a vector from 1 point to another such that this call differential change is 0 0 is called parallel transport sector if you move vector along a curve in such a way the 2 differential change in its combined its Contra variant component plants the Chris powerful symbol multiplying V times DX along a trajectory that being 0 is called parallel Lee transporting the vector along certain curd now funny thing about curved spaces is if you take 2 points and you start a victor over here and parallel Italy transport along a curve you get some of that here but you choose among the curve and Carola transported and general O'Connor different so the notion of parallel transport is relative to a particular Kurt is unless showed you about color and we talked about how cold around copper something like that every parallel transport sector from here vs. transporting it around the back of the call you would get a different final result payroll transport hot assad but young fats and of course when that happens is be cost is curvature in between so that I think it is easy leaders stay here is that if you look at what I said was for a vector field but now just imagine that you could imagine that it was a vector fueling could just imagine that the vector field was only defined along recur In fact we don't even after we can even asked how do we construct event along with curved such that everywhere parallel to to itself along the curve what we do is we solve this equation we solve the equation that when we move the vector we move it in such a way that it's cancels the president if it were a field we didn't test yet I would like to assure the vector field would be independent of the path prepare might be 1 path along which the Spanish is everywhere and another path along which it does when you get to me and you have a unique cancer 1 transport of the vector I would say that the sector was coal variably constant along 1 curve and not go very becomes constable America now would be a right with the finding something slightly different instead of saying a vector field Webster suppose we start with vector over here and now I would have show curve up with vectors it is but and filled up so roughly speaking as a vector field now along curve votes not worry about what is going on here and just require of the vector field along the curve for those caught but Cauvery Leconte and this is the equation says that there is get yah yah that's right core areas and you could prove that this equation conserves the weight the I didn't assume it but warned could prove it from here Young Club does it there will be we not but it would matter unified Nunavut later struck a thought specify curvature is in here but didn't make it nice surrounded so that has a finite curvature of the curvature of the tip of a sharp cold would be infinite so I want to avoid that not a lot not only is the curve is a Jew best that's exactly or coming to a bribes defined so what those slots see what's going on here let's take the call slice it an open up on the blackboard preparing twice an open up works from various media here is be out here now but stated curve that just goes around like that that corresponds to exactly the kind of curve drew over here right goes around and comes back to the same point over here at this point and this point OK l let's take a tangent vectors said it's pretty clear that this tangent vectors changing what would a vector which didn't change quite a vector which they change would look like this there's just like this for a comeback yeah come back to misdirection over here so the answer is not 0 in general be be tainted sector to a curve is not call variously constant but did factors knowledge is that this is not a geodesic call Vatican see just flat and everything out this is a circle however argued circuits novice shorter it's it's not the shortest distance between these 2 points this between 2 point that drawn back on the court and it would look something like this the shortest distance between those 2 . OK so now but now willing to write poetry said question of the in sector and the notion of a geodesic in tour Passaic Suri said I Abbott the notion of a geodesic geodesic is a curved ones defying it in 2 ways you can
define it as being the shortest distance the shortest occur between 2 points better yet a curve whose distance is stationary it could be the longest distance between 2 points at the distances stationery and then we will say that occurs due bastard but abandoned and don't necessarily better definition effectively gives of definition of that definition Mr. locally along the curve and require that the curve be as straight as possible al-Qaeda whether that are give you I'll give you some intuitions and give you a precise statement event each point along the curve got a lot the derivative tinted Vector 0 0 Van that occurs Estrada's possible but let them let's think go what this means I'll give you a year in tuition outright Ecuador's chauffeured supposed curve terrain and Meryl this is a two-dimensional example but there's nothing particularly two-dimensional about the notion of a year overdue Bessix now let's suppose you have a car you drive on this terrain also assumed that the size of car particularly the distance between the 2 wheels is small by comparison with any curvature of the car is very small compared to the hills and valleys and so forth and now take a steering wheel and point that absolutely dead ahead straight and start driving dunk Turner we'll make sure advance there was pointing your Brody mechanic straight ahead and don't deviate just go straight ahead of the curve that you will execute on this space will be a geodesic but another way to say the same thing is that the change in vector along with curve is constant change vector along the curve so let's define engine that it we have but it is it where workers Bonalbo pouring Togo we're aware struck 2 pins in that the right of the point and taken bring port is separated by little interval called D X and paint small and now draw a vector through those 2 points make that Victor B units at