Maybe Farsight can clarify how he feels about it but I don't personally make a distinction between "space and time" (when mentioned together) and "spacetime". "Space" on its own has a very different meaning. Agreed. I believe the problem is that motion requires time as a discrete parameter; it's a relation between space and time, which is why "motion through spacetime" is nonsensical. It's like asking what the slope of an entire graph is! Block time is the complete historical account of spacetime, devoid of the concept of "now" or having a "flow". SR was sufficient to kill any objective flow of time, and you've personally admitted that Physics cannot account for or model the concept, so the acceptance of this type of concept should be trivial.
As I said, even though all current Physics models essentially presume Block Time in their formalism, I'll give you a more specific answer. One consequence of Block Time would be that the past, present and future all share an equivalent existence. This suggests that "knowing" the future as we "know" the past is merely a practical difficulty rather than a theoretical impossibility.
The fact Farsight keeps calling spacetime a "mathematical space" should give you a hint. Depends. If you're defining motion as "displacement in space over time", then technically yes. But you could also imagine toying around with more general definitions, for example "displacement in spacetime with respect to proper time", which is why I said it depends on how strict you want to be. Heck, you could even just agree to accept the construction "motion through spacetime" as synonymous with "motion through space over time". It's not like English is the most consistent language anyway, and as far as I know nobody has ever bothered to define all these terms precisely enough to definitively allow or rule out "motion through spacetime". There's just never been any need. Depends again on how you're defining things. The way I see it, if you try to define "flow of time" you could end up with something meaningless (flow in litres per second?), tautological (\(\frac{\mathrm{d}t}{\mathrm{d}t} = 1\)), true but not especially exciting (\(\frac{\mathrm{d}t}{\mathrm{d}\tau} \geq 1\)), a reference to an unsolved problem in physics (the second law of thermodynamics), or just a reference to human psychology (our perception that we age along with the world around us, presumably related to the second law of thermodynamics).
To me you're just describing determinism here. To the extent "block time" means anything to me at all, I don't really see any need to bundle it with determinism.
Determinism presumes both a flow of time and a discrete system state! SR does away with both concepts in one fell swoop. Throw away that definition based on those faulty presumptions and we can recover (a slightly modified) Determinism just fine. There is a frame in which particle A is measured "virtually instantly" after its creation in an EPR experiment (therefore, the system state evolves into particle B being measured second), and there is another in which particle B is measured so (therefore, the system state evolves into A being measured second). Taken together, we conclude that A and B both existed "virtually instantly" after their creation. This isn't a problem because we've already granted the future's existence to be on the same footing as the present and the past, so it should not be surprising that it can affect the present similarly. What we're left with, if we were to map "causal linkages" in a spacetime graph, are fibers of arbitrary length running at restricted angles, yet with no sense of direction. In other words, the system state necessarily includes all of spacetime, which allows for a unique existence devoid of inherent stochastic behavior.
I don't see how determinism presumes a flow of time -- this is a metaphor, not a postulate. Newtonian mechanics constrains both all of past and all of future from any instantaneous snapshot of all variables (particle positions and momenta). Both special and general relativity constraints a portion of the past and future from various finite snapshots, which makes them more useful since we have only finite knowledge of the universe.
