Markus Hanke

Allow me to point out that the speed of light is not constant at all, it is invariant - these are not always the same !

I am sorry if my post came across as saying that there should be no questioning of existing models, and hence no further breakthroughs and developments. It certainly was not meant that way, rather it seems I didn't explain myself carefully enough. Existing models must always be questioned and continuously tested, and precisely because of that there will be further developments in our understanding; any other state of affairs would be tantamount to total stagnation, which is not what we want. However, what I was trying to point out is that not all questions that can potentially be asked will lead somewhere meaningful in the context of physics - specifically the "why" type of question lends itself to infinite regress ( as imatfaal has pointed out ), since every answer you give can be followed by another "why", ad infinitum. There may well be a few layers of "why" questions that have meaningful answers, but you need to cut it off somewhere, or else you are no longer within the domain of physics.

I should reiterate again here that what physics does is make descriptive models of the universe; it does not concern itself with the question of whether these models are the "ultimate truth" or not, only with whether or not they are valid in the sense of being good descriptions. Physics is to the universe what Ordnance Survey is to Ireland - a way to find as accurate a description ( map ) as possible. So, if String Theory turns out to be "true", then that means only that it is a good model. If LQG turns out to be "true", then that means only that it is good model. And so on, I think you get my drift. Of course it is permissible to go on and keep asking in what way a description is "the truth", but then you are no longer doing physics. Such questioning should be encouraged of course, if for no other reason than philosophical and metaphysical curiosity, but I also think it is fairly important to keep the boundaries of ( and delineations between ) the various domains of enquiry in mind. Certain domains permit certain types of questions to be asked, but will fail to answers others outside that domain - e.g. you can ask a physicist how clocks in different places are related, and he will be able to answer you, but if you ask him why an ideal clock reads time, then you will likely not receive a satisfactory answer, because in physics that is simply how it is defined to be. Likewise, a philosopher can write you an entire treatise on the question of what time is, yet he will generally be unable to calculate for you even the most simplistic of clock relationships. That is because these two domains of enquiry have different aims, ask different questions, and use different methodologies. Likewise, for the discussion at hand, there needs to be a cut-off point where we quite simply have to say "it's a valid model because it agrees with all available empirical data", and be content with that, because it achieves exactly what physics sets out to do in the first place. If we keep asking "but why is that so", then eventually there will come a point where no further answer can be given. That is not a failing of physics, but rather a manifestation of the fact that all domains of enquiry are limited in scope. Just my own two cents' worth P.S. I do not mean to suggest that we shouldn't question models in physics - far from it, since that is an essential part of the scientific method. All I really mean to say is that it is important to realise that there are questions that physics quite simply cannot answer within its own domain.

It would still be the same, but note that 9.8m/s^2 is coordinate acceleration, not proper acceleration, so its numerical value depends on how you set up the coordinate system. It is also only valid on the surface of the Earth, not anywhere else.

How exactly you interpret the effect of the wave depends on how you set up your coordinate system, but in the transverse traceless gauge the key to this is that the light traces out a geodesic in spacetime, whereas the constituent particles making up the arms of the LIGO setup do not, since they are part of a rigid structure. Furthermore, there are two perpendicular arms, and since a gravitational wave is quadrupole radiation, a passing wave will induce a phase difference since the two arms are effected differently. One way to think of it is a time-varying Shapiro delay - while the coordinate distance the light has to travel remains constant, the metric ( =proper ) distance does not, so in essence you are getting a fluctuation in the coordinate speed of light along one arm of the setup, but not along the other ( due to the polarisation of the wave ). This is detectable.

The motion of ( uncharged, non-spinning ) test bodies is independent from any measurable property other than its initial position and momentum. Therefore, it does not make any difference how massive they are, when it comes to relative linear acceleration between them. Do take note though that the 9.8m/s^2 figure is valid only on the surface of the Earth, and when both bodies are of comparable mass, then to obtain the figure one has to take into account the motion of both of them - i.e., we have a full 2-body problem, instead of just an isolated test particle in free fall. The numerical value itself will not be affected though.