and then take a limit it which the 2nd point approach is the 1st point that's cold the Tianjin vector now I need 1 more equation here remember that and where our case Charlotte so what she got there where did you construct attention that was very simple His eagled the D X and divided by something divided by the distance between those 2 points just divide by the distance between those 2 points and that's what we called DES Lenore Diaz's DS squares of supporter of distant GM in the X and PEX just provided by DOS that defined a unit vector you could prove it to you vector or you do that it's a unit vector at every point along space along a curve call the tangent vector points some direction between 2 neighboring points directions specify but to neighboring points and linked as 1 so that's the emotional attention back down and now about the motion of engine that there is constant that's a statement that plug the team's invective of V the right-hand side on the left-hand side of this equation should be 0 so let's write that equation plugging tea and that equation says that D T M quote but plus gamma and aura in T R D X and let's say what I did all I did was plug-in for V T a D DVDs and V Aditi entity as such said equal to 0 both each pouring and checks whether as you move along curve this quantity is 0 everywhere as along with current if it is the curve dude that's become response to setting a steering wheel dead ahead and moving in a straight line as you can as the motion you dissect right it will need a former writer slightly deformed let's start divide both sides of the equation by D yes but a little distance between neighboring points here with differentiating which means we take a will a double here DX on divided by D D yes says that derivative of he and with respect to distance along the curve misses the derivative of the Tianjin with respect to distance along the curve plus but suddenly quit minus is equal to minus Ghanaian neck and are TR but then the exit and by DOS Wozniak somebody yes M so I will equation
the only involves the engine back they also of course involves they Kristofer symbol but suppose we given that but suppose a given the core off or symbol and the equation of motion of G desert Justice equation Gamez are made up of a metric suffer another matter we know what the right hand side because they 1 what since he itself is a derivative which writers as the 2nd derivative these 2nd acts end by DES squared I've that P His DXN by Biesinger differentiated began the year square the 2nd derivative of X and along the curve is equal the minus Garrett and and Aziz look like anything familiar probably not but it is actually if we want to think of there kind along the curve some measure of time as we move along with curved than this thing would be acceleration 2nd derivative of the position with respect to Parts if we imagine that s is like kind will combat so fans will like time or if elsewhere increasing uniformly with tar and that this will be the acceleration the foot high acceleration is equal to something that depends on rhetoric and he and those somewhat easier who deal with them later says the look on a new equation acceleration is equal to something that depends on the gravitational field the metric is the gravitational field so missed the equation that replaces Newton's equation has also Our for the motion of a particle in a gravitational field some sent a particle gravitational field moves in the street as possible trajectory now most of the greatest possible a crevice trades possible trajectory based kind not just through space so now I have become the space so far only just starting the mathematics of curved bases curves spaces as remind would've understood mathematician remand was the 1 who invented most of this most of the mathematics for curb spaces ordinary Kurds spaces in which a distant was governed by the Pythagorean theorem and always the square of the distance was always positive cast east is space which also has a natural measure of space time and distance along curved spacetime distance between neighboring . 3 points Minkowski space are it is a space with the notion of distance between neighboring points in fact most of the square of distant distances usually defined the Romani in geometry in terms of its square on in
Michalski geometry again with notion square distance between neighboring points was a call the proper time between points the proper time proper time between the points is a letter also notion of neighboring dissidents sorry give it to point the space-time T X is based on distance between Roberts quotas based on distance scored a proper trial is equal to DT squared not plus BX squared minus the X squared and if the or more subsea y and z would also have mind as to why square miners deceased word that's called Bacau square was a proper time eh it's a conventional to rewrite as minus D yes squared sometimes use sometimes how I'm going to use biggest town but according to the convention dishes evil minus Genia no X new VX nerve was at Maine GE Melido is the metric of space X mule stands for hello LOX new since the tea x y and z and according the standard convention he is called X 0 0 and X X 1 why is next to and Zee is Equifax at convention so you'll you'll run from 0 to 3 the says exactly the same for exactly the same form using new window because its conventional when talking about space time use viewing the only thing is the metric tensor strike down record cancer of the metric tensor now has a minus 1 that corresponds to Didi squares notices from either side here 0 0 0 0 0 plus 1 0 0 0 0 0 1 0 0 0 0 1 that's the matrix OMB metric tensor Jimmy I was really read minus deeply square plus B X squared plus 1 square plus busy square the world yours doing here zeros of doing telling you that there are no cross terms DPPX PDY anything like that is a special coordinator he's a Minkowski coordinates and they play the role in relativity Cartesian Quarterdeck's and special relativity this is all there is to it in general relativity genial becomes are function of space and kind was just