## Slicing the spacetime "loaf"

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 non-physicist) 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
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 Blog Entries: 47 Recognitions: Gold Member Homework Help Science Advisor An inertial observer slices up Minkowski spacetime into constant-time-coordinate slices that are purely spatial according to him... phrased more geometrically, the spatial-slices are Minkowski-perpendicular to his worldline. (This notion of Minkowski-perpendicular 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...EfGDxdHn-2yCFA It turns out in [the spacetime geometry of] Galilean-relativity, 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.

 Quote by jtaravens 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….)
This "same now" as you are talking about is absolutely nothing physical but all in the minds of those who imagine some kind of frame (3-plane of simultaneity) to exist. Many might think that such planes are tremendously educational but personally I think they confuse more than they help in explaining relativity.

## Slicing the spacetime "loaf"

 Quote by MeJennifer This "same now" as you are talking about is absolutely nothing physical but all in the minds of those who imagine some kind of frame (3-plane of simultaneity) to exist. Many might think that such planes are tremendously educational but personally I think they confuse more than they help.
Are you saying that the 150 year difference in "now" doesn't really happen? What do you mean by “nothing physical?”

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!

 Quote by jtaravens Are you saying that the 150 year difference in "now" doesn't really happen? What do you mean by “nothing physical?”
The theory of relativity does not demonstrate that spacelike separated events have a particular order or simultaneity. Only by convention can one make the claim that these events are part of the same "now". The problem is that some teach this convention as one of the basics of relativity, hence much of the confusion.

Recognitions:
 Quote by MeJennifer This "same now" as you are talking about is absolutely nothing physical but all in the minds of those who imagine some kind of frame (3-plane of simultaneity) to exist. Many might think that such planes are tremendously educational but personally I think they confuse more than they help in explaining relativity.
You still have not addressed the question of how one could express the idea that the laws of physics are Lorentz-invariant (which is universally understood as a physical symmetry of the laws of physics, just like spatial translation symmetry or time-reversal symmetry) without referring to the idea of expressing the laws of physics in a family of coordinate systems related by the Lorentz transformation.

Recognitions:
 Quote by jtaravens 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 non-physicist) 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?
Yes, the moving train thought-experiment illustrates why, if each observer assumes that light always moves at c in their own frame, then different observers must disagree about simultaneity, hence slicing the loaf in different ways (each 'slice' represents a set of events which are all simultaneous in a given frame). As a simple example, if I'm at the exact middle of a train car and I set off a flash there, and there are two detectors at either end, then if I assume light moves at c in both directions in my rest frame, I must conclude that the two events of the detectors going "click" happen simultaneously. On the other hand, if you are on the tracks and see the train car moving forward, you'll see the detector at the front of the train moving away from the point where the flash was set off while the detector at the back is moving towards that point, so naturally if you assume light moves at c in both directions in your rest frame you'll conclude that the light must have caught up with the back detector before the front one, so the click of the back detector happened at an earlier time than the click of the front one.

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 systems--this is a symmetry in the laws of nature known as "Lorentz-invariance". If you made a different assumption about the speed of light in different directions, the equations would be different in different coordinate systems.

Mentor
 Quote by JesseM You still have not addressed the question of how one could express the idea that the laws of physics are Lorentz-invariant (which is universally understood as a physical symmetry of the laws of physics, just like spatial translation symmetry or time-reversal symmetry) without referring to the idea of expressing the laws of physics in a family of coordinate systems related by the Lorentz transformation.
Just say that laws of physics are tensor field equations on Minkowski space. This implies everything that's relevant without explicitly mentioning the coordinate systems.

Mentor
 Quote by MeJennifer This "same now" as you are talking about is absolutely nothing physical but all in the minds of those who imagine some kind of frame (3-plane of simultaneity) to exist. Many might think that such planes are tremendously educational but personally I think they confuse more than they help in explaining relativity.
I think they are tremendously educational, and I don't think it's possible to understand SR without them.

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.
 Mentor 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.

Recognitions:
 Quote by Fredrik Just say that laws of physics are tensor field equations on Minkowski space. This implies everything that's relevant without explicitly mentioning the coordinate systems.
Not explicitly, but if you want to define what it means to say the equations are "on" Minkowski space, and you want to define it entirely in terms of actual physical measurements, is it possible to do this without at some point describing a physical procedure for constructing coordinate systems in which measurements are made?

Mentor
 Quote by JesseM Not explicitly, but if you want to define what it means to say the equations are "on" Minkowski space, and you want to define it entirely in terms of actual physical measurements, is it possible to do this without at some point describing a physical procedure for constructing coordinate systems in which measurements are made?
I don't know. The mathematical model can be defined without ever mentioning coordinate systems (e.g. physical fields are sections of tensor bundles, which can be defined without coordinates), but as I have been arguing in a couple of threads recently (e.g. this one), a theory of physics needs something more than a mathematical model. It needs a bunch of postulates about identifications between things in the model and things in the real world, and I don't know if it's possible to make all the identifications we need without mentioning inertial frames. My guess is that it is possible, but it's just a guess at this point.
 Recognitions: Science Advisor 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 Lorentz-invariant? 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?

Mentor
 Quote by JesseM 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 Lorentz-invariant? The second implies the first, but does the first imply the second?
Not by itself, but we don't have to add much to the first to get the second. All we have to do is define the term "law of physics" to mean an equation constructed from tensor fields on Minkowski space (i.e. sections of tensor bundles of Minkowski space) using simple operations like tensor product, contraction, addition, and multiplication with a scalar field. (I said "like" because I may have left something out from the list). We can take a version of this to be the second postulate of the theory. The first postulate is that we take Minkowski space to be the mathematical model of space and time. The second is that at all laws of physics are equations of the kind I just described. Those two together imply that the "laws of physics" are Lorentz invariant.

 Quote by JesseM 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?
In my way of thinking about SR and GR, the idea that a clock measures a real-world quantity that corresponds to the proper time of the curve in Minkowski space that represents the clock's motion is a postulate of the theory.

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 $mx''(t)=F(x'(t),x(t),t)$ has exactly one solution for each initial condition).
 Thanks guys. I printed out some stuff on spacetime diagrams. I'll try to apply what I learn to my above stated problem. John
 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 two-dimensional basis that means the intersecting lines representing the velocity of light form the asymptotes of a two-sheet hyperbola. The velocities inside those asymptotes were referred to by Minkowsi as world-lines. 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.

 Quote by JesseM Not explicitly, but if you want to define what it means to say the equations are "on" Minkowski space, and you want to define it entirely in terms of actual physical measurements, is it possible to do this without at some point describing a physical procedure for constructing coordinate systems in which measurements are made?

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 space-time diagrams for some reason.