Quantum Entanglements, Part 3 | Lecture 8


Formal Metadata

Quantum Entanglements, Part 3 | Lecture 8
Title of Series
Part Number
Number of Parts
Susskind, Leonard
CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Stanford University
Release Date
Production Year

Content Metadata

Subject Area
Proof test Effects unit Bestrahlungsstärke Quality (business) Amplitude Crystal structure Force Gasbohrloch Wavelength Roll forming Turning Automobile Halo (optical phenomenon) Spring (device) Musical ensemble Tool Electricity Electric charge Hose coupling Linear motor Magnetism Magnetic moment Striking clock Big Bang Stock (firearms) Magnetspule Power (physics) Electromagnetic radiation Car dealership Excited state Audio frequency Gedeckter Güterwagen Computer animation Hour Mass Maxwellsche Theorie Book design Gas turbine Universe Remotely operated underwater vehicle Throttle Flatcar Microwave
Ruler Matrix (printing) Analog signal Year Measurement Summer (George Winston album) Crystal structure Interval (mathematics) Cosmic distance ladder Generation Separation process Roll forming Tool Transformer Magnetism Star Typesetting Cylinder head Ballpoint pen Tolerance analysis Electronic component Electromagnetic radiation White FACTS (newspaper) Spant Metre Brightness Hour Rotation Measurement Gas turbine Miner
Matrix (printing) Ruler Quality (business) Transmission line Magnetic core Schubvektorsteuerung Day Year Measurement Magic (cryptography) Interval (mathematics) Cosmic distance ladder Dyeing Roll forming Bahnelement Ship class Caliper Month Tool Transformer Quark Typesetting Electronic component Ground station Season White Paper FACTS (newspaper) Car dealership Metre Excited state Flap (aircraft) Angle of attack Series and parallel circuits Cartridge (firearms) Plating Measurement Remotely operated underwater vehicle
Weapon Matrix (printing) Prozessleittechnik Quality (business) Bomb Sunlight Geokorona Year Green politics Crystal structure Radial engine Interval (mathematics) Cosmic distance ladder Bird vocalization Roll forming Meeting/Interview Elektrooptischer Kerr-Effekt Eraser Pink Typesetting Winter Electronic component Paper Bloomery Map FACTS (newspaper) Finger protocol Angle of attack Sizing Series and parallel circuits Cartridge (firearms) Plane (tool) Flatcar Holzfaserplatte Manned mission to Mars Last Ground (electricity)
Roman calendar Geokorona Scale (map) Observational astronomy Interstellar medium Cosmic distance ladder Light Separation process Turning Roll forming Meeting/Interview Cardinal direction Tool Atmosphere of Earth Magnetic moment Sky Hall effect Vertical integration Dipole FACTS (newspaper) Excited state Direct current Signal (electrical engineering) Mint-made errors Plane (tool) Noise figure Storm Hot working Transmission line Year Force Crystal structure Interval (mathematics) Cord (unit) Mitsubishi A6M Zero Litter (vehicle) Automobile Hose coupling Typesetting Butterfly valve Stellar atmosphere Cosmic microwave background radiation Cosmological principle Two-way radio White Angle of attack Sizing RRS Discovery Universe Remotely operated underwater vehicle
Prozessleittechnik Effects unit Photography Scale (map) Finishing (textiles) Cosmic distance ladder Light Particle Bird vocalization Roll forming Turning Comet Continuous track Sky Season FACTS (newspaper) Cosmic microwave background radiation Weight Autumn Signal (electrical engineering) Mass Measurement Flatcar Day Measurement Year Galaxy Force Crystal structure Summer (George Winston album) Andromeda (play) Rack railway Apparent magnitude Volumetric flow rate Sea breeze Automobile Spare part Negation Hose coupling Star Array data structure Typesetting Silicon Stock (firearms) Digital electronics Inertia Data conversion Lead Metre Potenzialausgleich Hour RRS Discovery Universe Expansionsturbine Last RWE Dea AG
Ruler Capital ship Hubble's law Grey Bestrahlungsstärke Speed of light Transmission line Scale (map) Boat Measurement Force Galaxy Relative articulation Cosmic distance ladder Light Basis (linear algebra) Bird vocalization Separation process Lecture/Conference Packaging and labeling Continuous track Cannon Stream bed Video Typesetting Mechanic Textile Inertia Crystal habit Stromschiene Data conversion Car seat FACTS (newspaper) Cogeneration Metre Belt (mechanical) Standard cell Fahrgeschwindigkeit Cartridge (firearms) Flugbahn Measurement Gas turbine Control panel (engineering) Book design Universe Radioactive decay
Quality (business) Scale (map) Cosmic distance ladder Aerodynamics Bird vocalization Pattern (sewing) Roll forming Astronomer Water vapor DVD player Ground station Electronics Power (physics) Focus (optics) Quantum fluctuation Missile Wood Fahrgeschwindigkeit Particle physics Atomism Cartridge (firearms) Measurement Flatcar Hauptsatz der Thermodynamik 2 Hubble's law Hot working Doppler-Verschiebung Lapping Measurement Year Galaxy Crystal structure Force Thrust Rotating radio transient Spare part Electric charge Star Big Bang Pressure Cylinder head Redshift Burst (band) Suitcase Cogeneration Brightness Standard cell Schwache Lokalisation Theodolite Flugbahn Luminosity Universe Ground (electricity)
Hubble's law Hot working Speed of light Scale (map) Bubble chamber Year Galaxy Force Cosmic distance ladder Circlip Yardstick Canadair CL-44 Low-pressure area Big Bang Typesetting Ballpoint pen Electronics Power (physics) Molekülphysik Microscope Receiver (radio) FACTS (newspaper) Classical mechanics Cogeneration Metre Sizing Fahrgeschwindigkeit Atomism Food storage Universe Gas turbine Measurement Throttle Week Last
Ruler Hot working Grey Speed of light Geokorona Scale (map) Year Measurement Galaxy Summer (George Winston album) Force Interval (mathematics) Cosmic distance ladder Light Particle Separation process Colorfulness Volumetric flow rate Ship class Cardinal direction Mass distribution Material Electric power distribution Sensor Gas balloon Hall effect Gas FACTS (newspaper) Classical mechanics Cogeneration Specific weight Concentrator Metre Excited state Direct current Standard cell Fahrgeschwindigkeit Diving suit Mass Universe Level staff Noise figure Flatcar
Day Geokorona Scale (map) Year Measurement Force Light Energy level Particle Gear Turning Visibility Astronomer Limiter Spare part Steckverbinder Rail transport operations Atmosphere of Earth Kontraktion Knife Ballpoint pen Electronic component White Ozone layer FACTS (newspaper) Renewable energy Angle of attack Game Sizing Computer animation Cartridge (firearms) Absolute zero Halbwellendipol Hour Ford Explorer Plane (tool) Universe Remotely operated underwater vehicle Flatcar Ground (electricity) Last
Computer animation
I am mother and her
real this program is brought to you by Stanford University please visit us at Stanford died EDU Wilner reporters are
lowered their perhaps future little circles knowledge and the extension Rob just a tiny bit about the extension from general from special relativity to general relativity just a little bit and then have general relativity is applied cosmology were real space time was like as opposed to the flat featureless boring space of of special relativity space time of general believes active curve it responds brings yeah allows or kinds of wonderful things that happened such as the evolution of the universe Big Bang and so forth so tho we would just discussed that obeyed spend them a couple of hours saying if we can encapsulate them summerize basic very very basic cosmologic Serrano objections that and there are no questions about previous things got a lad Elliott I do not know well not this this is much arrears ah yes we assumed the awesome electric and magnetic fields we assumed that the equations for electric and magnetic fields Our rents variant of course not that when a long wait deriving the structure Maxwell's equations but we also soon the equations will linear In other words every single turned the equations involves at most 1 power of electric and magnetic fields it would know terms with electric field times derivative of the magnetic field so what that would be said that we would say that that means that field theory His linear you what that means a moment more specifically dealers said that the the field command only to the 1st power we also assumed that Maxwell's equations had did not have higher derivatives in them but there were no turns in the Maxwell's equations with 16th derivative of the electric field or something assumptions or with those assumptions where rents variants and the assumption of electric and magnetic field fit into a single entity that the transforms coherently is enough to prove Maxwell's equations statement that but if a furious linear what that means is slut suppose theories of some field of quarter tho some degree of freedom but it could be a field it could just be the position of a harmonic oscillator any quantity which does something interesting Yassin equations for it and there are solutions solutions will be of the form 5 of exon he well that's a field will be a 5 x With some definite function or the quality of what's qualitative exit that's a solution if the equations for a theory of linear Van among other things that means that is if you multiply a solution by any number it's also a solution of you have a wave you could multiply that way where 125 or anything else I'll just magnifier amplified make waveform higher and deeper without changing its other than stretch it and make it stronger but that's still a solo solution doubling the amplitude of a wave tripling away for multiplying it by a very small can't stand any number is also a solution report state the more enticing statement is that they have to solution was AMX and another solution B of X then the sum of the 2 of them is also a solution so what that means is that if you have 2 different waveforms for example where form this another 1 word for has a higher frequency smaller wavelength books like this if you add them up you get a wave which looks like a little hard to draw with the strike 1 way superimposed on the other just end them and that's also a solution so you could get away what implication of that for example is if you have a wave moonwalk coming in from the left another 1 coming in from the right Our if you don't have the left wait the right way of were simply moved to the right way moved the laugh and eventually just move on the last wave would move to the right and also just move on these some of the solutions after time would just be too waves moving past each other linear equations have this feature that waves which are propagating toward each other with just passed through each other without scattering without are getting stock without anything happened it completely transparent to whichever other they don't affect each other because the some of the solutions is still