I don't really understand what is meant by curved space. Why does mass warp space? Why does curved space alter the velocity of a massive object? Normally to change an object's direction you have to apply some force to overcome inertia. So how does curved space do it? What is space anyway? Layman's terms, please.
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Ultimately, the reason we use curvature to describe gravity is because that's what fits with our observations. General relativity is extremely good at predicting the motion of objects under gravitational interactions. If you want to get some intuition for why we choose curvature to describe gravity, the reasoning is basically as follows.
Imagine that you are in a free-falling elevator. You will feel as if you are weightless. There is no way you could tell whether you are falling in a gravitational field, or whether you are in the depths of space, far from any source of gravity. Any experiment you can do will have the same results in both cases.
Actually, there is only one way to detect the presence of gravity -- if you could look out the window and see another elevator falling beside you, you would notice the elevator slowly coming closer to you. This is because both elevators are being drawn toward the center of the earth. In the absence of a gravitational field, any two objects that are "free-falling" (have no forces acting on them) will NOT be drawn closer to each other. In a gravitational field, two objects that are free-falling (have no forces other than gravity acting on them), may be drawn closer together. This is called a tidal effect, and it is gravity's only effect.
A convenient way to describe gravity mathematically is to use the mathematics of curved space-time. The basic rules are simple. Matter (and energy and all that) bend space-time. Objects always "think" they are travelling on a straight path, just like you might think that you are walking in a straight line on the surface of the earth. However, if you look at someone else travelling beside you in a "straight" path, even if you start out moving away from each other, the curvature may eventually bring you back together (on the surface of the earth, you'd have to walk pretty far for this to happen, but it could happen). If you are unaware of the curvature, it might look as though you are being pulled together. That's how gravity works.
Some objections: This may seem very counter-intuitive. On the surface of the earth, we think that we feel the force of gravity. Actually, though, the force that we feel is the force of the ground preventing us from continuing on our natural, "straight" path.
It can be hard to reconcile this geometrical point of view with everyday experience. As an example, consider two massive balls, at rest with respect to each other in empty space. It can be hard to understand why curvature would cause these two balls to come together, since they are not moving through space. The key thing, however, is that they ARE moving through time. In relativity, space and time are not two separate concepts. Gravity curves both space and time, so as the balls move through time, the gravitational curvature will cause them to bend into each other.
Technically Natural
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So GR is a descriptive theory and offers no explanation of the mechanism which curves spacetime? – Deschele Schilder Aug 18 '20 at 05:29
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All scientific theories are descriptive. Newtonian gravity doesn't explain the mechanism that creates a gravitational field from matter, and GR doesn't explain the mechanism whereby matter bends space-time. Maybe future theories will lead to more insight. – Technically Natural Aug 18 '20 at 23:25
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You must have heard the phrase "matter tells spacetime how to curve, and spacetime tells matter how to move".
Now in reality, it is not mass, but stress-energy that causes spacetime curvature. Anything that does have stress-energy (and currently any elementary particle we know about) does have stress-energy and does curve spacetime.
If you would like an analogy, spacetime itself is the tracks for a train, the train cannot go another way, it must follow the (curvature) of the tracks.
Now imagine there is a little curvature on the tracks, maybe the tracks bend very little over a 100 miles, but if you are on the train, the tracks seem just straight, you do not notice the curvature locally. This is how in our normal everyday life curvature is, you can only notice local spacetime curvature (other then the fact that the ground is accelerating upwards) if you go close to a extreme object, like a black hole.
Now you are asking how does spacetime do it? How does it change the direction of an object without a force acting on it? Just like the train must follow the tracks curvature, any object (that we know of currently, meaning any elementary particle) must follow spacetime curvature, this is what we see from all experiments.
Suppose I'm orbiting the Earth. The spacetime curvature is controlling my motion i.e. I move in a circle centred on the Earth rather than a straight line because the spacetime in my vicinity is curved. This is an example of Wheeler's statement - the mass of the Earth curves spacetime and the curvature tells me how to move. there is an important distinction between acceleration due to an applied force and acceleration due to spacetime curvature. If I'm floating in space then I can let go of an object and it will remain floating next to me. This applies whether I'm orbiting the Earth or whether I'm floating in empty space far from any masses. My acceleration relative to a released object is called the proper acceleration and it's an important invariant in relativity. Any object that is moving solely in response to spacetime curvature has a proper acceleration of zero.
