There’s this three-dimensional analogy used to explain how gravity follows from the 4D curvature of space-time. One expression of this analogy can be found here:
The problem I’m having with this analogy is that it uses gravity, which is the very thing it’s trying to to explain, in its explanation… A marble thrown onto the bed moves toward the bowling ball (see link) because of gravity: without gravity, even if it could not hover over the depression in the bed’s surface but somehow stuck to it, its course could still describe the equivalent of a straight line across the depression even if it was moving slowly.
I hope it’s clear what I’m trying to say. My question is, is there a similarly simple analogy that does not have this flaw?
Yes. This relates to Einstein’s inhomogeneous space, and the gμν gradient in gravitational potential. It’s a gradient in the properties of space which causes the curvilinear motion that we label as “curved spacetime”. The curved spacetime is the time-integral of this spatial gradient. For a better analogy start with the cannonball in the rubber sheet:
Imagine you’re standing underneath the rubber sheet. Grab hold of the rubber around the cannonball and pull it down to give yourself some leeway. Now tie a knot underneath the cannonball, get rid of the cannonball, and let go. Now you’ve got a flat rubber sheet with a knot in it. The knot is a “stress-energy configuration”, and surrounding it is tension. The tension gradually reduces as you move away from the knot, so if you could measure it, you would measure a radial gradient. But note that this rubber sheet represents space, and there’s no stepping outside of it. Our “marble” has to be within the sheet, and a part of it. What we need is a ripple. This repesents a light wave, or photon - a transverse wave, also known as a shear wave. In mechanics a shear wave travels at a speed determined by the stiffness and density of the medium: v = √(G/ρ). The G here is the shear modulus of elasticity, to do with rigidity. The ρ is the density. The equation says a shear wave travels faster if the material gets stiffer, and slower if the density increases. In electrodynamics the equation is somewhat similar: c = √(1/ε0μ0). Here ε0 is electric permittivity and μ0 is magnetic permeability. So scale this up a dimension and flip the tension gradient into a pressure gradient and you’re there.
I can’t wow you with my knowledge of physics but I did see a show on Stephen Hawking where they explained the stable orbits of planets using curved space time. The example used was a carnival ride. Do you know the ride that looks like a series of toddler swings. The faster the sride turns the higher the swings went. Applied to the sun and the planets they did the following: The mass of the sun creates a basin from which the smaller planets circle. They maintain their relative position because they are in motion. However they are slowing down and will one day reach a point when the mass of the sun would overcome the planets motion and they would be dragged into it. Or maybe the mass of the sun is getting larger I can’t remember. Anyway I though that image made it simpler for me. This also explains the reason why comets aren’t sucked in. Perpetual motion was discussed in the show but I won’t bother explaining it because I would most likely get this wrong.
Only the explenation of curved space and time explains all the mechanics behind the illustration, why the illustration becomes obsolete, if a person only beholds the illustration it’s heavily misleading.
Thank you my friend, congratulations are in order: you’ve succeeded in stripping my poor brain of some of its already small understanding of the mechanics of GR. I am in complete agreement with you that the “rubber sheet” explanation/model is really rather lacking.
Having thought about it some more, I think it boils down to being really rather different from anything you’ve ever thought of before. Gravity in GR is the mechanism by which mass interacts with spacetime, and vice versa. Spacetime, however, is 4-dimensional. So the curving of spacetime by mass (into a dimple such as what the ball does to the rubber sheet) is the curving of a 4-dim beast into a further 5th dimension (don’t even try to visualise it - it hurts). All matter and radiation then follows geodesics in this new, curved, 4-dim spacetime. This is observed from our perspective to be the orbits of planets, the collapse of stellar dust clouds, the bending of light around objects and our own bodies being kept against the surface of the planet - all of these phenomena occurring to matter can be explained as this matter following geodesics (the shortest possible route between two given points) in spacetime.
Note that the rubber sheet analogy fails to work properly as it considers only the curvature of space; time is kept separate from the physical model, and included only when trying to show the trajectory of objects (such as the smaller ball “orbiting” the larger). In reality spacetime is 4-dim, and all of these 4 dimensions are curved by matter with mass, and all objects made of matter move through all of these 4 dimensions along geodesics at the same constant speed: the speed of light.
Sorry guys, I still don’t understand why, say, a light, slow-moving object would move toward, say, a planet instead of just straight past it if it got near it. Farsight’s pictures illustrate my problem: why would any object ever leave one of those lines due to gravity if it was on it?
The problem lies in that you’re trying to visualise something that it is impossible for you (or anyone!) to visualise. The rubber sheet analogy provides a fairly close idea of what it’s like, but as you explained it is false. The only way of truly understanding GR and the interaction between mass and spacetime is through the equations, and they don’t provide a way of visualising the mechanics.