a solution so our that's character of our was 1 of the features of Maxwell's equations linear narrative as I said what that to electromagnetic waves would just pass through each other about doing anything all the electoral moving magnetic waves can talk to each other so speak we can interact with each other but they interact with each other by virtue of interacting with electric charges and electric charge which happens to be president maybe a whole bunch of electric charges a whole bunch electric charges Our will be excise vibrated To have forces on them because of the electromagnetic field moving through the solar field moving through a bunch of electric charges will stop and oscillating and create a new wave of those new waves which were not present in the absence of the charges cannot be fill orders in any sense the son of original two-way his his newly created waves and so their aura nonlinear effects in electrodynamics but only because electromagnetic waves can affect charges In charges can affect electromagnetic waves and the net effect they can sometimes be for 1 electric 1 electric wave to affect another that's the classical theory of electromagnet novel a direct effect of 1 electromagnetic wave another but you could imagine what's right and let's write down somewhere near equitation and harmonic oscillator the harmonic oscillator the equations these 2nd acts by squared the acceleration mass is equal forests which is minus some spring constant 6 suppose I have a solution Of this equation and now I take another solution of the equation orders right down again instead of calling it a close X 1 X 1 another solution by deeply squared equals minus KX tool His cars wanted 2 different solutions of the same equation if I ended equations I just get that and the times the 2nd derivative of X 1 plus 6 2 is just a quarter might escape X 1 was still so you see directly from the form of the equations that the sum of 2 solutions as another solution but suppose I change the equation suppose I put here another turn what's Coweta P find X 1 squared off something else Q start P X 2 square these is Our both solutions of the same equation but the equation is now a non linear non-linear means it has quadratic things in the fields X squared you
a had these equations what you'll get am not times these 2nd X 1 plus text tool but he squared equals Martinez Katie kinds X 1 plus extra tool but the Leicester here he is not tax 1 plus 6 2 square it is not XY written it but it's not P X 1 square was PX to squared is not part the summer of X 1 X 2 squares so X 1 plus 6 2 it is not a solution of the equation when you have something quadratic simply just doesn't solve the original equation even OXY next to there's no doubt that you could edit things from Maxwell's equations which among here non-linear means that it would contain on the right-hand side of the equations things quite tragic in the electric field things quadratic in the magnetic field analogs of this year which would which if they existed and Maxwell's equations would cause electromagnetic waves not the rich other not to be transparent richer but 1 electromagnetic wave would affect the other and cause it to scatter course to were 1st the door of so you ask me where the you could derive Maxwell's equations from the principle of relativity along think that's roughly were to wear is is not nor was much more are it some of it coming from experimental physics such as the fact that electromagnetic waves dough freely passed richer but you a heads together a few assumptions we're not too many derivatives and the equations and that's enough but not you can derive I don't know any basic laws of physics you could arrived just From relatively alone without input of physical of experimental Bright symmetries constrained theories but they don't be become members of OK questions that let's let's move on to 1st of all a little little little bit about generative thievery but then on the cosmetology were real space-time is really light expanding space deflating space-time Our which cannot be described Barry special theory of relativity requires more general structure called a general theory by the special theory of relativity could be summarized by a very simple idea but a geometry of space time the and that geometry as determined by knowing the distance between every pair of neighboring space time point this is back about geometry in general about reminding geometry in general that if you know the distance between every pair of neighboring space-time points principle you could reconstruct the entire geometry expects Ala look I'll give you an example of a few minutes but but already nontrivial geometry but you represent small interval between 2 points by 8 coordinator set of coordinates with corn X mule were as usual you runs from 0 to 3 a another point x new plus VX where the school and the vote is called EX knew what the distance between those 2 points the distance between those 2 points of course it is what areas of Star Wars certain space time the ball back but how do you represent represented in terms of the coordinates of the sport in general relativity we wrote that the distance between those 2 points squared is equal the rule T-square minus X squared might BY square miners disease square which we can also write as VAX new VAX new another way that we for hours 8 a new rule the EX new VAX way a was just the matrix very simple matrix that ought to be closed Shall proper biblical the child proper time our where 8 was just a matrix warned minus 1 minus 1 minus 1 0 0 3 ways that we wrote the space time interval between neighboring point now after want statement of the general theory of of the special theory of relativity special theory is that he can ask where should be taken to be and In a variant of the geometry another weren't all observers generally they will not agree on what the is that we rotate coordinate tolerance transform coordinates disagree about the components of X but what they will agree about is the proper time between the 2 points proper time invariant that's 1 hour Stewart's once initial starting point for a sort of axiom Maddox description of special theory wrote to a 2nd statement which is now axiom that Europe is that these transformations that you couldn't do on ex-wife easy in which preserve the form of the interval Lawrence transformation if we look at Lawrence transformations rotations and so forth b tens of client coordinates has exactly the same the interval terms of young prime coordinates so that's a statement of symmetry about the basic proper time can be expressed In all old Lawrence framed by the same basic formula and then the final statement is that having found with the symmetry of the interval lawyers found out what these symmetry and what what family of coordinates always give the same answer here well then the rule is also paying all laws of physics are identical in those different frames of reference which are related by Laurence Tribe Mel Of course it is not true that every set of coordinates that you could think of interval as expressed in the same way we example just from ordinary space not space space if I take coordinates X and Y which are the usual kind of Ex-Im White the usual kind of Cartesian coordinate yet the separation between 2 points it's just the square 1st squares if I rotate the coordinates the X Prize in white primed by a standard rotational coordinates than the same interval is DX Times Square must be wide prime square born of I do something done like news coordinates use accord Nash where X and why they are not measured in the same units X is measured in inches and why is measured in the meters I have a point X Y and appoint a X BX White was BY was is the distance between those 2 . it is not DX square post BY square why not because you measuring the interval in In different in different units In particular supposing I wanted distance In into what's your but let me out before the centimeters here centimeters and meters 7 and leaders like factors 12 factors were 44 is Kanarese you might suppose I want the distance between neighboring points by 1 measure it in centimeters
cm mysticetus acts in this white What is a distance between 2 points is it is not the X square 1st you square but the X squared plus DYE divided by a hundred a hundred square while over 100 but family is that if you use the same unit Hawaii as X new measuring things and centimeters that would be DX square rusty why squared but if I use meters for the vertical axis here and I want to get back to centimeters a divide the white component of vectors by a hundred silica rewrite this was X squared plus for over a hundred squared off over 10 thousand why square so what I do some stupid like dealers awkward coordinates where Dixon 1 are not measured saying younger than my expression of distance it is going to have a former which will contain some funny numbers reflect like well on them and thousands here in particular won't be the nicest symmetric DX with what want the goal of further supposing I don't use orthogonal coordinates opposing I decide to choose coordinates and choose coordinate axes which required back and now my role is going to be the coordinates of a point on back and by moving along the x-axis not to the point where you drop of perpendicular but where you draw a line parallel to the Y axis likewise the y coordinate service would be x coordinate them as would be the white quarter then the distance between 2 points you don't have the standard Pythagorean anymore their triangles with triangles now here's X his this is not our not a right triangle sold the hypotenuse is not a high-power very distinct 3rd side of triangle the later this is definitely not be excluded was Y squared even if we use the same units on the X and Y axis is just not sure what what is it what is it while right down precisely what formula is but it will always be of the form of some coefficient X squared plus another coefficients BY square in court he was called incident they some number BY square but but then what's new with about a quarter systems which have angles in them like which I'm not rectangular which of course broke and perpendicular that their additional terms with core Sea Pines DXT wife products are Murphy different uh coordinates that's what happens if you have coordinator systems which are not off on which are not orthogonal war what Bago elements like that you could also write this another way you could arrive sealed 2 times BXP plus Nova Tool BYD acts as a whole bunch of coefficient so we could give these coefficients names we call this 1 1 for the fact that it multiplies DX square because 2 tool box that works well where we giving them different names which just quite quality to tool for a while like this 1 we could call a X and this 1 we call AYX but notice that a ex-wife AYX are exactly the same thing I simply topical efficient VXT wine and Split between 2 terms 1 which is DXT which be YBX Justice symmetrise just to make a nice finally I could display these coefficients aid as a matrix a 1 1 80 1 ex-wife 1 2 to 1 at a 1 2 8 2 1 8 2 2 just displayed on symmetric as a year as a matrix that Beatrix's called metric and this month completely encapsulates what we