called of X becomes a function of space and time not just a functions space a function of space time but 1 thing what did you say I am always is this a lasers matrix different then Delta Nu 1 1 1 1 away aliases as a minds want place but there's also an invariant concept in varying concept as this 1 has 1 negative Eigen value and 3 positive ID the magical space-time always has 1 negative I get value and 3 positive I was when I can spend much time dealing with that equations always are take care of all of that boat it needed vectors 1 negative Oregon value and 3 passes bath in needs as one-dimensional time 3 dimensions of space ifor right a metric of work for years to minus signs you would have square outside scheme that will have to positive and 2 negatives that would be allowed in relativity would correspond to a crazy space 2 kind directions at once and to space direction Einstein realized that the metric of space time should have 1 negative and 3 positive I you but this is no you don't need to earn money to worry too much about that other men that all of the equations of the same as we From that decisions like this you know I don't think so I don't think so not to my knowledge our death there somebody who knows a little bit about the history of geometry but to my knowledge no so I think Minkowski and buying orders from as I know the first one to explore these were young players on that a yard of like it is wrong the no I got it right the still minus signs these are both conventional minus team you know right young white just in order to keep to keep my notation consistent with where everybody else so strong don't Bert spoke at other matters said Everything is exactly the same way you just if you want to know whether the spaces flat was flat flat does not mean that there was a recording system in which the metric is a crocodile it means that recorded system in which the metric looks like as chronic Dr. except with 1 negative sign that the notion of a flat space-time flat space-time is 1 with 1 negative and 3 positive but which do exist court which the metric and before brought it is very simple form that's the version of a flap space time and space time cannot be brought this form that its car have you check with its current you do all exactly the same things here it could be cremated what you Kurt factory would be basically the
same thing with Our is specialists will let you have gas comp 8 new known as storm that's a good point on right if we would dealing with these coordinates we're right a them you know new DX known the general equation a general equation is to put the genie of exit the special case in which a dealing with these special court flat coordinator of the tawdry for use by coordinates it which is dealing with a special Minkowski coordinates which elect Cartesian coordinates is this is called a them you know time to pocket so that's what we inherited from special relativity is all inherited from special relativity OK OK everybody clear they mob work good thank you went Rock the top quite so for 1 0 do special relativity words would deal with the flat space with the flat space or going of looking at it in polar coordinates not polar coordinates as you Euclid were use them but polar coordinates as Minkowski would defined the purpose list of defying the notion of a uniformly accelerated reference frame special relativity there's a bit of a problem with the notion of uniformly accelerated reference that you have a butcher points which are a 6 separation and you start accelerating and you keep them at the same fixed acceleration or accelerate with this would you uniform form acceleration than they will maintain the distance between them but distances are supposed to transform when you are when you accelerate something so if we were to just stop these points moving with a uniform acceleration will actually discover that in the rest frame of the points the distance between them was like getting larger and larger larger and larger that would mean for example if were strains between them Hi that as they start moving in the tend to keep their distance uniformed the strain which stretch eventually break that's not what they normally would be thought of as a uniformly accelerated reference frame what's nice about a uniformly accelerated reference frame his knowledge of the distance between points stay the same but if you have strings connecting the points they wouldn't get stretched all sorts of things like another thing about a uniformly accelerated reference Raymond in special relativity if you wait long enough uniform acceleration will lead moving with great speed of light so you'll from acceleration to the extent that exists and makes good physical strength is not as simple as just moving these point all with the same acceleration I'm going to tell you how to construct what a relative this would call a uniformly accelerated reference but to do so I wanna go back a step forward or back 2 Euclidean space and talk about polar coordinates because these uniformly accelerated coordinate system is analog of Pola Court that's OK so let's start with ordinary space because when I want to introduce bombs full-court both accordance me an angle at a radius of there's a point that I can be characterized by radius and angle and some core equations withdraw from everywhere but but but but I thank you those equations you know that call sign square precise squared sign squared data is equal to 1 which is if which is the same as saying their X quartered plus Y squared physicals our squared finally tour equations do call sign of it is equal to Ito are a freighter plus Ito minus I favor over tool and science data equals E 3 I favor minus each the miners are they you could check the call sign squared besides Queiroz acquittal just simple identity for what possible say these are the basic equations governing public ah yes in this 1 but do and I am very good OK now what's the equation of a circle the equation circle is just hours equals can't stand a circle and then to the point moving around with uniform velocity uniform angular velocity with uniform angular velocity the acceleration of that point is uniform on circle is uniformly accelerations uniform on Darfur from what direction as a tourist . 4 per cent of the magnitude of the acceleration is constant however this have to do with relativity relativity basically the same equation to define a uniform accelerated point who if it if as tho I call was a uniformly accelerated pouring in and in special relativity at some point morning on it is accelerated its not moving with constant velocity constant