Indeed. And the problem reaches even further than initially meets the eye, because if you eliminate local Lorentz invariance, you inadvertently also eliminate the CPT invariance of quantum field theory. That means that, not only would you bring down relativity in the classical realm, but you would also bring down the entire Standard Model with it.

Nothing is "getting curved", that is the point. It just so happens that the relationships between measurements taken at different events in spacetime are the same relationships as those between points on an appropriately curved manifold. Hence, curved manifolds are good models to describe gravity. There is no "fabric" here that gets mechanically distorted in any way.

Because they have energy-momentum. For ordinary bodies, their mass would be what makes up most of the contribution, but if they have a substantial amount of angular momentum or net electric charge, or if they carry a magnetic field, then that would play a role as well ( albeit a small one ). What's more, if the bodies have spatial extension ( i.e. if they can't be considered point-like ), then their shape and internal composition would also matter.

I can think of a good few places where you might find those. Sorry, couldn't help myself. I'm too old now to pretend I still care about being politically correct

The term has a mathematically precise definition, so I'm not sure what you are getting at. The word "curvature" is actually just an umbrella term for a number of related, but quite distinct mathematical concepts. Yes, pretty much. I'm not sure I would agree with that. Even something as simple as a sheet of paper is intrinsically flat to a pretty high degree of accuracy.

That is not what happens. Time dilation is a relationship between different clocks in spacetime, it is not a physical "change" of some sort that "happens" to a clock. False. Two photon clocks at different places, when compared, are subject to time dilation just like any other type of clock. There's nothing particularly special about light at all. You can't eliminate time out of the equation. In fact, it is the principle of extremal ageing which determines most of the dynamics of GR, i.e. the behaviour of test particles under the influence of gravity. The geodesic equation is nothing other than a mathematical statement of that principle. Also, remember that locally your theory must reduce to Special Relativity, so time is an integral part. That doesn't make any sense, I'm afraid. The curvature in your example is just the relationship between different events along the light beam. There is no mechanical "bending" involved, since the light is not subject to acceleration at any point on its trajectory. I have tried this for literally years when I first learned GR, and I can assure you that this won't work, because all analogies are necessarily limited and severely flawed. GR seems counterintuitive to many people precisely because they get stuck thinking in terms of analogies, not because the model itself is in any way extraordinary. Ultimately - and I might make myself unpopular by saying this - the only way to understand GR is to actually abandon all visualisation aids and analogies, and go and learn the mathematics behind it. The point here is that there is no need to master those mathematics - you need to only get to a level where you can understand their meaning, so complete mastery isn't required. That is perfectly achievable for the vast majority of people, even those who aren't mathematically inclined, but it does take patience, effort, and perseverance. The answer is - the metaphor you are looking for simply does not exist. You cannot capture all relevant aspects of a non-trivial 4-dimensional concept by drawing pictures on a screen; sometimes even I wish that was possible, but the raw truth is that it's not. Yes, you can come up with certain visualisation aids for certain aspects of the model, but it is crucially important to understand the limitations of those analogies - and that's where a lot of people fall down, and where the "counter-intuitiveness" comes in. They confuse the analogy with the model itself. GR is not a rubber sheet, it is not an expanding loaf, and it is not ripples on a pond. Full GR is simple geometry, but it is not the geometry we are used to from everyday human experience, but a generalisation of it. I should explicitly point out here that the "usual sense of the word" curvature is not the type of curvature that is used in GR at all. When the average person on the street mentions curvature, then the mental image that is invoked is one of some kind of embedded surface bending within a higher-dimensional space - like the surface of a 2-sphere bending in 3-space for example ( that's where the rubber-sheet analogy comes from ). This type of curvature is called extrinsic curvature. However, GR uses an entirely different notion, called intrinsic curvature - this is defined by what happens when you transport tangent vectors along closed curves. This type of curvature is called "intrinsic", because it makes no reference to any embedding into higher-dimensional spaces, or any measurements that are external to the manifold itself. Instead, it is entirely determined by the measurements within the manifold. There is no good way to visualise this, and if one tries to force the issue, then that's where the counter-intuitiveness comes in. For example, people see a cylinder, and they intuitively say "oh, that's curved !"; and so far as extrinsic curvature is concerned, that is correct. However, GR considers only the intrinsic geometry of the surface, and it turns out that the surface of cylinder has no intrinsic curvature at all - it is perfectly flat, and that is true whether or not the cylinder is embedded anywhere. People find that counter-intuitive, but only because they do not know the correct meaning of "curvature". Hence, the only way to really understand GR is to go away from the analogies and visualisations, and learn the actual meaning behind it.