http://en.wikipedia.org/wiki/Cauchy_surface Note a particular author name in the history list Please Register or Log in to view the hidden image! Farsight has some weird obsession with saying "There's no motion in space-time", as if there's something weird about the relativistic formulation. Actually, as Rpenner has said, Newtonian mechanics has the same thing, that if you consider a continuum of snap shots of the whole of space and pile them into a 4 dimensional construct then you have a 'space' (space-time) which doesn't evolve in the same way space does. In fact, there's no parameter to describe how the system evolves. Space-time wasn't a new concept in special relativity, space-time exists within Newtonian mechanics. The difference is that in special relativity people in relative motion will disagree on which directions in space-time are spatial and which is temporal, while in Newtonian mechanics everyone, regardless of motion, agrees. Personally I suspect part of Farsight's dubious comments about space and space-time comes from the fact mathematicians and many physicists say 'space' when referring to any 'arena' they are considering. A topological space can have absolutely nothing to do with space (I'll use italics to refer to the usual notions of time and space else this will be chaos) but specific examples of topological spaces include manifolds, the mathematical formalism is most space-times. But since time and space directions in certain Riemannian manifolds (like those relativity works on) can be 'rotated' into one another the distinction is dropped by many physicists and space-time manifolds are just called spaces. Many mathematicians don't care what physical meaning you can attach to such spaces, space or time or even multiple times, so they make the distinction even less often. Then there's things like string theory where space-time is a union is usual 3+1 dimensional space-time and a compact space, which is entirely spacial. Some papers will use the phrase space-time to mean two different constructs (3+1 and 9+1) in the same sentence, often because they are talking about compact spaces which make up the compact parts of space. So given Farsight's 'limited' understanding of any of the details of relativity (even Newtonian mechanics) it's not hard to see where confusion could arise.
Thinking about it...you might be right. Perhaps Determinism doesn't imply a flow, only a state; I guess it depends on which definition we use. Cauchy surfaces (thanks AN, fascinating!) appear to be bidirectional, but Causal Determinism claims (or at least implies, in common usage) that future states do not share the same status as past states in terms of existence. This existence disparity is what implies a flow of time.
I find it quite satisfying to manufacture a "novel" concept in my mind and then discover that someone in the past contemplated the same thing. I feel like I'm touching a mind across time. AN, can you expand on what "initial conditions" means? What I want to know is, how do Cauchy Surfaces handle counter-factual definiteness? In other words...do we presume that the CS includes information which may be present but unknowable?
Er, no. Determinism is just the idea that you can recover the full state of everything in spacetime from just partial information. For example, Netwonian and relativistic mechanics and classical field theory are all deterministic because if you know everything along any two space-like surfaces, you know everything in spacetime. This has nothing to do with "flow of time" or "discrete system states", whatever the latter is supposed to be.
Everything along two space-like surfaces IS a discrete system state, right? (Shouldn't that be 3 in spacetime?) Anyway, what you've given as a definition Determinism (while differing from what I gave earlier) is interesting. If we can recover all information from partial information then is isn't partial (more specifically, our partial information set is complete and the rest is redundant). No real consequence, I just enjoy thinking of things from a new perspective. Here's another example: I "knew" this but never thought about it explicitly in this manner. A given spacetime interval, while invariant, is simply claimed to be made up of differing proportions of time and space by different observers. Perfect
Depends. I don't know what you're calling a "discrete system state". The 2 comes from the fact that equations of motion in classical physics tend to be second order differential equations, so you need everything specified along two different hypersurfaces to fix everything. In normal usage this translates to specifying the initial conditions and initial derivatives, which you can think of as specifying everything along two infinitesimally separated hypersurfaces.
Reimannian geometry reflects the essential nature of a gravitational field. Plot gravitational potential as a curve, then zoom in on an "infinitesimal region", and it's still curved. If it wasn't and was a straight gradient instead, then when you zoom back out the curve has gone and you're left with a straight gradient. That's a uniform gravitational field, which cannot exist. 3+1 dimensions reflect the real world of space where light has freedom of motion in any combination of three orthogonal directions, whereafter we derive the time dimension from that motion. One of cause and effect. People tend to think light curves because it moves through curved spacetime. It doesn't, it moves through inhomogeneous space where the coordinate speed of light varies, and veers as a result. He didn't. See his post and note that he said We've had this discussion before. We spoke about it previously in Lattices and Lorentz variance. See this post where prometheus said I'm rather confused by your remarks about waves moving though space. He also said Ok, I grant you that "real" space is not flat but it is locally flat, meaning that I can define an observer that experiences no gravitational forces.. That's wrong. Gravity is where spacetime is not flat. It's a crucial distinction. Spacetime is the all-times block view. There's no motion in this arena. Any equations of motion you derive are equations of motion through space. It's the real speed of light, or rapidity if you prefer. The locally-measured constant speed of light isn't, because you use the motion of light to define your seconds and metres, which you then use to measure the speed of light. Thus you always measure it to be the same. You can see that the speed of light varies with position using optical clocks at different elevations, which simplify to parallel-mirror light clocks: |---------------| |---------------| The space is inhomogeneous, the vacuum impedance Z[sub]0[/sub] = √(μ[sub]0[/sub]/ε[sub]0[/sub]) varies and c = √(1/ε[sub]0[/sub]μ[sub]0[/sub]) varies too. That's what Einstein kept saying, but it's nowadays dismissed.