velocity would mean that had not the erection would be fixed could it has no velocity over here it's moving straight upward this time Hi Mrs. X at this point the velocity of 0 at this point is moving forward as it moves up and up and up to get closer and closer to the speed of light but never exceeds it and the question is what is the equation had we What's the analogous equations Ferrer on for hyperbole is basically as a hyperbola let me just right books call sign and signed data are replaced by the hyperbolic versions hyperbolic call sign and let's call the hyperbolic angle omega omega is what increases as you move along promote its car angle plays the role of the angle in you could face a measure ofthe along the curve yeah along hyperbola misses hyperbolic signed Omega the definition of hyperbolic co-star inside very similar this is equal to Ito the each of minus major all over 2 very similar to miss a start that 1 and this 1 here is Omega minus Ito minus Omega over the symbol Aurelia means divided by 2 know why in this 1 no hyperbole and hyperbolic geometry feel check very easy that Koch square all Omega my scored is what but while the synch called cinch an hour's service equation and now the folly
saying state analog of this equation X physical too far crash Omega and T and teach the vertical coordinate physical tar center maker this is it is actually define our animated is the definition of armor maybe if you like it's a coordinate transformation to coordinate transformation from ordinary ex ante flat space sex until you're just a Minkowski coordinates ex ante it's a coordinate transformation to honor an Omega which 1 do do you think is like trying To tale well if you're on the horizontal axis where car show make it is just 1 of the horizontal axis Castle makers just wondered if you increase are you just going out along this axis Our is like a space court just going out along this axis you increase Omega from this point you traveling upward in an analogy with traveling around the circle traveling around the circle the allergy is moving upward alomar hyperbole so I'll make it is like trying to time like coordinate and are space like what now just arrows there's a uniformity of the circle at any point you could define are constant circles you could define aura in this case is constant on hyperbole was so his ahead a block Oracle's tool is irreparable Oracle's 1 is inoperable Oracle's 3 and finally if I take X squared dynasty squares but along 1 of these hyperbole it's just given by our squared times cost squared minus each square but that just 1 so I hyperbola it is this is this hyperbole is just fixed are constant are let Omega vary the angle vary and that defines a hacker black square modesty squared equals or scored so that defines E court was coordinates in special relativity are the closest thing that exists to uniformly accelerated court this point here was accelerated trajectory this point accelerated trajectory the distance between hearing here it is the same as the distance between here and here Pentagon check is the same as the distance between here and here that means that as freedom as the poor frame of reference rotate rotate it not rotates but accelerates as friend reference accelerates the distance between neighboring neighboring particles of think of these as a sequence of particles the neighboring particles Davis what's unusual here and different ordinary ordinary accelerated frame of reference is it be acceleration along these trajectories is different what's look worker trajectory which comes in very very close in here very small it makes us suddenly a sudden change of direction a sudden change of direction that indicates that this trajectory has a much larger acceleration in incidentally the acceleration of proper acceleration the actual acceleration of observer would feel everywhere is along the trajectories uniform saying acceleration here this year as here as here just in the same way that a particle moving a curfew refused saying acceleration as here this year the acceleration along the trajectories uniform the a proper acceleration proper acceleration was a proper acceleration lean and mean acceleration felt by somebody over here it on job 1st feels being being accelerated over here uniform all these things along hyperbole was everywhere is the same but the acceleration is different here here here and here these message is that if you wanted the fighting uniform acceleration you have to have you you acceleration which is constant time is OK but you necessarily have different accelerations at different points of space different aura but after never exceed the speed of light and the proper distance the proper distance between points stays fixed Lemmer proper distance of Riyadh but that call the hyperbolic angle call the hyperbolic angle it is our megawatts on mango doesn't grow from 0 to 2 . 