No. GR is consistent in an arbitrary number of space and time dimensions. IT is clear, however, that we live in four space-time dimensions. The point I was trying to make, initially, is that the idea that time as a dimension, ``on the same footing'' as space, is somehow very central to GR. If you take away this central pillar, all of GR needs revising. The two theories describe different regimes. String theory can never displace GR, as string theory never makes a prediction that is in conflict with GR---indeed, when one examines string theory closely, one finds general relativity is there ``for free''. No---the important point is the sign (i.e. signature) of the metric.
BenTheMan: Do you have a link or a reference for the following? I know of neither a GR formulation using more than a 4D space-time continuum nor of any observational data for it to be consistent with. The following is correct in that time as a dimension is central to the GR model. However, GR does not treat time the same as a spatial dimension. It distinguishes between space-like & time-like Intervals. All the formulae of GR treat the spatial coordinates in a symmetric fashion. They can be exchanged (via renaming or a coordinate rotation). The time variable is not treated in a manner symmetric to the spatial coordinates. It cannot be exchanged with a space coordinate. In GR, an Interval is a 4D distance between events. GR uses Differential Geometry in a curved space to model the laws of physics. A central formula in differential geometry is the metric equation, which is the formula for distance between two points. It is a function of coordinate differences (Delta’s below) In a 2D flat space: Distance[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup] In a 3D flat space: Distance[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup] + DeltaY [sup]2[/sup] On the surface of a torus: Distance[sup]2[/sup] = [R + r*cos(Lat)[sup]2[/sup]][sup]2[/sup]*DeltaLong [sup]2[/sup] + r [sup]2[/sup]DeltaLat[sup]2[/sup], where Lat is Latitude & Long is Longitude. R & r are radii. Note that for a flat or Euclidean space, the metric formula can be expressed as a function of coordinate differences only. For a flat space, the formula is the same everywhere in the space. For curved spaces, the formula always includes coordinate values: The formula has different coefficients for different locations in the space. (See torus metric above). In the Special Relativity model, Interval[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup] + DeltaY [sup]2[/sup] - DeltaT[sup]2[/sup] is analogous to the metric formula for distance in Differential Geometry. I do not know the GR metric formula Special Relativity applies n the absence of both gravity and accelerated motion. It is a flat space model. General Relativity applies when there is gravity or accelerated motion & the metric formula is more complex. GR is a curved space model, which approaches the SR model as gravity and acceleration approach zero. In classical physics, the distance between events (an ordinary distance) is the same for all observers, independent of their relative motion or the presence of gravitational fields. The same is true for the time interval between events. In GR the Interval is the same for all observers, independent of their relative motion or the presence of gravitation fields. The spatial distance & the time interval between events is not the same for all observers. Both are affected by relative motion & the presence of gravitational fields. The last two paragraphs (above) indicate one of the most fundamental differences between classical physics & Relativity: The invariance of the GR/SR Interval replacing the invariance of the distance & time intervals between events. Id est: The loss of classical notions of absolute space & absolute time. BTW: SR/GR Interval = zero indicates two events on the path of a light ray. A negative interval (Time-like) indicates events which can have no cause/effect relationship. A positive interval indicates events which could have a cause/effect relational ship.
Please Register or Log in to view the hidden image! Where does it say that these are limited to four space-time dimensions?
BenTheMan: As far as I know the subscripts in such tensor equations are limited to 1 thru 4 for GR equations. The GR metric equation is for a 4D space-time continnum only. Could you provide a context for the equation you posted? Such equations typically mention the range of subscripts/superscripts. BTW: I have no clue relating to the equation you posted. From the little I remember of tensor analysis & differential geometry, it seems a bit strange. R appears with & without supscripts. With subscripts it is a Tensor or matrix. Without subsdcripts, it is an ordinary variable or a constant. I have seldom seen equations written using the same letter for two very different mathematical entities. Might there be a typo? Differential Geometry is not limited to 4 dimensions. However, there are few books/papers which go into much detail (or provide examples) for anything other than 2D surfaces embedded in 3D spaces. Differentiual Geometry applies with any number of dimensions. I suppose that some GR tensor equations can be viewed as applying to more than a 4D space-time continuum. However, I do not think that any such hyper-dimensional tensor equations can be viewed as modeling some type of laws of physics. A model of reality (EG: GR) must have real world quantities, fields, et cetera corresponding to the various terms & variables in the equations. Since we seem to live in a 4D continuum, I find it difficult to imagine what a 5D or 6D version of GR would be modeling and what would correspond to the various terms, variables, et cetera..
Taking GR to higher dimensions is an idea dating back to the 1920's. Oskar Klein and Theodor Kaluza found that by doing so, not only can you create a picture of gravity consistent with GR in 4 dimensions, but you can also deduce Maxwell's equations and the existence of electric charge as a consequence (see Wikipedia for references). Einstein adopted this as his primary technique in searching for a unified theory.
