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Markus Hanke last won the day on September 18
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Well spotted đ Never consciously registered with me. What did register though was the fact that all drive systems appear to be facing rigidly backwards...kind of hard to steer or decelerate that way đ Same issue in Star Wars and many other movies as well. The other issue of course being that it is very difficult for screenwriters to even conceive of something that is truly alien in that sense, because we are trapped in our own frame of reference. Most scifi creatures are just variations on various animal/plant life, or on mythological archetypes such as ghosts etc. Itâs very hard to imagine something that is scientifically feasible (at least in principle), and at the same time entirely alien. PS. As a young kid I used to be captivated by Space:1999 and The Tripods. Nowadays these seem rather old fashioned and poor quality, but back in the day they were great! Star Trek and Star Wars as well, needless to say.

Curvature in spacetime is shown as a "fabric"
Markus Hanke replied to pmourad's topic in Relativity
Correct  but with the caveat that the concept of âgravitational potentialâ can only be meaningfully defined in certain highly symmetric spacetimes, such as Schwarzschild. It is not a generally applicable concept. Indeed. It vanishes locally in those regions, but not globally. Yes, correct. Think back to your math lessons in high school  remember how you drew simple graphs such as y=x^2. No question that the graph is globally curved. But now imagine you were to choose some point (eg x=2), and zoom into the graph there. What happens? The more you zoom in, the flatter it will begin to look. Itâs just like that. This takes a while to really get your head around. Iâm sorry I wonât try to offer a proper answer here, as typing LaTeX code on an onscreen phone keyboard is just too cumbersome and time consuming. What I will say though is donât focus on the components, but on what objects you pass to the tensor, and what you get out as a result. Rough outline: Imagine you have two test particles, whose world lines are initially parallel. Now choose a point on one of these world lines  take the unit tangent vector at that point (which is physically just that particleâs 4velocity). Then, still at that same point, take the perpendicular separation vector that connects it to the other particleâs world line. Now imagine the Riemann tensor as a machine with four slots (the four indices). Input the tangent vector into slots 2 & 4, and the separation vector into slot 3; leave the first slot empty. The output of the machine then is a new vector (because we left one index open)  it tells you how fast the separation between the test particles begins to change, and in what direction (relative acceleration between the test particles). Writing this down in math notation immediately gives you the geodesic deviation equation, bearing in mind that we need to use a covariant, not ordinary, second derivative. So the Riemann tensor is a machine that takes the tangent vector on one world line, and the separation vector between them as input; and produces as a result an acceleration vector that tells you how that separation between particles changes over time (geodesic deviation). This deviation can be a combination of any space and time direction, and can be complicated  you can get the world lines twisting around one another in a helix configuration, and all kinds of fancy stuff like that. This is really most of what there is to it in a GR context  Riemann has other uses as well (eg one can calculate tidal forces from it), but I wonât get into this here. It does in fact reflect all possible degrees of freedom of gravity. The individual components of the tensor represent tidal effects between various combinations of directions, but I really donât think itâs helpful to try to look at it this way  it wonât help you understand. Better to think of it as a machine with slots that take an input, and produces an output; with the indices of the tensor being those slots. Any tensor can be conceptualised in that way  eg the Ricci tensor takes a futurepointing timelike unit vector into both slots, and produces a real number that is the rate at which a small volume changes when in free fall. In vacuum R(u,v)=0, so in vacuum a small volume in free fall is conserved (but its shape will get distorted, which is described by a different tensor). Inside a matter distribution, neither volume nor shape would be preserved. (Note carefully that this geometric interpretation only holds so long as there is no expansion, shear, or vorticity, which is true for most simple spacetimes) Hopefully this makes any kind of sense to you. 
Curvature in spacetime is shown as a "fabric"
Markus Hanke replied to pmourad's topic in Relativity
Yes, thatâs right. As pointed out earlier in the thread, spacetime is only locally flat. If the accelerometer is too large, it will begin to measure tidal effects. 