2 separate for Simitis affiliate with Yao Yao proper acceleration yards of the derivative with respect to proper time proper acceleration would be be 2nd ex New not by DP squared up by DES squared where would be on what Dudley tell square Excuse me because square that's an awesome proper acceleration you measure the acceleration not with respect ordinary time but with respect to the proper time if the rate of change Of the Yom room of the proper velocity which is DX by decoupled with itself with respect to a crew now everywhere is the same along curves but differs from 1 point 0 a further at you are further out you are the smaller the acceleration straighter the last curve the trajectory in fact factor come right an equation for the acceleration don't for the proper acceleration certain the proper acceleration ADR long-sought trajectory depends on all in 1 thing it depends only on ah for Kate depends on are in units which is global 1 the acceleration less let's take a particular curfew and core capital R capital are represented distanced from here capital on of the Capitol hours just a particular value over Laura along that hyperbole the acceleration is given by 1 all over Capitol aura the smaller capital arm that it's actually the same things going on here a small little are from the bigger the acceleration needed the hold it in place pockets of the accelerations 1 or more on this doesn't make good units sent crude high with units of acceleration length provided by Times Square write a prank but strike that has linked squared provided by times Square comes 1 overlay OK so acceleration has units of 1 comes velocity squared if we work in general units not the the speed of light people wanted the acceleration here musical to see squared off that means that for a fixed on the acceleration is very very large you have to go to a very very large all our before a before you're talking about a moderate acceleration because of the square of the speed of light your cushion young hours the exodus of the liquor that's exactly right and this acceleration is exactly the acceleration of ordinary acceleration and you experience point will inordinate that curved space-time trajectory the more curved over
here versus you're here more acceleration so I'll use equals 1 generally but keep in mind that this acceleration is very big omelets are itself was very each they were tour will forever Gianna work they're just that was correct that is correct that is correct a uniformly accelerated reference frame is defined relative to a particular origin this traffic that's right now yes is a slap better find no real physics depends on her table with and were introduced in the somewhat arbitrary coordinates and not arbitrary coordinates with a right the equation of motion of the dude will see the equation motion of dude that's looks like a particle falling in a uniform gravitational field marked only if we don't really work it out in detail roster with doors will get there will get yes you're right this is 1 of many possible coordinate systems where you could put the origin any place you like and for that particular coordinate system we have a well-defined maker but we could move the origin if we like but that's not what's not what's keep the origin fixed OK so here is the acceleration our now let's talk about the neck with Saarbrucken metric 1st the metric in ordinary poll coordinates metric and polar coordinates ordinary distance square that's a goal to be a martyr squared plus what would remember something evolving favor squared office fears is sphere the plane to think about the right assist are square dictator square plus Belo square playing this just says that for a fixed separation Bader the actual distance along here increases linearly with are so forgiving theta the dissidents associated with these data increases linearly will not therefore square evidence distance increases quadratic this is electric of the flat plains in polar coordinates no notice 1st of all the metric is not the chronic doctor Dr that's 1 thing that has ah square dictator that's our scoring Prueter right two-dimensional met for 4 hours so look like this would be the crocodile pair of hazardous or square here position depend white because the court curve space not curve completely flat space curved quarter 3 M a large here is cow squared physical tho are squares deal maker squared Miners br square now take in only 2 dimensions for the moment than going to ignore y and z will lead wines now would be thinking about particle falling in a gravitational field along with vertical axis will be the X-axis and why you will be via the coordinates and they won't matter for the problem discussed the problem discussed it is just major time and distance from the origin time and distance from the bathymetric I don't see what that metric has to do with gravitation that because their gravitation supposed have something to do with a metric so their remember as remove all work here the acceleration is changing if we go very very far away yachts let's think about place far enough away that acceleration is pick an acceleration a good acceleration about 10 meters per 2nd per 2nd us but far enough out here that the acceleration G 10 meters per 2nd per 2nd accelerating the right are where was our formula are formula was that the acceleration is equal to 1 or more of our time see square however said that equaled digiti so that means ah is equal to see square Omar G. we have to go very very far away measured in were CDC is a 100 6 thousand 3 times and 3 times a year leaders per 3 times tend to be a itself 3 times 3 is 10 times France 10 16 the number here is kind of the 17th this is and little thing is the 62 at the end of the 16th meters before we find the place which is accelerating the gravitationally ordinary gravitational acceleration but let's go out there moreover if we don't move too much along the art direction this acceleration won't change much so let's introduce wet about 2 weeks on a certain distance are given by this we set sequel 1 of course envisages philology but we have to keep in mind that 1 over G corresponds to a very very large dissidents who won his thing which is big it's really see squared let's go out to a distance are where the acceleration is G and then think about region a regional which is closed this is corresponds to be near the surface of the year not allowing the things to change too much so sugar mobile Lou away from our call the distance along that why a call the wine next time so the distance from this hyperbola space like distance from this hyperbola His wife of the specific are below at that at that place no words I'm writing that Arthur little our business eagled the big garlic keep thinks plus why but making no coordinates these coordinates a