Yes, that's right. The term you were looking for is "metric signature", and only manifolds with a signature of (+,-,-,-) or (-,+,+,+) are useful models for spacetime. Purely mathematically though, you can of course have many other types of manifolds - the Euclidean space we are all familiar with from high school geometry is an example. Or you could consider a manifold with two time directions, like (-,-,+,+). But those are not suitable models for spacetime, because they don't accurately reflect how events in the real world are related.

In real terms, a geodesic is a curve the geometry of which is such that at each and every point the proper acceleration vanishes ( in technical terms : it parallel-transports its own tangent vector ). That means quite simple that, if you hold an accelerometer in your hand and free-fall along a geodesic, then the accelerometer will read zero at all times. To put it even more simply - in free fall, you are weightless. [latex]a^{\mu}=0[/latex] That is precisely the geodesic equation, though at first glance it mightn't look the same as what Mordred has written. In actuality, what is there are events - they are points in space at a given instant in time. What GR does is describe the relationships between these events - how far they are apart, what is the shortest connection between them, etc etc. In real physical terms, these relationships are simply measurements taken with instruments such as clocks and rulers. Clocks measure distances in time, rulers measure distances in space, and accelerometers measure acceleration. Taken together, those give us relationships between events. In the simplest case, events in spacetime are related in the same way as points on a 4-dimensional flat manifold are; this corresponds to the situation in Special Relativity. However, if there are sources of energy-momentum present ( planets, stars, electromagnetic fields etc ), then it turns out that the relationship between events is more complicated - these relationships are now the same as the relationships between points on curved manifolds. Curvature is just a measure of how much measurements taken with rulers and clocks deviate from reference measurements taken on a flat background. As such, what GR does is recognise that the relationship between physical events in spacetime happens to be the same as the geometric relationship between points on certain curved 4-dimensional manifolds - hence, such manifolds are adequate models to describe gravity, and how it relates to its sources. They provide the map that describe the territory. Don't think of curvature as a physical distortion of some mechanical medium, but rather think of it as a change in relationships between events. These changes can be measured by comparing rulers and clocks in different places; it isn't just an abstract concept, but something very very physical.

Join two Krasnikov tubes with opposite orientation together - effectively you would get a CTC, without having to get singularities or event horizons involved. This is an interesting topological structure - I am sure that there is some principle of nature which would prevent this from actually occurring in the real world, but so far as pure classical GR is concerned, this seems like a doable ( and survivable ! ) concept.

There is no possible scenario in which any experiment in any lab we currently have could produce a macroscopic wormhole ( or any other significant distortion of spacetime, for that matter ) - even our most powerful particle accelerators are too "weak" by many orders of magnitude to generate the required energy densities.

To be honest, I have no idea what you are talking about Gauss's theorem is applied in three dimensions here, not just "horizontally", or else we aren't dealing with a sphere at all.

Well, I think this is already encapsulated within the question as to validity - a model may be perfectly valid today, based on empirical data available today, but cease to be valid tomorrow based on new data that has become available, and which we didn't have before. That is why no model or theory is ever "proven", it is just continuously tested with increasing levels of accuracy and sophistication. Physics is an ongoing process based on repeated application of the scientific method, not a rigid belief system.