Curvature in spacetime is shown as a "fabric"
Markus Hanke replied to pmourad's topic in Relativity
No, this one is new to me. Thanks for bringing it up. Having skimmed through the link, my first impression is that this formalism is not nearly as elegant and intuitive as the standard one (and full equivalence with GR is yet to be shown). I kind of fail to see the advantage, though the point about substructure is interesting. See studiotâs comments on intrinsic vs extrinsic to begin with. Furthermore, there is not really any force involved in gravity  when you have initially parallel test particles in free fall, and attach an accelerometer to them, it will always read exactly zero, so no forces; nonetheless in the presence of gravity their geodesics will begin to deviate. Good question! This point is a bit subtle, and really the answer should be âboth of the above, depending on contextâ. The physical manifestation of curvature is geodesic deviation  meaning that initially parallel world lines will begin to deviate as they extend into the future. It is thus necessary for world lines to have at least some extension in spacetime before âparallelâ and âdeviateâ even make sense  you canât speak of parallelism at a single event. Thus curvature has measurable meaning only across some distance. Iâm highlighting the word âmeasurableâ because counterintuitively the mathematical object describing curvature (Riemann tensor) nonetheless is a local object, like all tensors. For clarification on this point, refer back to the example about calculus in my previous post. However, there are also scenarios where the effects of gravity are in some sense ârelativeâ. Consider a hollow shell of matter, like a planet that has somehow been hollowed out (not very physical of course, but Iâm just demonstrating a principle here). Birkhoffs Theorem tells us that spacetime everywhere in the interior cavity is perfectly flat, ie locally Minkowski. Thereâs no geodesic deviation inside the cavity. Now letâs place a clock into the cavity, and another reference clock very far way on the outside, so both clocks are locally in flat Minkowski spacetime. What happens? Even though both clocks are locally in flat spacetime (no gravity), the one inside the cavity is still gravitationally dilated with respect to the far way one! This is because while both local patches are flat, spacetime in between them is curved  if you were to draw an embedding diagram, youâd get a gravitational well with a âMesa mountainâ at the bottom; and the flat top of that mountain sits at a lower level than the far away clock, thus the time dilation. So in this particular case one could reasonably say that gravitation effects are ârelativeâ between local patches. Or you can put it like this: both regions are Minkowski, but one is more Minkowski than the other The isnât very intuitive, but mathematically perfectly consistent  if you look at the world lines of the clocks, youâll find that while they appear parallel in space (theyâre simply at rest wrt to one another), they deviate in spacetime. In GR it is crucially important that one fully understands local vs global, or else thereâll be no end to misunderstandings and problems. This point is where most, if not all, apparent âparadoxesâ in GR arise. In general, no, itâs not a scalar  itâs a rank4 tensor field, the Riemann tensor. However, you can choose to look at only certain aspects of curvature, such as how volumes change (rank2 Ricci tensor), or how areas differ from Euclidean counterparts (rank0 Ricci scalar), or the average Gaussian curvature of a small region of space (rank2 Einstein tensor). But to capture all aspects, you need the full rank4 tensor with 20 independent components. Tensors are not invariant, but covariant  meaning their individual components do vary in just the right ways so that the relationships between the components remain, hence the overall object is the same for all observers. Remember a tensor is all about the relationships between its components. 
Curvature in spacetime is shown as a "fabric"
Markus Hanke replied to pmourad's topic in Relativity
Do you mean Euclidean geometry by âold geometryâ? If so, then yes  Euclidean geometry cannot guarantee that the spacetime interval is the same for all observers; physically, this means that all observers experience the same physics, irrespective of their states of relative motion. Thatâs what we see happening in the real world, so we need any good model to reflect that. Minkowski geometry (and Riemann geometry in GR) do this natively and very elegantly. 