sort tailored to be around this point which is moving with the gravitational acceleration G the neighborhood of that point now right we write the metric rewrite the metric rewrite the metric our state liquor Leary right ready just rewrite it D. Kyle squared physical squared that's a slew of square but that's big are squared plus 2 are why little square at all times be Omega square my str squared so now I haven't even more complicated looking metric year buds so little more complicated not much inherited focus about the region of OR so let's divide out far from this Friday this formed 1 plus 2 white over are plus Y squared is over or squared are school here and factories are squared out of here and there were multiply this are squared deal maker squared minus squares 1 last step on call on times Omega bigger our times Omega on any given new name I'm Cohen little chief I've called the original time here has been achieved but topped called this thing will be wearers persist we and now I get a metric that looks like this 1 to Winola are what about this this is much smaller than its remember that Arsenal enormously big
are was how big a enter the 16th the leaders and while I am assuming that while using or a couple of hundred feet or something we're talking about falling in a gravity talking McCall forming a gravitational field so why is a small number compared da y squared of ask where is much much smaller while are a small 10 feet around 100 meters how that that fall 24 miles that tiny compared to attend the 60 meters shrank and NY square of our squares even much more somewhere dropped at home John the hope onto staff and the end of the day we get metric apart from this which is small just looks like Goodall metric bt squared minus squared it just looks like a space and time tomorrow as ordinary way be space squared minus D Sergey Times Square might space with over the loo correction a little correction and that little correction is what accounts the gravitation in this accelerated reference that 1st of all keep in mind that we're really talking about flat space so far we have not entered the city curvature was really so any gravitation that we find here is the center of the same fake gravitation that we find in the accelerated elevator were studying physics of accelerating court system it's the elevator polled in this direction he where we expect to find we expect to find it in that elevator there's an effective gravitational field effective gravitational field is associated with this so he is associated with us over here and let me tell you what the connection is now and in the world maybe instead of a young baby cast of telling you will do was worked out the equation of motion of a particle in a metric like site killing and that's all I was so it's really simple all was Oregon how harm is related to the acceleration when we write down said that the acceleration is C squared all aura the 1st see equal to 1 but that's equal to 1 and all GE GUC square on the gravitational acceleration we chose are so there was accelerating with 10 meters per 2nd per 2nd at Davis this equation in 9 seat was 1 units this is just 1 hallmark 1 of is Boris G. report I work or we have we have just 2 y terms G never seen the expression white times Jean studying gravitational uniform field that potential energy white and ashes are quite the potential energy it's the potential energy divided by the massacre to score the gravitational potential right it is white times G is the gravitational potential and this is 1 plus twice the gravitational potential at extremely general in any kind of gravitational field as long as the gravitational for you moralist constant with time doing anything to radically relativistic this coefficient multiplying DT squared is always 1 plus twice the gravitational potential while I close gravitational potential other than that it just looks like a gravitational potential because of my work out of the equation of motion of a particle in this metric I will find the equation emotion essentially as long as the particles moving slowly as long as the particles moving slowly as long as as good Newtonian approximation as long as things I'm not to relativistic there will see that that particle folds along the Y axis like a particle uniform gravitational field so is a basic metric wanna find out how particles moved in that sector what the role of particle but I don't know yes sorry it did see the article be oral why you're absolutely right we could set the sequel DeWine LYZ because br is equal to be 1 report scored right so riding the metric in the white sheet coordinates this is only good for the vicinity would make some approximation here we threw away a terms Mr. was very small is only good in the vicinity of of his distance from the center our was only good the region where the acceleration is really duties says they will let Eddie F. F. because could approach yes until he sets this is a confrontational trade out EDS because exhilaration that that correct could they will always be privileges but they want Bassam truckers that Of course you got so this a uniformly accelerated frame everywhere but that the same but that's exactly the same ordinary 6 as a uniformly accelerated frame with the origin or he is uniformly accelerated frame larger over here are the largest of the budget different elevators whose Florida at different heights at a particular instant that's what's going on here are and we can think of this curve here is being the floor on the elevator elevators being pulled in that direction now sort of elevator outside elevators pulled this way somebody's experiencing gravitational field being held there I'm sorry arms our right star of elevators being pulled that way we can take this quarter net all are to be the bottom of the elevator and the height above the floor of the elevator will Carling Wadi Wise a high above the elevator and here's metric were delivered use metric of space time in the elevator a uniformly accelerated elevator we had to do a little bit of work to get their act but that's where leave metric tensor in