Space is that black stuff between the stars, and time is what people say clocks measure, even though they clock up regular cyclic motion through space. Spacetime is the mathematical abstraction employed for calculations, wherein in SR macroscopic motion through space affects the local motion we label as time. However people tend to think of spacetime as space. For example, the other day I referred to an article about Gravity Probe B. See the bit that says The space-time around Earth appears to be distorted just as general relativity predicts. That's actually wrong. It's space around the earth, not spacetime. If it was spacetime, the earth would be a frozen blue-and-white worldline streak in a block. That's not how the world is.
Yes, that's what I thought your position was, and I don't disagree; space and spacetime are different beasts, and I also agree that "motion through spacetime" makes no sense. I think in your example given above, though, we could say that spacetime is distorted only because its geometric distortion remains over time, where the distortion is extended, surrounding that blue-and-white tube running through the 4D block.
Which is? Riemannian geometry doesn't concern itself with gravitational potential, per se. This mistates the important property of smooth curves (\(C^1 \quad \textrm{or}\quad C^{\infty}\)), that their curvature has a characteristic finite scale and well below that scale the curve is exceedingly well approximated by a straight line. Understanding this is required for understanding Riemannian geometry, the geometry of smooth manifolds. Here Farsight talks about the mathematics of "zooming" in and out as having the potential to change the mathematical object which models Newtonian physics. And yet, Newtonian physics students routinely approximate a local gravitational potential as flat, because at typical human engineering scales and tolerances, it is. But satillites obviously don't fall from the sky because humans "zoom in" on a portion of the potential that causes 1 kg to weigh 2.205 pounds of force. This does nothing to address Riemannian geometry. The characteristic curvature of space-time near the surface of the Earth has a radius of about 1 light-year. Bullets and apples fall about the same amount in 1 second because their world lines are both about 1 light-second long and a curvature of about one part in 31 million (split between space and time) is why in that 1 second they both drop a little less than 5 meters in space. \(\frac{R_{\oplus}^2 c^2}{G M_{\oplus}} \quad = \quad \frac{2 R_{\oplus}^2}{R_{\tiny \textrm{Schwarzschild}}} \quad \approx \quad 0.9694 \, \textrm{light-years} \\ \frac{1}{2} \times \frac{\left( 1 \, \textrm{light-second} \right)^2}{0.9694 \, \textrm{light-years}} \quad \approx \quad 4.9 \, \textrm{m}\) (This is a toy calculation which treats gravity as geometric and obvious reduces to Newtonian's expressions in this very Newtonian case, when fully multiplied out.) For atomic processes taking \(10^{\tiny -8}\) seconds, the flatness is 100 million times more extreme.
That's fair enough. You could start with a block of spacetime, and mark it out with a uniform grid pattern. Then when you insert the blue-and-white tube/worldline of the earth, the grid is no longer uniform. Mind you, I think the typical bowling-ball-on-a-rubber sheet depictions cause some confusion here, see images. They show a flat horizontal slice of our block-spacetime grid, but distorted downwards instead of outwards.