Curvature in spacetime is shown as a "fabric"
Markus Hanke replied to pmourad's topic in Relativity
There isnât really any kind of âactionâ in the mechanistic sense of the word. Itâs just that test particles and their world lines are themselves part of spacetime, so they cannot do anything other than follow its underlying geometry. Thereâs no duality of any kind. See below analogy for clarity. As Iâve mentioned in my last post, there is information, in the form of the metric which determines the relationship between points. So it isnât a âzero setâ. This is true even very far from any sources  even spacetime without gravity has geometric structure that is different from that of Euclidean space. This is (eg) why you canât accelerate to the speed of light  the fundamental reason for this is geometric, so geometry has real measurable consequences. Itâs exactly like the calculus you learned at school  the derivative of a function is defined at a single point, yet gives you information about the slope of the entire function. Thatâs because what it really does is tell you about the relationship between neighbouring points on the graph of the function  how it changes from point to point. If youâre given just the (local) derivative, plus boundary conditions, you can reconstruct the entire function, even though any one single point of the function is just an (x,y) pair. To give an analogy (!!!)  suppose you have two people starting out on different points along the equator, and flying north simultaneously at a constant altitude. When they start out on the equator, let them be  say  1000miles apart. What happens? The further north they get, the smaller the distance between them becomes. Eventually theyâll meet at the pole. Why? There is no detectable âactionâ or force between the two planes. Each plane starts off at 90 degree angle from the equator (so their trajectories are initially parallel), and they always fly straight (thereâs never any detectable change in direction from their initial trajectory). Yet they approach one another. Thatâs because they are both confined to the surface of the Earth, which is a sphere; so they must follow its intrinsic geometry. The metric governing this has real, detectable consequences. There is no detectable information about this at any one point on the Earthâs surface. This is because the geometry concerns relationships between points, so what you do is take measurements of path lengths, areas, or angles. For example, youâll find that the sum of the angles in a triangle on Earthâs surface is no longer exactly 180 degrees  itâs possible to directly measure this deviation. But you canât do it at a single point, you need to measure across some distance. Thatâs because the effects of a nonflat metric are accumulative  mathematically, you integrate components of the metric to obtain path lengths. To put it differently, the metric defines an inner product of tangent vectors, so itâs a local object, but with global effects across the manifold. Similar principles are true for curved spacetime as well. You can measure path lengths through spacetime pretty much directly (Shapiro delay, PoundRebka, gravitational wave detectors,...) and find that they differ from what youâd expect in a flat geometry. You can also directly measure angular distortions in the geometry, ie gyroscopic precessions, frame dragging etc. Gravitational light deflection is in effect a demonstration of the angle sum in a large triangle being different from 180 degrees close to a massive body. And so on. 
Curvature in spacetime is shown as a "fabric"
Markus Hanke replied to pmourad's topic in Relativity
Yes  but you can see how test particles in free fall (ie only gravity acts on them) move in space as they age into the future. In particular, you can see what happens over time when test particles initially move in parallel. Thatâs because in a small enough local area, spacetime appears flat, just like the surface of the Earth looks flat if you only look at a small patch of it. Global curvature emerges from the way many of such small local patches are assembled. To put it differently, a single point contains no information; but thereâs information in how such points are related to one another. The physics are in the relationships, not the points themselves. 
Thatâs an interesting perspective, Iâve never looked at it quite in that way. Thank you for bringing that in here. Given this, how would you characterise the relationship between QM/EM that governs the interaction between individual H2O molecules, and NavierStokes that describes the dynamics of very large ensembles of such molecules (ie flows of liquid water, for example)? The dynamics at play are remarkably different, it seems to me. I donât quite get this example, since the scale this process (molecular vibrations) happens on is the same both here and far away. Theyâre just separated in space.

Curvature in spacetime is shown as a "fabric"
Markus Hanke replied to pmourad's topic in Relativity
As studiot already said, you only need to know about this tensor if you want to look at spacetime in the interior of an energy distribution, where it functions as the source term. In vacuum it is identically zero. Curvature itself is described by other tensors. 
How can Space, the volume of the Universe, be bent?
Markus Hanke replied to Conscious Energy's topic in General Philosophy
Relationships between events  measurements of space and time differ in the presence of massenergy, as compared to some faraway reference. The mathematical object that describes these relationships is called the metric.