distance between neighboring pairs of points are In terms of 4 coefficients laid out as a maker this matrix is called Net right now if we're dealing with ordinary space of the blackboard flat space uniform space everywhere the scene and we choose our coordinates let's say an angle but which shows them by marking off distances which are equal along the x-axis distances which are equal along the Y axis then these coefficients will just be the B constants because our the big constants all over playing again if we did something foolish like site to to use coordinates which lump up over here and have a very different basic another words with the it the spacing but 1 a horizontal axis years X is Mexico's 1 x equals to this is Texaco's 3 4 or 5 6 we've made me coordinated measure very nonuniform then it's completely obvious that these Aileen's will not be constants because depending on where you are the coordinates mesh might be finer coarser and so forth the result of that will be there general views arbitrary quark crazy coordinates which are curve and all kinds of complexity where the spacing between coordinates is non-uniform then the general rule would be these coefficients will be functions opposition what functions a position not any functions a position if we're talking about flat plate but a fairly a fairly wide class of of functions can describe the interval between 2 points in what I could call crazy coordinates coordinates which curved back season which uniform but nevertheless the basic geometry of the blackboard is determined by knowing the distance between every pair of neighboring points With points could be described by coordinates X and Y at the saving his drew in special relativity that if you you lose the right kind of coordinates that means coordinates obtained by Lorentz transformation from each other within the space time interval between 2 points always of the same type was of the same form if you use some caddie want this coordinates which are curve which nobody can rent you from doing their thing we can always use in accordance with light the general formula will be very very similar what would he arrived on the general formula in any coordinates of exactly the same quantity would be how squared the sunset of coefficients usually called Genia position or local the generally be functions of position and then you'll have the X new VX just in case it was obvious this can also be written a R J X by BXJ some bovine injuries so this is a very very general way of writing the space time interval between an arbitrary poor pair of points and if you know what you no later than you know the spacetime geometry completely if you know the spacetime geometry completely that does not determine what Jr. nowhere why not because you could go always going change coordinator to go is going change the coordinates in space time and some peculiar way and change would Virginia know it so Gino determines the space-time but the state of the geometric properties of the distance between neighboring points and so forth but not the other way the geometry doesn't completely determine what you OK any questions about their these symbols on a blackboard spot is just a more general way of writing a spatial and temporal relationship between points and of course there are many possibilities for the choice of the functions G mu now let's go back but ordinary space from magic space which is not as flat as the blackboard which is Kira the 1st day would curb doesn't mean if I take this piece of paper stretch it out my flap on the surface here will agree that that is papers for relations between points for example the fact that that triangles but angles
which for 180 degrees and so what that's all determined by knowing the relationship between distances of neighboring points now I think this piece of paper and I bend nice and gently Inter City into a story that followed but not close it with just a somewhere that I have stretched arm outstretched arm only dented tree that is not what's called the fact most explicitly if you take a flat piece of paper the former without anyway stretching it you don't change the route you don't change the distance between any pair of neighboring points the distance along the piece of paper the distance being calculated moving by cranky insect moving along the piece of paper would not change why not because I haven't had anything fact a creature that lives on this piece of paper that can't look off the paper are can't walk on 1 side can work on the other side only use all winter motion within the piece of paper it was not detect the fact that we base the piece of paper every pair off points would have exactly the same distance as it had before and mathematicians do not refer to this process as curving curving space is what happens you can not flat surface he went
out the formula stretches without changing the distance between neighboring points so for example that the hemisphere of a rubber now hemisphere's Marsala hemisphere rubber a Surface Surface overblown and I tried to cut blond hair so it's only a hemisphere and I tried to put it down on the table here without stretching it obviously can't do it I'll have to take the boundary of the semicircular circular of circular boundary of the piece of the bloom captured about half to stretch it I probably have the squeeze the center of it push it down the plane our the distinction between bendy will quality and curvature real curvature curvature is that which is there when you cannot perform something to make it flat without stretching if that's called curvature again if I take some curved ceremony giving examples a series of very good example of what the surface of a sphere is a very good example workers you cannot flattened Iran playing without stretching of forming it course don't you won't that but that's the reason that you can't draw maps that show the bomb stretch green and make it huge we have our Mercator projection looks because Africa only about one-tenth the size in the area and because these fears Kerr the fundamental occur now leased space time or the spatial this but go back base others sphere but thank us we can coordinate we can but court songs fears such as that's on the surface of the earth longitude and latitude their angles wanted recalled later the other could recall Fly polar endeavor and as a new tho angles you could also expressed the distance between neighboring points beer In terms of in terms of Little differential expressions like the main difference for leader the fact is no way of cordoned the ties in the sphere will allow the components of the metric to be constant space they will necessarily be vary from point to point space that's the test it's not an easy task to implement the general was a better test but the right test for whether a surfaces curved or not is whether you confined coordinates which make the coefficient of the magic constant inspects if their constant and space it means at worst your coordinate taxis what tilted relative to each other but nevertheless the spaces flat it's impossible defined coordinates where you could flatten now out will you cannot where you can represent the metric as being constant then you have a fundamentally curved surface that you cannot fly so curvature is a mob structure for flattening out a surface and parents making metric constant matrix that's emotional curvature in practice the examples of the dollar very simple while nice example a funny set of coordinates that you could use on the plane polar coordinates public order sorrow at home you represent points by angles data and radial distance all Pink supposing I want to talk about these distance of querulously not armed yes let's take a look interval here this Dr and some beef what is the distance between this point in this point right right down cancer for you could check yourself be answers very simple its DRS squared plus Our Square D square why is it are squared Davis Square played was just be a change in angle between these 2 . usurious south a right triangle here 1 side of which is Dr for the other side is not be leader forgive the food you move the bigger the side of that trial the size of this triangle here is not favorites authorities data so Pythagoras Theron tells you that this is square on they square is an example where the coefficients of a metric Cass business our signed form of Sun's last year they made their to use those it is a listed society it seems while playing here is basically not only on the house will be played in the House small dictators go far away the distance grows so the distance along the beach they decide he is proportional now better off when always Her and so now we have a metric which we can display in the matrix form 1 over here we are squared are squarely here multiplying be squared 0 0 so his In hint a simple example of a court system with a flat black board is displayed with the metric which is not constant everywhere but what I can do is River Back to work through X and Y coordinates x equals arco later y equals are assigned later a rewrite this expression and lo and behold it just becomes DX square past square In which case the metric in these coordinates is just good although 1 1 0 0 sold the flat plains for the fact that this is a flat plane may not be so obvious when it's displayed in this peculiar way here not Theatre Square toasty y squared square the status quo at all but nevertheless the factor to confine coordinates the flat about means of the blackboards flat even displayed it in some curvy
