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Slicing the spacetime loaf 
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#1
Aug708, 01:21 PM

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I’m having a little trouble understanding an analogy in Greene’s “Fabric of the Cosmos.” I would be grateful for any attempts to explain this or to point me to some examples on the web.
I’m reading about spacetime and the Greene equates spacetime with a loaf of bread that can be sliced at different angles. In Newton’s view, two observers at each side of the “loaf” would experience “now” at the same time which looks like a straight cut through the loaf of spacetime. Relativity says that if the observers are in motion then the cut through spacetime is at an angle and thus the spacetime loaf would be cut at an angle meaning that the observers “now” would be significantly different from each other. Greene gives an example of you and “Chewie” both sitting on a couch. Chewie is 10 billion light years away. Ignoring rotations of planets and stuff, if both you and Chewie are stationary then you two can say you experience the same now. HOWEVER, if Chewie gets up and walks at about 10 mph away from you – his now will be about 150 years before yours. If Chewie walks towards you – his now will be about 150 after you. (I may have mixed up the moving toward=after your now; moving away = toward your now but regardless….) I can’t wrap my head around this. I understand (pretty well for a nonphysicist) general time dilation effects and the basic ideas of relativity. But I can’t figure out how and why the angle of the “slice through the spacetime loaf” changes forward 150 years or back 150 years by Chewie moving towards or away from you. Is this just an extrapolation of the moving train with two people shooting at each other when a flash occurs on the train and the observers on the train say the shots occur at the same time and the stationary observers (off the train) say one fired before the other? I appreciate any help you guys can give. I’ve been puzzled by this for days. Thanks! John 


#2
Aug708, 01:46 PM

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An inertial observer slices up Minkowski spacetime into constanttimecoordinate slices that are purely spatial according to him... phrased more geometrically, the spatialslices are Minkowskiperpendicular to his worldline.
(This notion of Minkowskiperpendicular is related to "constancy of the speed of light" postulate.) In general, the families of slices are not coplanar... so simultaneity is relative. Look at Fig 60 on p.123 of Geroch's General Relativity from A to B http://books.google.com/books?id=Ukx...EfGDxdHn2yCFA It turns out in [the spacetime geometry of] Galileanrelativity, the families of slices of inertial observers are all coplanar... corresponding to the notion of absolute simultaneity. The however, doesn't mean that the families of slices are the same. Picture a deck of cards, then picture the same deck beveled. 


#3
Aug708, 01:52 PM

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#4
Aug708, 02:17 PM

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Slicing the spacetime loaf
I'm still pretty confused. I was thinking along the lines of typical time dilation effects where motion is going to make people age differently  I assumed that the aging difference would be similar to this changing of the "nows." A kind of warping of time or something. Please excuse my ignorance towards the mathematics and physics terms  I just picked this book up b/c I became fascinated with special relativity. I have a Ph.D. in molecular genetics but sadly during the whole stretch of schooling I only had to take 2 semesters of undergrad physics and my math runs dry after calculus II! But I’m trying! Thanks! 


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Aug708, 02:26 PM

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#6
Aug708, 02:36 PM

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#7
Aug708, 02:45 PM

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Of course nothing forces you to assume that light moves at the same speed in all directions in every frame, you'd be free to pick a set of coordinate systems where light had different speeds in different directions. But the physical appeal of this assumption is that when you do design each observer's coordinate system based on the idea that light moves at c in every coordinate system, you'll find that when you express the equations representing the fundamental laws of physics in terms of each coordinate system, they end up obeying the same equations in each of these systemsthis is a symmetry in the laws of nature known as "Lorentzinvariance". If you made a different assumption about the speed of light in different directions, the equations would be different in different coordinate systems. 


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Aug708, 04:13 PM

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#9
Aug708, 04:20 PM

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I do however agree with what you say about the "same now". It's not like we have to use inertial frames, and it's only when we do that we get Greene's result. 


#10
Aug708, 04:30 PM

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jtaravens, are you familiar with spacetime diagrams? If not, then you should drop everything else and learn about them first. If Chewie is at a positive x coordinate and is walking away from you (who I assume is at x=0) with speed v, then the slope of his world line is 1/v. His simultaneity lines must have slope v, because otherwise the speed of light wouldn't be 1 in his coordinates.



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Aug708, 05:25 PM

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#12
Aug708, 06:14 PM

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#13
Aug708, 06:30 PM

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Another question that occurs to me: is the notion of living in a spacetime with a minkowski metric physically equivalent to the notion that the laws are Lorentzinvariant? The second implies the first, but does the first imply the second? A metric in spacetime basically just tells you the proper time along an arbitrary worldline, so would it be mathematically possible to come up with laws of physics such that the proper time along any physical clock would match what you'd expect given a minkowski metric (so that if you did use an arbitrary inertial coordinate system, the line element would always be ds^2 = c^2*dt^2  dx^2  dy^2  dz^2), and yet other aspects of physics would differ from one inertial coordinate system to another?



#14
Aug708, 07:25 PM

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Postulate 1 gives us a mathematical model of space time. Postulate 2 defines the term "law of physics" and says that we will only have to consider equations of a certain kind Postulate 3 tells us what clocks measure Postulate 4, 5, and so on (I'm not sure how many we need), tell us what other gadgets measure. (#2 in particular might seem like a strange thing to postulate, but I don't think it's any more strange than e.g. Newton's second law, which really just defines the term "force" and at the same time says that the force F is such a nice function that the equation [itex]mx''(t)=F(x'(t),x(t),t)[/itex] has exactly one solution for each initial condition). 


#15
Aug908, 08:24 AM

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Thanks guys.
I printed out some stuff on spacetime diagrams. I'll try to apply what I learn to my above stated problem. John 


#16
Nov1611, 02:19 PM

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Brian Greene, with his analogy, relates the length of a loaf of bread with time and its height with the single coordinate representing all the spatial coordinates. Although the loaf of bread analogy flips the orientation of the time and space axes used in the Minkowski diagram, the mathematical relationship does not change. Basically in the Minkowski diagram the speed of light forms the boundary lines that intersect at the origin of the light cone. All other velocities are inside the the cone. On a twodimensional basis that means the intersecting lines representing the velocity of light form the asymptotes of a twosheet hyperbola. The velocities inside those asymptotes were referred to by Minkowsi as worldlines. All events trace out world lines and determine a past, present and future. He showed further that compared to a stationary person, a the frame of reference for a traveler must change. It becomes an oblique coordinate system compared to the stationary orthogonal coordinate system from the viewpoint of the stationary person. The angular differences between the coordinate of these two different coordinate systems is found by taking the arctangent of the ratio between the velocity and the speed of light. For example, if the traveler is going 60% the speed of light, you would take the arctangent of 0.60 to find out how to rotate from the stationary axis to form the oblique axis. This is the same as saying the slice of bread is cut at an angle. The angle changes when one observer is traveling relative to the other person staying stationary. Strangely, the traveler, if he could observe the stationary person, he would see his frame of reference as orthogonal and the stationary person's frame of reference as being oblique. It is kind of mind twisting, I know. I hope I have not confused anyone further than they may have been already.



#17
Nov1611, 06:36 PM

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Great to see you back in the mix, JesseM. You and Fredrik have got things on the right track here. But, some of the members here really don't like spacetime diagrams for some reason. 


#18
Nov1611, 08:36 PM

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