the vicinity of a point accelerating with the speed of gravity sorrow the acceleration of gravity on New Year's field but OK now what are the rules about particle motion could can most of you probably know that the rule about particle motion of particles geodesic not yield 6 of spades the kid that success based on another word we take the metric of space time whatever it is only got exactly the same operations and here is the equation of motion of a particle this is equation emotion was the equation of motion which says dude that go straight ahead but not straight ahead in space now but straight ahead in space time but otherwise the equations of saying that so much right now there were slightly differently these 2nd acts in but we don't want to to armed OK squared this is called a proper acceleration as long as the elevator is moving slowly roars down here it hasn't got it hasn't been accelerating long enough to get up to speed of light then the proper time the ordinary timer essentially the same and this is just ordinary acceleration this could be what would we choose X to choose extra BYD a white component of acceleration along direction so this would be the white component of acceleration but it's a left-hand side and the right inside minors gamma them In component stands for why this is white component of motion so we're miners gamma why 0 why direction and R and T T it is D X all by s X end by OK now fortunately there's a whole bunch of now and and OR
could run all Moller possible coordinates 1 through 4 so it looks like what was basically can such contributions but most of them are extremely small as long as the elevator is moving slowly and as long as d objects that were interested in namely the object the particle who scored twice as long as all of the emotions law the only 1 of these combinations is significant how big is he seemed IDS a slow-motion essentially 1 his essence theology Bacau Excuse me this should be town powerful relativistic case these other components of 4 velocity but the components of 4 Molossidae particle which is moving slowly the only component that's important is the time component kind component of velocity was at the scene were the prime component of ORC Watts is what this is for the derivative of time with respect to time this case through plan suspect the proper time the proper time saying so of the tools entries here with and in standing for time those of big and they just about warned that both equaled 1 was about a space component provided by D town will last portion of the actual ordinary spatial velocity was filming the spatial velocity a small small compared to the speed of light so the only important contribution he comes when honor and in are kind components and that gamma time time peace 2nd wife peak hours is equal from early this gamma warned that must be the gravitational force the Cristo awful symbol must be the gravitational force must be the year of derivative of the gravitational potential energy so let's go back the definition now here it is up here here's a Chris awful similar were interested in that where we need we need a 1 was time components and 1 white comport stores 0 we get what I have here a minute ago but he was a ski there was 7 they not it is not right now Our next yacht OK a throwaway some small things OK so GDP and was as pedia this year was supposed to be wide so the GDY something GAY something but they are only GDY something is GYY and that's 1 only remind us for our rules signs because we're tired not bid UYY is eagled the plus 1 so the only Jedi the only GE that appears here is G why why only component of the metric the that has a white index this was your wife really get Healy a parcel G wiII something with respect to was the star of these abuses most be time components post BY component GYR With respect to trying as for the same thing GYR 0 there is no why are component such zeros only 1 component is only wanted this 1 of you work about a gives you might derivative G Orin S. was supposed to be kind components services tight time divided and what is In the end is supposed to be wise this co-star staff symbol is nothing but the derivative a minus the derivative 1 Hey after was a one-half on him of derivative of the kind time component with respect Why with a factor warned him that what's on the right-hand side of year now but figures should be plus derivative From gene prime time with respect Why does is look at all familiar that the right-hand side of the equation emotion contains a derivative with respect for wine potential energy right forces Eagle derivative of potential energy so somehow GTT must be the potential energy but look at His GTT right here duty to this terms constant doesn't have a derivative but this trend does have a director with respect worried the derivative this term of respect the wine just to so this is just equal to all it to a minus surplus it is a modest minus cancels reserve is visit was a fear the was canceled and there is a minus sign on onto turret to remember where the rally Yarborough minus sign came from the fact that would be How squares muggers and my GTT the derivative GTT with respect Why is just 2 G curators have Morozov equal to its go from minus G S equation of motion of a particle in uniform gravitational field so we would go we went through what is a rabbit complicated derivation mobile we learned as a following that 1st of all of was and metric can have fairly complicated structure and arbitrary coordinates a uniformly accelerated coordinates has extra turned and the equation of motion for Odessa at least as long as things are going slowly and along with Tony an approximation equation of motion for a geodesic Newton's equation eyes in a uniform gravitational field uniform gravitational field constant acceleration and that's what we expected but to do it up properly using metric and using samples symbols of all the things that a fairly complicated procedure Einstein of course of guest guest all of this is the opposite direction he didn't know about Crist offal symbols you know about he knew about uniform accelerated coordinate systems and some along here is somehow the place where we started that don't know exactly how we started but along here this was the observation that that