linear funny quarter but ideally and can't take us here if you were to take a sphere plus work out the Magic atmosphere that we have 1 coordinates which I guess is Coca-Cola wrangle and its angular distance along this year the 6 more arbitrary Corey over here lawyer was whether tool right to lose the angle from the equator undertaker begin from the north but latitude angle from the equator and just well be angle from the north of simplicity they differ by 90 degrees and the other and the other variable as a new tho angle which is but Of course when data and the other 1 35 suppose that take to neighboring points characterized by D data were the interval being characterized by a 5 areas before is this much is the site of a right triangle and defy characterizes the side of the tribal now let's suppose for simplicity this is a unit sphere that means a sphere of radius 1 just simplest if it's a sphere radius 1 How forested man favorite simply measures distance along with if I tell you to walk along a line of constant latitude a line of constant latitude from will not a line of constant wanted to excuse me a line of customer longitude and I tell you the walk a certain number of degrees or a certain number radio than the distance that you walk is absolutely proportional to the angle to the number of radio and it doesn't matter where you are the same number of radiant corresponds missing this sold the unit sphere we are at distance along with me along with rated direction is just Dean they at the distance so long and so they use Pythagoras is theorem and tells us that the distance longer DS squared His latest square but what about square is defying the actual distance along these other leg of the triangle here not up near the north given define Paul corresponds to a small town near the equator the same defying the same Interpol is a larger as we go all the way down to the South Pole tickets small organ emphatic they horizontal size Of these triangle it is trial he is defined poems signed data when you're up at the the sign a fate is very small favors equal Zero up at the North Pole it's also equal to 0 down near the south pole signed figures signed a favorite is 0 down South Pole this is actually the distance signed the fate of times defied the distance along the horizontal said here then using Pythagoras Iran who discover that the distance between neighboring points is status quo plus signed data squared defy square This is a metric of a sweeter for two-dimensional sphere data squared plus sign square defy square a symmetric in the usual as a musical and polar coordinates of longitude and latitude was this 1 other spheres of radius warned what about of radius all are the only difference would be wooed multiply the whole thing by are square since this the square of the distance it ought to be quadratic in the radius of the sphere so we double the size of the sphere we double every lane and that means the square of the slave we get multiplied by 4 so 1st sphere radius are we now worked out with the metric tensor is a metric tensor would be on squared ah squared signed data square go science data squared 0 0 so again this is not so different from the Polo coordinates the leaders remind you for the polar coordinates we had with the our squared plus are squared our status square the storm from more was any all were there terms which is can't stay and as a was turned down which depends on sigh somewhat similar OK this metric can never be flat and that there is no set of coordinates that you couldn't find the cord ties sphere so that you make the coefficients of the metric tensor constant impossible to do that simply because you cannot flattens fear onto a plane without stretching Our if you the answer would this is it fundamentally curved surface the fact that occurred this could be decapitated by creatures moving around on the surface without ever leaving the surface or looking along for example they would discover that triangles on the Have the sum of their angles larger than a 180 degrees because the trials sort of Bauhaus Powerball out Our making these some of the angles larger saw a creature would walk Estrada's include along a line make any angular turned here later of angular turn come back to the same point man discovery the some of the angles Of his closed triangle would not be 180 degrees OK so that the air and therefore would discover they lived on a curved surface without ever leaving without ever getting off the scene there was banned some higher dimensional steps that's the notion of a metric tensor that's the notion of distance or between neighboring points the notion of curvature which I'm not going to work out a mathematical expression curvature sufficient for all-purpose is just to say if spaces curve yet cannot get read Of these
variations variation from point to point in the coefficient of metric tensor same thing is true in space and that was the new ingredient the Einstein introduced into the general theory of relativity the idea that the proper time between points 1st of all can be represented by a metric tensor metric tensor which varies in space that in itself is nothing new White because even if you took ordinary flat space-time expressed in funny coordinates you could make the metric tons of vary from point-to-point but the new ingredient was that space time can be Kurt curve means that there is no wake up on do no better choice of coordinates will you could undo the fact that the metric space time varies from point to point Our I want to take it deeply into the general theory of relativity what we're going to do is explored 1 or 2 simple geometric situations the car mop in cosmologic and I'm not even going to the please state what the geometry of various kinds of cosmology is armed and that explore the that and see what kind of a world we live in the what now we're talking about space time we're not just talking about space but based upon the kind of cosmology is forever be interested in cars Marla jeez which kind which don't vary from point to point in space Samoa every point in space is exactly the same as every other point that's a course natural if you're in the middle of a starts quite different than being good In some ways interstellar space but on a Hall of the Universe is homogeneous which means the scene from point-to-point some facts was right now some facts about the universe on all of the universe is homogeneous homogeneous does not mean a flat as an example geometry which is homogeneous we have this fear the sphere and every point is the same as it every every other point sought a homogeneous geometry birds butterflies geometry homogeneous means homogeneous in space from point to point in space none space time things change with time the universe grows so homogeneous in space some out points yea Obama holding on scales larger than the intergalactic separation things are really wasn't as yet litter plus then I will see it as a sign of pictures says about God yes yes yes you just have to get that's still small-scale structures that's still small-scale struck now job Yao Yao Yao OK so let's so we can talk about size skills scales of a couple hundred million light years I remember me the 10 billion light years error of the force we see so on a scale of all 100 million light years maybe a few hundred million light years our things things flat so that's it can't but a hundred size and scale of 100 of the size of observable universe everything becomes rather homogeneous now throughout telescopes and we can see that the universe of various all kinds of the universe does not seem to come to an end anywhere is adjusting to go on demands of flooding know it's a lot bigger we know that's a lot bigger than the portion city and on some sufficiently large scale but it appears to be home a largely that I could change that could change some astronomical observations might discover particularly in the microwave background on the biggest skills on the sky it might be discovered that there is a little bit of variation across the left hemisphere might look a little different than the right atmosphere that could happen this is not a principle Of the despite the fact that it's called the cosmological principle that doesn't mean it's a principle of thought because of that has to be true it could change that but on at the moment as far as we can be is pretty homogeneous on large enough scale and deal with it on large the 2nd flat fact is that they are not space time distinguished based in space time and space at any given instant on large enough scales is flat what that means is if you take every instead of time in the expanding universe you take a triangle the any triangle composed of the Stratus lines it you could make straight lines back but some of the angles of a triangle a enough the 180 degrees all the spatial relations and instant of time are consistent with space being slapped her lawyer said well on a small area we will follow up I know what you can't collect you can that you can opt for an instant of time know anything about 3 points which is separated by a large distance but you can but put so what Straus based space-time time vertical spaces horizontal as it is 3 points in space at the same instant of time 1 tool and the 3rd 1 forms triangle back over here OK so if we happened to look at the world at the same instant the time there is no way that we steal 3 points but we can collect data later From the 3 points by collecting data from the 3 points love we can correlate information about these triangle and moral leaking collected from different perspectives and then Breyer signals together learn a great deal by looking back at the strangled learned a great deal about what we what with the shape of a triangle was at that time so it correct cannot you cannot know anything basically you can order anything except right now here and now but you can't collect signals fact I can know anything about you will right at the since the end of time I only know that you
would time awards light signals something from silicon look back and Carly data and and that of course is what we do we look out of the sky and make measurements of galaxies bestow a little bit indirect but when we make geometric statements were making statements about looking back on the sky and reducing what geometry of space must debate right so a 2nd statement homogeneous everywhere the same more than met based base flat that's not the same as they space flag is space lattice saw who wore a finish dead last taste it it but this time it was I was on if you follow 2 points was the photo points followed 2 galaxies as time goes on the distance between those galaxies increases were part but there's a standard description of such a space time 1 which is expanding as time goes on but which is homogeneous and which is flat and is rather simple seem to understand Our described in terms of the space-time metric a proper time as described in Items the quarter so we take the usual coordinates x y and z the and to it T X Y and Z and 3 that's just like special relativity time except the mandatory considers more complicated in fact let's see what we can deduce from the fact that spaces everywhere is the same in the flat we can deduce that any given instant of prime we can find coordinates which just allow it to be represented In terms of Cartesian court so that means that weekend we expect that it be of the form the X square plus BY square cost easy squared the distance between 2 points but we should account for the fact remember incidentally than the special theory of relativity face comes in with a negative sign social really Premier minuses just like in special