No, look. What we call “gravity”—the fact that objects seem to attract other objects—is supposedly no force but simply a distortion of the fabric of space-time. But the whole bowling-ball-on-a-bed example is nonsense, as the only reason why the bowling ball creates a dent in the bed is gravity. And the reason why a golf ball rolled slowly across the dent will move toward the bowling ball is also gravity. But we’re talking about a dimension in which there is no gravity: what the idea of the curvature of space-time supposedly does is explain what we call “gravity”, here in our 3D world, in terms of a higher-dimensional (4D?) world in which there is no gravity. The bowling ball example and the like are mere deferences of the question: if 3D gravity is due to 4D gravity, what causes 4D gravity? 5D gravity? Etc. etc.?
Okay… No slightly more complex analogies, either? I guess all the lines in Farsight’s first picture would have to lead directly to the bottom of the dent.
You confuses 2 concepts …you read things extremely litteraly.
Your first claim is that in space there are no gravity in itself, then you can’t mix the thought of applying gravity to disturb spacetime, thus the fabric of space.
It’s like air doesn’t smell, but if someone farts close by, it may smell …then you can’t understand why air now suddenly smells as the first statemen will conflict with other observation.
I think you address two different issues - the difference between the theories of ‘gravity’ and ‘curved spacetime’ on the one hand, and the failure of a model to describe the latter theory free from the restrictions of the former theory. The second issue seems legitimate to me - the model is confusing, but as for the first issue I can’t get my head around your questioning.
“a distortion of the fabric of space-time” is “a force” - at least the first term refers to the same thing as the second but from a more objective, comprehensive, less subjective, limited perspective; 4D instead of 3D. Time is integrated into the model. This goes beyond the conceptions that we are able to deduce from our physical orientation.
Einstein induced this theory, using as an assumed premise the unity of everything - which as a scientist (a religious one) he ardently wished to discover, describe. He basically created the model. You ask for a more complex metaphor - but the theory is itself a metapor. It “forces” the idea of time into a model of physicality, using the image of a quality that we know only as spatial - “curvature”. So, in order to grasp it, you have to think creatively - i.e. falsify the literal meaning of the language you’re using. n
The cause of the failure of the model is that it is too crude, too literal, does not call on the imagination.
“Gravity” is a term to describe from a 3D perspective, the “activity” of physical bodies. This activity only implies time, so does Newtons model - Einsteins model includes time. So “gravity” means in 3D what space-time curvature means in 4D.
That sounds nice, but doesn’t really say anything. If you can imagine it, you must be able to portray it, in theory (i.e., of course you may lack the necessary skill and/or material or something like that).
And of course thinking of 4D as 3D is metaphorical. The thing is, in science a metaphor only has value insofar as it can be taken literally: otherwise, why invoke it at all? For the beauty of the image? But that is art, not science.
[size=95]She found Ulysses among the corpses bespattered with blood and filth like a lion that has just been devouring an ox, and his breast and both his cheeks are all bloody, so that he is a fearful sight; even so was Ulysses besmirched from head to foot with gore.
[Homer, Odyssey, book 22.][/size]
This example may help explain why Odysseus was covered in blood, but it will not help explain why driving sharp objects violently through living flesh tends to get blood all over the surrounding area. It does not matter whether it’s Odysseus using his arrows and sword, or a lion using its teeth and paws. Likewise, it doesn’t matter whether what we call “gravity” is a 3D or a 4D force. Fact remains that it cannot be explained in terms of 3D other than as a “force”. Therefore, examples like the bowling ball example are completely worthless as far as understanding gravity is concerned. And they are misleading, as they make one think one now understands something one formerly did not understand, in terms of something one already understood. “Yes! I now understand gravity in terms of—gravity…”
Only the word ‘curve’ is merely a metaphor - it suggests the math.
Of course, in the formulas, the word ‘curve’ does not appear, only the type of math that we would also apply to a curve. That the non-mathematical formula seems circular does not mean that the mathematics is meaningless. I always assumed that the mathematics of this are in fact meaningful, that they allow for technology / method, for which the older model is not sufficient ground. But I don’t know that for a fact.
Imagine a swimming pool. Every morning you swim from one end to the other in a straight line. But one day in the dead of night I truck in a load of gelatine powder and tip it all down the left hand side. This starts diffusing across the breadth of the pool, imparting a viscosity gradient from left to right. The next morning when you go for your swim, something’s not right, and you find that you’re veering to the left. If you could see your wake, you’d notice it was curved. That’s your curved spacetime, because the pool is the space round a planet, the viscosity gradient is Einstein’s non-constant gμν, you’re a photon, and left is “down”.
As to how the gradient attracts matter, consider a single electron. We can make an electron along with a positron from light, via pair production. Since the electron has spin angular momentum and magnetic dipole moment and can be diffracted, think of it as light trapped in a circular path. So if you’re swimming round and round in circles, whenever you’re swimming up or down the pool you’re veering left. Hence you find yourself working over to the left. The same principle applies to protons, they have spin angular momentum etc too, and they can be annihilated back to light just like electrons. Matter is made of electrons and protons etc, and that’s why things fall down.
See ag-physics.org/gravity/ for another take on it. I don’t agree with Albrecht’s “relativity without Einstein” or the exchange particles, but it’s the same general picture.
Anyway I though that image made it simpler for me. This also explains the reason why comets aren’t sucked in. Perpetual motion was discussed in the show but I won’t bother explaining it because I would most likely get this wrong.