motion is due de 6 dude 6th in a particular metric and electric of uniform acceleration just looks like so if comeback with come around full circle From the 1st lecture where we talked about Her accelerated elevators giving rise to gravitation and we've come around full circle but so far we haven't gotten to real gravitational fields is not real gravitational fields because they really correspond to flat base up there if we want to take this metric and calculated B curvature tensor it would be exactly 0 indicating that they do exist coordinates with a metric as a simple form like that so this gravitational were experiencing here is really exactly the gravitation due toward an accelerated frame of reference nut Due to real gravitating matter assigned guess what we know what to say effective real gravitating matter would be instead of putting 2 wiII fans little G. What's the gravitational potential and say we have a gravitating object and again use Y coordinate over here now the for 1 right so we can it we can expect the more we study of actual gravitational field are gravitating object when get Newton's constant here divided by white America and that's the shore chilled metric but not quite a workout ashore chilled metric as good no he made a mistake with the potential energy of Arabian and over gravitating object try again for you have sovereign right so get but that's a weird fears that means that the kind the of greens when White is the largest a small of God because it's positive but something crazy happens at the point where surrogacy about to Jeter's Fuji and the rest of the gravitating object something happens when is equal to 2 GM is coefficient is 0 something very peculiar the young officers to do the horizon of a black hole that cracks are plug-in card Colombian a gravitational potential he guessing guessing that the same kind of thing would happen the same car thing what happened and it does GTT feeds GTT will be wiII that is just the derivative the gravitational potential so are equations to work out and look like a falling gravitational field we want to plug in the gravitational potential here GM over y but then we find something really on this coefficient vanishes somewheres vanishing is the horizon of the black hole next time we're going to get in to the subject of a short shows metric will not get the right now we need field equations to derive if we don't we haven't discussed field equations yet we only discussed the geometry discussed the emotions of a dude a gravitational field and so this is just a little demonstration of how dude Bessix give rise to new equations and then we jumped ahead said OK let's its guests that the general and see something like that but immediately we discussed we discover something odd in that on this the orders of the black hole but OK Issaquah Fisher tonight too young that try start will act that it should be what that where what's a big but that no no no no no the metric is magic is not replicas of sex return here now is unrelated to the original metric regional flaps priest metric was just a year and that's correct if you were to done the calculation of the original slot space coordinates you would have found no force because he ate is constant and has no derivative has focused off so if you were to work in the original coordinates you would find no gravitational force that's not surprising if you're an outer space and you have an elevator in elevator was not being accelerated no force things move on straight lines in that frame in accelerated frame reference particles will trajectories and those career trajectories are we cannot really current trajectories but that they look like curved trajectories and coordinates and the emotion EU zone is governed by Newton's equations will write him side received the fact that the crisp are full symbols are not cancer that's important if they will attend they if they was 0 in 1 frame they would be 0 on every frame and since his derivative Is the gravitational force it would tell you that of gravitational forces 0 1 frame 0 in every frame but that's not true the gravitational force and they None accelerated framers Iran the gravitational force accelerated frames not 0 that's the equivalent of support groups eradication the acceleration so it's important that this thing on the right-hand side is not cancer if a war and was you own 1 frame that would be 0 in every 4 a bank that the idea of of the value of the force Newton's equation depending on what frame of reference you give a little more appeal for wiring ETA Utah they will make acceleration much that arose only in order to be working in a reasonable nonrelativistic their framework on the yacht was in order to compare with nonrelativistic we don't want the acceleration to be so big that a small amount of time we get up near the speed of light we want the acceleration by chose I decided just couldn't be ordinary gravitational acceleration here see squared along well that's just the way it is if you these coordinates the place where the acceleration is modest is very far away so theft or dead occasional fewer lyricist pretty modest the walked strong recursion fuels rule for more please visit us at
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Metadaten

Formale Metadaten

Titel General Relativity | Lecture 4
Serientitel Lecture Collection | General Relativity
Teil 4
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15036
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 15, 2012) Leonard Susskind moves the course into discussions of gravity and basic gravitational fields. The Fall 2012 quarter of the Modern Physics series concentrates on Einstein's theory of gravity and geometry: the General Theory of Relativity. This course is the fourth of a six-quarter sequence of classes that explores the essential theoretical foundations of modern physics.

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