theory wrote but we should possibly take into account a question of unit if you want till measures based in centimeters brought a asked for the distance between points in meters kilometers all miles the only difference would be be that you would multiply this by some conversion factor and let's call conversion factor AT square In other words if we were measuring let's measure the distance between OK let's measure the distance between points by the number of galaxies along more less straight line between 2 0 . it we have 2 points in the universe far from each other the there are many galaxies in between end roughly speaking the distance between galaxies it is all the same about about 2 million light years between neighboring galaxies roughly speaking and that will be rough on purpose we simply represent the quantities particles by asking how many galaxies on their galactic spacings are the between well since the galaxies seperate with forum at different times this measure of distance will correspond actual different measures in meters and for that reason we have to keep track of the fact that our coordinates maybe we may be working units at a given time which are not meters which are not centimeters but which corresponds to a measure corresponding this summer number of galaxies between points or ever would throw in a factor it's simply a conversion factor unit conversion factor From whatever the units of x wines off the actual physical distances this it AP Here is constant space it doesn't vary from point to point the space it may vary in time and might vary in time for example if a galaxies a separating but callers point x equals 1 scholars point back sequels to assist the name of the point at this point out here forever out at Texaco minus 1 actually make it a three-dimensional array coordinate attractions space which X begins with the galactic expansion but to look at how we discover that the galaxies are separating from each other we imagine some call what nets which a rectangular Cartesian coordinates but the coordinates are spreading together with the flow of the galaxy says the galaxies expand forest space expands the distance between neighboring points in his grid is growing at the end of each year and a half he does not depend on way you Warren space that's the statement that spaces flat vector space is flat means that space can always be represented by a constant set of coefficients and fact postcards instead of coefficient could be taken to be the year of the ordinary Caucasian car at space What about trying time is exactly as it is in the special theory wrote that EDT square but was right that's where the rewrite it did cow squared physical weekly squared same as a special relativity plus or minus functional time which represents the expansion of the universe it increases with time find usual BX where plus Y square plus so this kind of metric represents here we could see that he is the galaxy and we emerging a mesh of coordinates embedded with the galaxies but as the
galaxies expand the coordinate mesh expands with that means that the coordinates so we assigned to the galaxies don't change with time and again the fact that this is the galaxy appoint DEC's equals 1 Mexico 0 Wykle's 1 this is the Galaxy lexical 5 and white will 7 that is preserved with time because the coordinated increases with viewers but we have to account for this expansion where to account for the fact that distances actually expanding with time and that's and by a kind dependents the coefficient anybody the call now the stock has for the scale factor which called the scale factor and thing which does not I said depend on space but it does depend on time and it represents the fact that elite 2 fixed points in space fixed me to say that they have fixed coordinates the distance between them increases in time this it is the basic but Our was set up for essentially all of observational cosmology that the university ascribed by aII metric which looks like so so we 1st tried a couple of things about as close that of the world will vie for us to get from last season they work as they say in what changes the rack that SEA is the the threat of lead later that they taxes as Beatles leading points but this also true that you should not be last year but what want mm thick as hell together by molecular behind need stake is held together by electromagnetic bonds which is strong enough dog over calm the expansion of the universe that the had very good question the gallon I don't spending galaxies are small enough in concentrated enough that their own gravity holds together and if I even causes them to shrink with little bit a star shrinks with the distance between galaxies on the average is large enough that gravitational interactions between them are fairly negligible and survey sort of a baby expand with the ambient background expansion but the individual galaxies a small enough gravity is pulling them together but they the it's an awkward situation where art with some scales of man and distance are such that things are being pulled together and other scales of distance and mass are structured that day that drowned but Our that things are basically just flowing with laws that were not saying that he was not those merit Great Great Hall of the People vast a slack is that there's thought in process of doing a discovery was made that things were the way people what they discovered but there has to be more mass of the galaxy holding it together and the apparent massacre see-through telescopes the sky but looking optical light and that of course was the discovery of Dr. matters but Dr. energy but by but once that's accounted for want to do it once in 1 Galaxy you do another time another galaxy and you do a whole bunch of cross-correlation checks and so forth once you've done that you know that gravity is strong enough with the Galaxy the hold together not strong enough were written earned our neighboring galaxy and not that of course there are some exceptions are alloy neighboring galaxy the Andromeda just happens to be close enough and heavy enough where heavy enough that the force of attraction of gravity happens To be pulling because in the fall toward each other I'm just happens that those unusually massive galaxies and not to logic until the start of an exception galaxies but comet together but on the average the galaxies part and on the scale large enough that the gravity is weak enough there just all separated from each other that what were sent but but Oh yeah the yet neighboring galaxies no or examples Bromley average even neighboring galaxy where there what a scene that it had made what this is final both be precise precise a year a scale factor Is Not for so a scale factor scale factor has units only doesn't have units forms I'm not sure mean better call the forest could it be to To some force the causing things to learn but for the a breeze so let's let's not worry about forced more and calm the effect of gravity for others it is a pity that is Luna it's so that you see me down you you should an object from the surface of the Canon cannonball except the revelry at 50 thousand miles an hour the past greater is there is an easier moving up him His graduate closing all of its inertia because it will intercepts
navvy reasonless that shout about that that was that is not statement but once shall out whatever mechanism chatter about the fact that the Earth and the Campbell receding from each other is not due to any force From mistake Aristotle may think that motion was the report it's too To inertia and on the other hand Our really is the 2nd thing is going on as the cannon Wong was out it decelerated before serve gravity decelerated it may be enough to celebrated make a fallback although it may not be enough to decelerated because the fullback depending by the British leader case it decelerate the force of gravity causes the velocity of change with time because asymptotically if you're really does escape it keeps going with the uniform velocity approximately as it's moving the party decelerate the U.S. is just like that there's an expansion of and until recently it was thought that that expansion is decelerating expansion is like shooting an explosion at a cannon with keeps it going is not force which keeps it going is just the inertia but it was expected that that inertia would be fought by the gravitational attraction be things between things not necessarily that meant that things would but that they would be solely decelerating expansion will be decelerate that turned out to be true but let's say what we can say on the basis of respect but let's talk about the expansion and the distance the kind dependence sucker paradox is that are separated by a certain distance let's take the the deal along the x-axis simplicity of our galaxies along the x-axis they don't have to be along the x-axis just taking it for simplicity along the x-axis and let's take separation between now and the call what separation would not be actual distance but the court Wordnet separation caller dealt X belt X does not change with time Dr. X the coordinates Our in bed in an expanding this is X equals 1 and this is X equals 5 forever and ever eternally from now on this galaxies Mexico's 1 galaxies Mexico's 5 Texas for and will stay that way forever but let's not specified which is called the OPEC's what the distance actual numerical this In meters for example or whatever units for use between equals between those 2 points defined it we have multiplied by a has the conversion factor From these labels which are simply labels the actual metrical distance is measured in where units real physical units we up off talk about meters or abuse of the unit leaders but the actual distance is a kind of excellent call that these dissidents that's the actual and 80 is a function of time OK what's the velocity of recession between those 2 how fast separating from each other in meters per 2nd that is simple we just differentiate D with respect the prime B is the dissidents if we differentiate it with respect to time for these 2 points and give us a velocity between now know that that's thinks Delta is just 4 units in the case of X equals 1 Mexico's private never changes so we differentiate with respect kind the only thing that changes it is a 8 expansion factories increasing it has a derivative so we can say that velocity which is the kind of speed the speed of expansion between 2 points is just Eagle Tool time derivative 80 are going to use the notation kind derivative just putting a stop on top of a friend time derivative it is often I think would probably use that before I hope we have not got means IDT so most is the time to remove 8 times don't banks at the loss 4 well you this let me multiply and divide it by an 80 where dealt X firms that Beat the distance arrived at a very simply very simple rule that velocity of separated the velocity of recession between 2 objects is proportional the to the distance between them trying something which doesn't depend on way you want it depends on when you are but not where you a dog over a member a does not depend on physicians base it does depend on time 18 . divided by a a doctor by as the coefficient relating velocity the distance another name for How will the Hubble expansion I usually sometimes called the Hubble constant but quite obvious that in general 8 of 80 is not a constant in general depends on time AT and In general both depend on time so the Hubble expansion rate the thing that usually called capital H ferocity is distant times age but H is nothing but the ratio although AT the time derivative of a itself most important fact about it is it doesn't depend on where you wash it doesn't depend on which pair galaxies you're talking about every pair galaxies at a particular instance of time will really will recess from each other will have a recession for a velocity of separation which is proportional to the distant