General Relativity is a four dimensional theory of gravity. The fact we can extend it to higher dimensions doesn't alter this fact. The resulting theory would be an extension of a 4 dimensional theory.
This is Einstein's equation, the equations which defines GR. Anyway, there's no problem taking mu to be arbitrary. Indeed, when Einstein was discovering general relativity, he was troubled by the fact that his theory never predicted the number of space-time dimensions. The number 4 (=3+1) is an input to the theory, not a prediction. If you want to model our universe, you should use D = 4, but if you don't care, then you can use arbitrary D. Anyway, the point I was making is that GR, as a model of our universe, only works in D = 4. Saying that brushes over the fact that GR falls apart as a description of our universe if we don't use time as a fourth dimension. Einstein's equations are based on the fact that time and space are two sides of the same coin.
Thread Cleaned. If you want to talk about all of the reasons why GR is wrong, go find your friends in pseudoscience. Further posts will be deleted and the posters warned or suspended, as appropriate.
BenTheMan: Curved space using time as a dimension is a model, not a reality. GR is an excellent model & even though difficult to understand, it more easily understood than models which do not use curved space with time as a fourth dimension. There is a fundamental difference between the following two statements. The mathematics of curved space is an excellent model of the physics of gravitation. Space is curved by gravity. Time & space are not two sides of the same coin. You can travel in any direction in space. Travel in time is severely restricted. There are other differences. As mentioned in a previous post, when using spherical coordinates, the angle coordinates differ in fundamental ways from Radius coordinate. Similarly, the GR space coordiinates differ in fundamental ways from the time coordinate. Time & space are not equivalent.
The second dimension requires the first dimension. The third dimension requires the first two. Does time require three dimensions?
This is turning into a debate on the philosophy of science, which is not really (I think) the point of the OP. GR is ``just a model'' in the sense that evolution is ``just a model''---by all accounts, GR is a better model of Nature than evolution, because we have a mathematical formulation of GR. If you have an alternative model of nature which treats time as anything other than the fourth dimension, you should write a paper and propose a test to distinguish your theory from General Relativity. Until such a time that general relativity is proven wrong or inconsistent, the answer the the OP is an authoritative : ``Time is the fourth dimension''. Period/ And, as I mentioned in a previous post, this is a poor example. I can always perform a coordinate transformation in which the three spatial variables are treated the same---this is called cartesian coordinates. However, I can never perform a coordinate transformation that changes the signature of the metric.
This has more to do with the fact that 3 comes after 2, which comes after 1. There is nothing deep here.
Third dimension. You need the other two, to make it the third. That's counting, agreed. But it's not just counting. You need the other two. But you can have time on its own, or with just one other dimension, can't you? So it's different in nature from the first three.
Thanks. I get your drift. 3 space dimensions and then you start again. 1 time dimension. I'm not being awkward by the way. I thought that because they shared a common description, dimension, that the 4th dimension would be a continuation of the first three. Do some people believe that time has extra dimensions like space? We certainly experience it as linear, much as the first space dimension. And what about the other dimensions that make up the extra dimensions talked about in mathematical theories? Are they extra space dimensions, time dimensions, or something different entirely?
BenTheMan: To start this post, I want to make it very clear that I consider GR to be the best model we have for Non-quantum physics. I am very much aware that GR treats time as a dimension. The following are quotes from some of your posts. I have bolded some phrases/statements. Two of the above suggest that you view time as more than an extra dimension. You seem to consider it equivalent to one of the spatial dimensions ( Id est: Interchangeable with one of them). This I consider totally invalid. GR does not treat time in the same manner as any of the spatial dimensions. There is a flavor to your posts which indicates that you view GR as more than a very useful model. If you view it as more than a model, it seems necessary to accept the implications of such a view. In particular, the model views reality as static geometry. There is no motion: There are only unchanging World Lines. In this view, there is no past, present, or future: There are only geometric properties of World Lines which are static curves in a 4D space. Do you hold the views of the previous paragraph? If not, you are not in full agreement with the following from one of your posts. The word period indicates that there can be no further discussion because the statement is an absolute fact. I agree that GR does a great job of modeling reality. It in highly consistent with all the non-Quantum observational data we have. However, I consider it to be only a model. I do not consider our universe to be static. I believe in something very similar to the commonly held notion of there being an unchangeable past, a fleeting present, & an unknown (but somewhat predictable) future. If you accept the following, you are replacing the notion of motion with the concept of a static reality & giving up the common notion of past, present, & future. Time is the fourth dimension. Period BTW: Einstein once consoled a recently widowed woman with a letter which included the following. The above is a paraphrase from memory, not a verbatim quote. It strongly suggests that Einstein viewed the GR model as being much closer to reality than I. That view & his disagreement with various aspects of Quantum Theory are the only issues on which I disagree with him. Considering who is was & who I am, I sometimes wonder if my view of past, present, & future is an illusion & I wonder if someday, “spooky action at a distance” will be explained by some variation of hidden variables or some other interpretation of QT.