There isn't even really any such thing as gravitational potential in GR. It's a relic from Newtonian gravity that's only sometimes used in weak field approximations. No, you've already been corrected about this before: in Riemannian geometry, it's perfectly possible to have homogenous spaces with non-zero curvature. The sphere and saddle space for example. Can you point to a uniquely distinguishable point on the sphere? No, that's just your own pet theory, and would be 3 dimensional space + motion, not 3+1 dimensional spacetime. It's called that because it's there right from the beginning in GR, before you even start talking about the motion of anything. This isn't even a correct restatement of the mainstream understanding of GR. In GR, light travels along geodesic paths, which are the closest things to straight lines in a Riemannian manifold. I've also corrected you about this before. It's only correct to say you'll end up with a curved description of the path of light if the metric components are inhomogenous, meaning that your coordinate system (not space) is inhomogenous. Because the metric components are coordinate dependent, you are completely unjustified in talking about them being inhomogenous in any absolute sense. There's only one coordinate independent relation I know of relating to the paths of light (or geodesics in general) in GR. It's the Jacobi equation: \( \frac{\mathrm{D}^2}{\mathrm{d}\lambda^2} J \,+\, R(J,\, \dot{\gamma}) \dot{\gamma} \,=\, 0\,. \)Intuitively you can think of it as a differential equation describing the convergence or divergence of light rays (among other possible things) travelling very near one another. Unlike your reference to metric components, this equation is coordinate independent, and it relates the behaviour of nearby light rays directly to the Riemann curvature tensor R. The only thing you could criticise prometheus for here is where he says "[space] is locally flat", and even that's only if you're really being a jargon nazi. It's especially silly because prometheus explains what he means by that, saying "meaning that I can define an observer that experiences no gravitational forces" which is true: a freefall observer in GR locally doesn't experience any gravitational pseudo-forces. The curvature only shows up over sufficiently large distances as tidal forces. When prometheus said "[space] is locally flat" he was referring to the fact that Riemannian manifolds locally resemble (flat) Euclidean space (or in GR specifically, flat Minkowski spacetime). This is practically a defining property of Riemannian manifolds. Spacetime is no more an "all-times" view than it is an "all-space" view. Spacetime doesn't forbid you from being interested in what's happening at a particular time any more than space forbids you from being interested in what's happening at a particular place. Again, this is a linguistic issue, not a fundamental point about motion and spacetime. No, that's always c = 299,792,458 m/s. That's what an observer will always measure the speed of light to be locally. By definition since 1983, but even before that, that was the only real speed of light in GR. No, rapidity is something different. Nope. Only the metres, and only since 1983, and we only use that definition because we think the speed of light would be different anyway. The whole point of invariance of c in relativity is that it is independent of the details of how we define our units. Even the new definition doesn't guarantee by itself that if you do an actual experiment, you'd measure an invariant c. The reason is that if you even used the SI definition at all, you'd use it once to calibrate or measure the size of your equipment. So when you measure the speed of light immediately afterwards it's no surprise that you get back exactly the same speed that you just put in. But that doesn't explain why you still measure the same speed of light with the same apparatus 6 months later, when the Earth is moving the opposite way on the opposite side of the sun. You're ignoring the experiments that found an invariant c before 1983. It doesn't matter what Einstein said. As I pointed out, to the extent those quantities vary, they're coordinate dependent. This is not out of hand dismissal. If Einstein really did place much fundamental importance on those quantities, we have good grounds for disagreeing with him.
This has been thoroughly debunked by now. You think people tend to think of spacetime as space, no matter how many times people prove they understand the difference just fine, and explain our use of language in posts like [POST=2894464]this[/POST] one. This doesn't mean anything. "Frozen" implicitly means "not changing with respect to time", which is not what spacetime means. When you look at the Earth and you don't see a worldline, that's normal, because "look" implicitly means look at a particular time, which means just a single point along a worldline. Asking what an entire worldline visually looks like is a contradiction in terms: when you look at something literally with your eyes, you are not taking an "all times" view by definition. This is just so silly. You keep picking fights with the vocabulary people use, and ignoring that this is all specified very precisely in detailed models that are defined elsewhere. When I read a post by AlphaNumeric or prometheus talking about motion or worldlines or spacetime, I don't care too much about the details of which words they use, because I know which models they're referring to, and I think in terms of those models.