and multiplied by the Hubble constant or the Hubble parameter which is closed they if he would like to they give is the time yes right right now you video which Iran's here think is the right question if you look back and you see a galaxy far enough away far enough away and you want a new measure its expansion should you you lose 8 . 0 8 today or should use it them here says that but typically aid over 80 In changing relatively slowly in time so that large large pieces of the universe be Hubble constant doesn't change much with time but that's did that is taken into account when people look at the recessional that the recession will velocities have to be slightly modified from the book formula to account for the difference of time directly looking at things will work with the question of where they live as you see more subtle but you see equals 1 1st of all the time and space have the same now don't yes but yet nominal think of excellence as not having any dimensions of all it is the label SYDNEY Fred Charlie and so forth Mexico's 1 Mexico's political 3 carried the completely dimensions of course there is that's but somewhat ambiguous you could choose 8 Kabibe dimensionless subunits of late but some arbitrary I like to think of it as just saying Exs labeled the points and boat carrying dimensions and the dimensions of dissidents in a server to get report OK this is the Hubble along varsity at any given instant of time is proportional to districts with the Hubble constant itself with a book is it surprising not what it says is the fastest the vastly move further you get that'll really thought that the vastly move the further you get our things 1st things are moving fast get further apart but there's a detailed formed rector of scored habit when you think about it it really could hardly be any other way well actually I guess you could you could contrived but not with a set of assumptions here the Harbaugh was just 30 In special relativity Hawaii it took satisfies deep how we call 0 along a light gray Interpol light gray XTY EDT so for the proper time 0 why is let's move along the x-axis units in which the speed of light is 1 then the X will equal the D. C at a speed of light of course the speed of light a the exit CD but see is set equal 1 as on all my equations then BX equals T and he squared minus the X squared equals 0 so I like Brady is a peculiar path through space time where the proper time a warm the panel along the path of the library 0 that characterizes the emotional like rays that need proper distant a longer light rail the proper time along the right library is 0 not going back and assuming that it's true and the general theory of relativity which it we can read a lot from this opposes a light moving along the x-axis forget y z then what is the equation of a light ready the equation of library it was just decay Oracle 0 or the equals a of TV kinds VX but Our which like that the other ways we could divide by 80 if we like it's do duet right yes while still a light gray moves would be that's not just given by C but C. divided by this makes it look in the coordinates so we using as a light slows down with time wiII a gets bigger and bigger armed Mr. said that velocity of a library the by shrinks with time now what's really going on is not polite raise slowing down is if you go far enough away things are moving out faster and faster and so were from your perspective but but this is what a library this is Our emotional library in these coordinates which remember these coordinates are not measured in meters In these coordinates x simply dimension and the light moves amount the distant X at a given amount of time it was a distance which depends on the scale factor so you keep track of things in because Margie by remembering the light rays move on clothes novel finds no lines mean they have 0 proper time and you can calculate How do I really would move through the by knowing that it's the X is always equal to the blighted by scale back and only scale factors there because as time goes on a given DX corresponds to a larger and larger separation be crazy if library always moved with the same DX forgiving seat because of late kind of given X is a huge huge this soros that's taken care of but putting scripts so that's the that's emotional library allocate capped we know what a if he is for a particular situation we could calculate the motion like Our so
this is a spacetime geometry they have to of certainly not the general wrote the general spacetime geometry of general but it is on a home B fairly general description of almost all versions of I know of our large-scale cosmology because knowledge scale large enough but the overdue could ignore the variations from point to point it's more complicated if space workers space could be be assessed fear it could be a hyperboloid it be something else want space appears really to be flat expo EPA's BY square was easy square which makes the formulas broke the next question Mr. Watson answer questions just about as geometry but the most fundamental questions my warm it is what is a gift to you have vary with time I was a gift does it growing a shrink keep growing there's a growers power of time growers logarithm of time grow if it grows exponentially with time How was it behave rat is determined by gravitational force gravitational burned them and its roughly the same question or a version of a question of if you see a missile moving away from you How was move more from you would find that depends on its equations of motion the depends on gravitational dynamics depends on articles maybe and all that stuff so that becomes the next question how does any of these very were not work that out for 1 example well I think maybe we'll work out 2 examples where that precedent were there the charge of electrons very relief it is not the to same focused on that question missing focus on it know that I have to there is a pattern we also Free dance through charge of electron doesn't it to charge of electrons dog varying with time and we noticed there consumer awfully important right but from from a geometric point of view from the point of view of the geometry of spacetime we got the only question but a central question will be how it thievery if have hefty didn't vary return would have just plain old flat space-time we get rid of this AD by changing the units but changing units we could get rid of this a very would only be because varies with time our space time have a year among the trivial geometry it would just be playoff flats based on the city was constantly right now I did not mean to imply that the a variation of a with retired necessarily the most important question cosmology are but let me tell you if we discovered 8 didnt vary retired we might not change atomic physics d'amato remain unchanged particle physics very much but we sure change Cosmo Margie could be something totally unrecognizable from the present respect when I am not going to talk about any way experimentally detect a recession velocities and so forth that's over part that outfits thrust of her head in the Europe All-Stars and also has also but if 1 of them but you see before you can't stop thinking about actually you really do measures Our you really have to have some theory behind you know what it is you measure that's true could measure a local velocity of nearby in nearby galaxies were wrapping sophisticated but if you want a very very distant galaxies in you have the worry for example of the Hubble parameter changed with time as you point out so you better have some basic theoretical structure to begin with before you can even ask questions of basic geometric set up when talking about a geometric set up and not be too mad these way of measuring space time distances astronomical scales is as a suspect most of you have some idea of how astronomy measures a burst into velocities and so forth distance very easy way to measure distances are the easiest way to measure distances to a distant galaxy just look luminosity telescope of David's far Bright near our use simple law that these luminosity terrible luminosities with the square of the distance because like spread out that's the dumbest spoke problem From may not be the best way to measure distances stars but roughly speaking that gives Europe pretty good idea How far distant galaxies how measure the velocity measure their velocity by the redshift of their atomic go transitions but Doppler shift and most together tell you the Hubble so hobbled union aright numerically was off by a factor of 10 some mark on his own constant by incidentally there's an interesting surpassing the Hubble constant worry Constance with pride was really work and that distances were related by a fixed number age of velocity then you could safari questioned running backward we take to galaxies a certain distance apart rather backward they come together How