Dinosaur--- I think I will quit responding to this thread, because there is nothing new for me to say. Let me conclude my position as quickly as possible. First, we both agree that GR is the only theory of gravity which works. We also agree that GR is a model of our universe if (and only if) time and space appear on equal footing in Einstein's equations. There are some misconceptions about what ``on equal footing'' means. Let me explain with an example. In the Heisenberg equation Please Register or Log in to view the hidden image! time and space appear differently. For example, time has only one derivative acting on it, whereas space has two derivatives. This is one of the motivations which led Dirac to develop his equaiton in 1920: Please Register or Log in to view the hidden image! Dirac's equation is consistent with special relativity, and can be made consistent with general relativity. Schroedinger's equation cannot. This is because in Dirac's equation, time and space appear on equal footing: in this example, they have the same number of derivatives acting on them. Now, Dirac's is a very beautiful equation. Notice that some mus are up and some mus are down, and that there are an equal number of mus. This means we should properly write \(\sum_{\mu}-i\hbar \gamma^{\mu} \partial_{\mu}\psi\) If you look at Einstein's equations above, you'll notice the same feature, that the up indices and down indices come in pairs, and are thus summed over. This is a hallmark of a realtivistically invariant theory. Schroedinger's equation does not share this feature, and cannot be put into a relativistically covariant form. Second, the hair-splitting between ``GR is right'' and ``GR is just a theory'' always gets under my skin. The difference between the two statements is trivial, and is largely a matter of semantics. Insofar as we can design experiments to test the nature of the world, GR is the simplest theory which fits the observations. Moreover, there are no predictions of GR (in four dimensions) which cannot be tested for and found. When a scientist (and not a philosopher) says ``GR is right'', this is what he means, and this is understood by everyone except philosophers and people who try to confuse laypeople. Third, I've never said that time and space are ``interchangable'', and have even gone out of my way to point out the fundamental differences in the two, down to the mathematical treatment in Einstein's equations. When I said that they are ``two sides of the same coin'', I had the Lorentz transformations in mind. GR is based on the concept of Lorentz invariance, which means that all equations describing nature should be fundamentally invariant under the Lorentz transformtions. It is a typical textbook chapter to prove that Dirac's equation is Lorentz invarant but that Schroedinger's equation is not. I want to relate a simpler example below, which will (hopefully) clear some things up. Let's consider, briefly, the relationship between time and space translations, vis a vis Special Relativity. I want to establish, categorically, that the line between time and space is not a firm one. Consider two frames, O and O'. Let us define a ``boost'' as a coordinate transformation along some axis, at some velocity. Start O and O' off in the same reference frame (i.e. they are co-moving, or something), and then boost O' relative to O. This is an elementary calculation---you have length contraction, time dialation, etc., which most of us here at least have heard of. Now, let's do a different experiment: from the same starting point, boost O' along the x axis, and then boost it again along the z axis. Now we can ask: how are O and O' related. As it turns out, O and O' are related by a rotation. So the question is, how does this make sense? Rotations are not really dynamical things---I can write the transformation describing a rotation without reference to time at all. The same is not true for the boost---the boost transformaiton requires that we give one frame some velocity relative to another frame, and velocity carries time dependence. If you stare at the equations for long enough, you find the solution: boosts are somehow ``rotations in time''. Then the fact that two successive boosts (in different directions) is equivalent to a rotation is not so mysterious, then. This is what it means for space and time to be on equal footing. So to reiterate. 1.) GR is the only theory which describes our universe, at scales much longer than the planck length. 1b.) In some regimes, Newton's laws work; however, Newton's laws can be seen to follow from Einstein's equaitons, given the appropriate boundary conditions. 2.) GR predicts that time and space are not independent entities---boosted reference frames can be related via rotations, which implies that time and space get ``mixed up''. 3.) Debating over whether this means that ``GR is right'' or that ``GR is a very successful theory'' is better left for philosophers of science, and not scientists. Anyway, I'm late for work. If you have any questions about what I wrote (regarding the example), I'll try to answer them. As far as arguing with you over what constitutes ``right'' and ``just a theory'', and trying to convince you that abandoning the four dimensional interpretation of ``space-time'' is tantamount to abandoning general relativity, though, I'm finished.
This seems to be nothing more than semantics. Let's say scientists develop a model, call it model X, that matches all known experimental data at each and every length scale, to the maximum accuracy our measurement apparatus allow. Whether you choose to say "model X is how the universe works", or "the predictions of model X match exactly with the laws of the universe" is immaterial. There is no special "God book" which tells us us that the model truly is a reflection of reality or not. If you're concerned with the physics, it doesn't matter whether you say "space-time is curved", or "the mathematical model of curved space-time matches experimental evidence". Anything beyond that is metaphysics.