We are far away now from anything that can even remotely be considered physics. The role of physics is really quite simple - it makes models of the universe around us, not more and not less. Given any particular model, there are really only two valid questions we can ask : 1. Is the model a good model ? 2. Is the model a complete model ? The first question is ( given access to the right technology ) easy to answer, by comparing the predictions of the model to empirical data ( i.e. measurement results ). The second question is somewhat more involved, but can also be answered by examining the constraints a model places on the space of all possible preparations and measurement outcomes; this is what happened ( e.g. ) with the controversy about hidden variables, and Alain Aspect's experiments. That's all physics. What is not physics is any number of other questions, such as 1. Is the model the truth ? 2. How does the model relate to reality behind the measurements ? 3. Was what the model describes somehow designed ? 4. Why is the model the way it is, and not somehow different ? And so on. Don't get me wrong - these are valid questions in their own right, but they are not question that physics is designed ( or claims ) to be able to answer. They belong to different domains of enquiry. I think it is very important to keep a proper perspective on this, and not confuse physics with something else, such as metaphysics and philosophy. Yes, there are parallels, overlaps, and cross-references between these disciplines, but confusing them and muddling them together leads to nothing good. Just my two cents' worth.

I don't know if you would find it instructive, but I have ( some time ago ) written a "First Primer" type article on General Relativity on my own website : http://www.markushanke.net/general-relativity-for-laypeople-a-first-primer/ It's largely non-mathematical, and focused more on diagrams and explanations. Perhaps it can be of help to you. If you want to delve into it deeper, there are also more mathematical articles on there.

The source term in the gravitational field equations isn't mass, energy or even relativistic mass - it is the full stress-energy-momentum tensor. This is a frame-independent quantity, so all observers agree on it.

That's not correct - in fact, if the body is spherically symmetric and the field is irrotational, then, in three dimensions, you are automatically dealing with an inverse-square law. Since this implies that the field must have vanishing divergence outside the source, a perfect sphere is exactly equivalent to a point source, so far as the form of the force law is concerned. As swansont has correctly pointed out, this is precisely Gauss's theorem, which is in turn a special case of Stoke's theorem, which follows from elementary topological considerations.

I think that, within the context of physics ( as opposed to other domains of enquiry ), the numerical outcome of measurements taken is precisely what constitutes objective reality, and that premise is what the scientific method is based on. One can philosophise about some type of "hidden" reality somewhere "behind" the outcome of measurements taken, but that is at best metaphysics, at worst full-blown philosophy. This may be a tenable point of view if one constricts himself to the classical domain only, but it demonstrably fails if you look at the bigger picture and consider the quantum domain also. Here, if one simultaneously assumes both Einstein locality and counterfactual definiteness ( i.e. the notion that there is an objective reality independent from any measurements performed ), there is an upper bound as to how strongly measurement outcomes can possibly be correlated - that's precisely the Bell inequalities. It is experimental fact that in the real world those inequalities are violated, which means we need to give up either Einstein locality or counterfactual definiteness ( = realism ) in order to correctly model the universe. It is difficult to imagine how giving up Einstein locality can be meaningfully reconciled with the observed causal structure of spacetime - especially given the fact that Lorentz invariance, and hence CPT invariance, appear to be fundamental symmetries -, so my argument stands that we are pretty much forced to let go of the idea that there is some type of reality separate from what we can measure, unless some future model ( quantum gravity ? ) completely overturns our understanding of spacetime and causality, which of course I can't rule out. None of this is really relevant for GR as such, but it does become relevant when one looks beyond GR at the bigger picture.

While I understand what you are trying to say, I nonetheless disagree with you. "Reality" in the context of physics is what we can measure with our instruments; in the specific case of GR, reality is what clocks and rulers measure. We can very easily compare our map ( = curved spacetime ) against real, physical measurements taken of space ( rulers ) and time ( clocks ). That is why I mentioned the Pound-Rebka experiment, since it is a nice case-in-hand to demonstrate this.

@StringJunky : Thank you To put this even more succinctly - if you don't believe that the contour lines on a map are "real", then strap a 50-pound backpack onto yourself and meet me on the summit. We'll talk then