long does it take for them to come together that's easy you take that to either you divide distant along was take greater distance those at the workers at out a distance Eagles velocity times prime yet so how long will it take for those galaxies collide with each other if you write backward while the answer is the distance between them provided by their velocity if the velocity we uniform another words if the Hubble constant were actually constant Our if the harbor but not the Hubble Castle comes if the velocities were constant if the velocity of a given of this start he was constant with time and we ran backward we could ask how In time does it take before they collide with each other and uh I should said of the Hubble constant that was wrong if the velocity was constant other words if every galaxy was receding with a uniform velocity that didn't change with time then how long would it take to galaxies to be on top of each other With this so would be the distance between them the right about velocity but of the velocity is equal to the distance on the Hubble expansion parameter the streams distance provided by H. times distance was a suave 8 each is Of the Hubble expansion parameter is it I think that it would take if you win the galaxies backward but 2 neighboring or too distant galaxies to be on top of each other notice that that time doesn't depend on which galaxies with talking about it doesn't depend on how far apart they are right now because that cancels out what remains the is equal to 1 over of the bigger the Hubble constant the last time it would take With if we ran backwards the 2 galaxies call richer doesn't know which County any parent galaxies the be sitting on top of each other in a time of war order were a workaholic as if to to go well of course is that told them on top of each other the will be on top of each other no sir might crash yet but this does give you some How far back in time before pressure before has long this incidentally it is no different really saying I look out and I see something moving away from me with this the velocity the woods distance I can roughly estimate how long ago in the past that must've been ejected proceed moving more certain velocity and what's a certain distance I can make a good guess about how long ago it was sitting in my lap here so it's it's really no different than that it's a crude and rough estimates of how long ago the Big Bang wars by measuring be velocity of distant galaxies measuring the distance we get the Hubble parameter the inverse of the Hubble parameter is we weakened quality age of the universe it's 1 measure of the age of the universe nobody knows where when the universe began Howard began but this interest age is of water at the time of the big bag with Ewart incident we expect them forward you expect would you expect the real time was head along the the well we we haven't even mentioned forces here this would happen if the velocity was absolutely uniform so OK so think about that object that's moving away from we measured velocity sometime in the future receives moving away from us was speed want to know how long ago if there were left ear What can answer that question by running it backward take take that same object reverse its velocity and asked how long it takes the form of theater the answer is a less time than if a will move in with uniform velocity be because the earth attracted because it to accelerate or yeah takes less time so this In these standard old-fashioned Primakov Montes this estimate the Of the age of the universe was a little bit or fear numerous will have to be less old man Our but not by a lot not that easy Keller said but this can only be carried out later on well below right at some point this theory correct Our crash means exactly what as you as you run backward all really b backwards of what happened as you were always the Galaxy so the materialized fluctuations the relative degree of almost complete home maybe by attracting the matter attracting clumping himself after I and I don't think you would that is not what you would see few literally backwards if you run a real event backward you don't get it the see know the backwards over forward motion on it if you that young young young young frightening this various ways described it sometimes it's called the cycle of thermodynamics underscore the increase of entropy sometimes it just call shit happens but some measures as the players as thousands of songs that area says yes there is no other workers easier over a new jazz resulted in case he is or whether not is that now is not as a sort of a review
of only as a a really important physics only ratios of admitted that Mitchell quantities so what you could say is please I I think that that that is a very good point and we should talk about that Our the distance between galaxies divided by the linker meterstick is increasing and that's really all you could say the daft what equivalently meterstick secrets now when you take a choice but if it's Standard & meaning it's another of convention but we like to think that these parameters of atomic physics and molecular physics of all these things which hold meters sticks together is fixed we like imagine that a massive electron fixed and all fundamental parameters Our fixed numbers the speed of light or what the but a particular dimensional numbers I would not like to do it I can say the radius of the proton decreases with time as you know instead of saying universe expands it is a matter of pays more than anything else of historical preference historical our convention that week kind of think on the microscopic comic God distances as fix things parents they of course determined was a meter wiser meters dekametre along well was a mere meters a distance from a mostly why is it so big in the comic units because it takes a lot of Adams to make an oral it was about a biology but that biology we like was change time and store will we use conventions which say that the size of an ad on the size of a proton and so forth it is determined by a fixed set the parameters that enshrined In year in some ways of piracy where they keep the meterstick these days and and finished use also use those units back convention numerous expanded if somebody big enough for looks at the universe From a large distant perspective and likes to use the distance between galaxies is basic unit that observer will see everything shrinking shrinking will say Adams gets more sordid is a matter that reform that the apart From apart from coordinate transformation novices metric in some specific set of coordinates for the right statement is that the phonetic given these 2 things the budget can be brought to this formed by an appropriate choice of words yes yeah that's about sent a derivation but I did do but is not terribly this you want us off their lows last answers what we look at everything today we calculate we measure what the Hubble constant is today today and then we run at backward we backward With today's parameters that will tell us how long it takes for the universe collapse something much more of the Hubble constant opposed varies with time there is a time in power and the fact is and the increases you go back for the Big Bang bubble curse was much larger earlier times in this today so alone and loss cities were much larger so even now the distance between the galaxies may have been much smaller the velocities of rejection of those same galaxies lottery that news of the Hubble parameter must be much larger the test but are saying is but we look at things today the clippings telescopes we measure today's Hubble constant we know what it is that tells us today with the ratio of the velocity and then using assuming that the velocities uniform doesn't change with time we can ask along with taken for college when we enter that a little bit of information from gravity as we did if we did backward then we would be all over under not overestimating the time because gravitational attraction would cost speed up Charles Our Vermont and others with great precision so we know what the age of the universe numerous meaning the time until say galaxies would've been sitting on top of each other its 13 . 5 billion euros threats let's a receiver will don't find our what is doing with time for 1 of closest to measure and you could look back through telescopes vary very distant past you could look at principal Yukon look at 2 galaxies you could in principle tried factor don't do it this way but you would look at them carefully enough to see that the receding from each other a little bit of a long long in the past and so in that way you could measure the relationship between velocity and distant long in the past and determine how the Hubble parameter changes with time and if you know that could also figure out how the scale factor changed but course that's not the way it's not the way it's actually done well was more sophisticated than that but the 1st thing is what'd you expect what'd you expect on the bases the basic Newtonian physics on a 0 gravity the answered it roughly analogous to the same question it you would ask about the outgoing cannonball houses distance of the cannonball is now is just the distance between a pair of nearby galaxies I was very with time varies with time in a manner which is entirely analogous to the the bowling ball not bowling ball cannonball dissidents were received from work guerrillas there no equations governing forces and acceleration determine how the cannonball would receive from
and so the exercise of figuring out how any varies with time is a simple exercise Newtonian physics and Charmian sequels and make show you how works a simple example let's a draw point out until they are our in an approximation with the universe is very big in the distance between galaxies is small and everything is homogeneous in uniform we can't think of a galaxy is being smeared always based very homogeneous gas gas particles the guest of particles of moral standing still must the whichever but their flowing with the flow and the gas particles expanded let's say this is us over here there is no reason why we should be concentrating on us and will discover in the equations don't depend on hold asked whether it's us over here for a summer these 5 billion light years away the equations how the universe expands should not depend on who we choose to be asked but let's shows us anyway as sober and now that take some galaxy point X was put on the x-axis would prove the X-axis we are at x equals 0 by definition is a galaxy at point X X is labeled its distance is 8 times X distance distance here the actual distances in meters His 8 times next year we are over here and let's solve Newton's equations how this galaxy what are the forces on the Galaxy Galaxy is red galaxy over here well there is of course the force of attraction galaxy but that's not all is a lot of other galaxies here in between was forming yes all of these galaxies it is also galaxies where all these galaxies gravitationally attract the galaxy that I've called X so it sounds like a very very hard problem to calculate the motion of X but Newton would not a founder of our problem he did I don't think he said of this exact problem but he would not have the he would have you lose a fact that he discovered about gravitational forces is basically a fact about inverse square forces is a statement of your have spherically symmetric distribution of matter spherically symmetric about the origin and the uniform distribution of galaxies historically symmetric sustained in every direction the Mets forced foreign object it is the 1st dual top all of these matters is side this fear about center another reason next forest Poland the galaxy toward the center of coordinates past is due to all of the matter inside the Sphere and all the man outside the sphere doesn't contribute anything as 1 state the sorrow in calculating how X moves you could forget all the material on the outside just doesn't make any difference on the other hand Newton also discovered something else he discovered that you spherically symmetric distribution of man's like the son object due to a spherically symmetric mass distribution on the inside of this here is exactly the same as if all the mass for concentrated right at the center so this means we always have to to do the find out how our galaxy and distance x moves Mr. the study its motion in a fictitious problems where all of the material we must bounded by X all things closer the next all concentrated right about and they just pretender was a problem in which a cannonball was ejected from a from near and that will give us the exact correct and that's that that's the thing that Newton were not sure if he didn't solve that problem he would solve our problem very easily so let's work it out well I'm which With article work it out next time will work out in detail these equation of motion aid increases with time and work out the simplest cosmology which called matter dominate because MultiFinder Howie increases and serve discovered 1 of 3 basic laws of cosmology that the verbal talk about next time the question now because the 1 more class vicious staff we're a rebuke of soap was gone about it not to look where all of that well the uterus based must closet it's this where yeah but at says are we were really a lot we see a color said the survey said all that hurts life it will Chen said know what I think is becoming as well but not like earlier work out the that next time let's look at and then there are sort out as
a all that OK so let's are it is question on wet let's flotsam library refused to show With years the now what happens the scale factor very early in time to get very small In fact puts just assume it goes 0 some simple way column and let's plot would like ready looks like when any is very very small these DX by is very very large so light gray suit light race starting from this point over here this is equal 0 x equals 0 AT is also very very small 6 with 0 OK so lightly starts very fast measured in because 80 is very small and small distance so it goes in a given interval of time is very large so it starts moving very fast but see that means fast means almost horizontal and Senate speech would mean completely horizontal then a starts to grow 0 co-op closed the quarter velocity it's not the ordinary last when the velocity of light Pierce a slowdown from light the light slows down according to some loss that depends on how a varies with time and now here we are right over here we are and here's today spaces flat on on forever and ever no matter what this could reshape light every displaced over here and backward we will discuss some point will come to some point that's the point which sends like a loss from people 0 that gets the West now that's it like that we see came from the point uh from donor will say about it except that we run the backward remember its apparently slowing down slowing down slowing down so always start we can run at backward and will eventually intersect people 0 and the oils area except to work out some examples of how like propagates so there will be some place that belied originated if you won't work at will work out an example with a specific of tea or try calculate how far away that life originated from but and how long it took to get us are running out of steam sword or T there is is a vote they wait for the Laurel everything started out completely homogeneous started out completely uniform but extremely dents the tabletop he went on and on forever now that probably not really true but just a matter tabletop is extremely big but extremely debts but but so the molecules that way molecules think of molecules that we see today is being can billion light years away from us are really had the minus 10 centimeters course Everything that today's observable universe is observable universe was trainee tiny patchy and then the hall brought tabletops vowed to expand but no shelves no shelter with no sense in which some matter was ejected from a point 1 have formed the shelf just everything started to uniformly expand but and so it continued the former completely homogeneous space but the separation between points just got bigger and bigger think of a beloved figure will balloon expanding are you so the balloon was small you go about it then expands the same about foreign apart but below uniformly filled with dots it's not that office's are over here we see Dobbs on some shelf said balloons uniform that's the nature the geometry with talking about homogeneous everywhere the same but uniformly this time suffered that not so well find out is what it is the thing is that all that time that of the rule of thumb is that something about eating measure by is for but there is still a major alert came up here they were they only of that was bad the beauty of the where they there was there but their Apple oasis is the way were worse for work force From the force between a galaxy of our galaxy it is affected by the expansion closer it is a strong but we were aware that the consequences of that where the fear was much smaller member now only things within the sphere affect motion of the sect's wrote us when all of us were much smaller so that that St. galaxy was burned through onslaught also backs a lot of bad things are giving way is about
last forces in cost more TV is all but was it makes be even acreage that they know the observational evidence so bucket From what which begin with beer 0 curved 0 flat and if you on lodge Plane known know the schoolyard try to look around you can't tell of fear the of flat White because you're not sampling a large portion of it To tell don't yes budget review triangles will be able appeared were the only way you could tell the sample a along enough portion of the so that you can actually see after the same is true with the uterus if you only get to sample a portion small by comparison with the radius of curvature of the operating turn here the radius of curvature and clear if the radius of curvature just radius unless you consider ample part of Europe the comparable to the radius of the year years you will not see better particularly if you're a little bog moving around on the surface of the you have to explore a region which is more or less some find a fraction of the entire ratings the year Earth In order to observe the fact T the aren't the same is the universe you have to annex Explorer a portion of the universe which is comparable not necessarily comparable to the some reasonably reasonably large fracture of the radius of of curvature of space spaces also may also be heard if you explore what to small fraction of it you can't detect the curvature prior to the statement today what what you might be able to for example is you could put combat gear you might be able put limits on how big Europe by exploring some triangles and no not out on a football field assuming the football field over was nice and flat are you could put some bans on how big the year affairs you could by measuring the angles of triangles and so forth you would find it very close to being flat and you would say Well the deviation from a 180 degrees is less than point dipole 1 and therefore the Europe has to be bigger then the size otherwise you have seen the year of the funding triangles same thing you could do by exploring a region of space of size billions of years measure triangles various astronomical ways measure of very very large triangles compute the angles and so forth and they ask whether on the scale that you measuring his any visible curvature the answer is not on the scale that we measured up till now we cannot detect the presence of curvature of the universe it appears to be flat on a scale of 10 20 billion light years meaning of as saying as it is for the Earth it means that if the university's Kerr its radius of curvature is much bigger than the visible portion laugh How much is much bigger 10 times at least at minimal may be much more than that but absolute minimum be universe 10 times larger in a radius and the portion that we can state and so that means that we can really detector curvature worries is a really that might be there what happens if it is it starts out very curve of the curvature is larger skirt a small sphere is highly as a sphere expands the curvature gets smaller and smaller until series as big as you if we can barely see the curvature so the numerical measure of curvature of the surface of the earth all decreases as the earth gets bigger and bigger same thing is true of numerous curvature of the universe decreases as the universe expands and apparently the universe universes expanded so much that even if it were Boulter begin words if it's expanding won't her day-to-day as it may be because it is best it will be the start of a brewery euros is normal study as fair game they days call ozone levels were renewal of the lease we could on the expansion or contraction of the universe is controlled by this of take the universe could be flat it to be positively currently to be negatively curved whether it's expanding and contracting is not determined by that it is determined by how it evolves if if continues to grow at expanding their it grows and 3 contracts the we tracks after a while it it's contracting the universe is shrinking there is a connection which is no longer apparently believed to be true but there was an old connection between the curvature of the universe and whether it's Curry collapse again on that has changed and I will tell you about next time but not spending is expanding contracting is contracting I would say there's a component which is contracting out that's not the way I think about it and think about expansion and contraction corridors contraction could follow In the same way the ball from the air which is receiving could fall back down that's a sense the question is this particle he and moving away fast enough the that all of man's within a year Is it is sufficient to pull it back down which case escape or is moving slowly enough circle mess with him will pull it back in which case the universe will quit order hours just between on the knife it appears to be just between on knife-edge between now right people
gained owner who the preceding
program is copyrighted by Stanford University please visit us at stanford .


 1060 ms - page object


AV-Portal 3.9.1 (0da88e96ae8dbbf323d1005dc12c7aa41dfc5a31)