We, and all objects we currently know of, exist in spacetime, and must follow its curvature.
Árpád Szendrei
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New to the term "stress-energy". Looked it up, and found it a.k.a: "Energy Momentum Tensor". It was described as: density and flux of energy and momentum in spacetime.
Paraphrased, is just another way of saying: energy and momentum are being transferred through spacetime.
Therefore, based on the original question, would it not make sense to ask: how does matter transfer its energy (kinetic) and momentum to a gravitational field?
– spaceface Aug 17 '20 at 00:01 -
Timed out on editing my comment.. Q: Is gravity an energy-momentum field? If so, how is it generated across space (spacetime)? Thanks. – spaceface Aug 17 '20 at 00:11
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@spaceface yes, the static gravitational field is based on stress-energy (or the energy momentum tensor) correct. You are asking how it is generated. Now there is something we call the intrinsic property of elementary particles, one of these properties is for example EM charge. That is like asking how is the EM field of the electron generated. In reality, we do not know. All we know is that we see in the experiments, that the electron does have a static EM field around it, and it is an intrinsic property of the electron. – Árpád Szendrei Aug 17 '20 at 03:50
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@spaceface Similarly, we do now know how mass (stress-energy) of the electron or any particle, generates a static gravitational field around the particle, we just see from the experiments that it does. – Árpád Szendrei Aug 17 '20 at 03:51
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I guess Im going to have to put a pin on that one :) So the tensors describe the various vectors (space+time) along which kinetic energy and momemtum change (flux) the gravitational field, but we dont know why they do? – spaceface Aug 17 '20 at 12:34
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Would this be correct? Term "curved space (space-time)" is a mathematical construct that describes the motion of free-falling objects (along geodesics) within a gravitational field, because Einstein remarked on the Equivelance Principle that free-falling objects are indistinguishable from other inertial frames of reference (non-accelerating).
Why is a person standing on the surface considered an accelerating, non-inertial frame of reference? Still struggling with this.. obviously.
– spaceface Aug 18 '20 at 00:50 -
@ÁrpádSzendrei When talking about electrons we can't see in experiments that they have a gravitational field around them. The field generated is way too small for that to be observable. – Deschele Schilder Aug 18 '20 at 01:42
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@spaceface "Why is a person standing on the surface considered an accelerating, non-inertial frame of reference?", because the person on the surface is being acted on by a force, that is, the force that keeps the person on the surface and not falling deeper. That force is caused by the solidness of the ground, that is caused by the strong, and EM forces that keep the matter intact that makes up the ground. So the person on the ground is not free-falling. – Árpád Szendrei Aug 18 '20 at 03:57
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@ÁrpádSzendrei - That just tells me there is a constant force being applied to keep it in place. But how does that result in the free falling object accelerating towards the earthbound object at 1G (9.81m/s2)? – spaceface Aug 18 '20 at 04:18
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@spaceface the free falling object has something called a four velocity vector. The magnitude of this vector has to be constant. The object, even if it is initially stationary relative to Earth, is moving in the temporal dimension. The object can be stationary relative to Earth, so the spatial component of the four velocity vector does not change, but the object has to move in time. But the object is in the Earth's gravitational field. This slows the object down in time. The temporal component changes, so the spatial component has to compensate, because the magnitude has to be constant. – Árpád Szendrei Aug 18 '20 at 04:49
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@spaceface Thus, the object will start moving in space, towards the center of gravity. Why does it accelerate? Because it moves deeper into the gravitational field, slows down in time even more, the spatial component has to compensate even more, it has to accelerate in space. – Árpád Szendrei Aug 18 '20 at 04:51
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That is the answer I was looking for! I think I got caught up in discussions about switching between frames of reference (inertial, non-inertial) without finding an answer to acceleration. Thank you. – spaceface Aug 18 '20 at 05:04
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Above my head, but thought you might appreciate this article, which also suggests an answer to the relationship between Mass and Gravity.
"In 1995 Jacobson argued that entanglement provides a link between the presence of matter and the geometry of spacetime—which is to say, it might explain the law of gravity." - article: "What Is Spacetime" by George Musser (Scientific American 2018)
– spaceface Aug 19 '20 at 01:43 -
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Somehow a similar topic: https://physics.stackexchange.com/questions/581151/why-would-an-orbiting-electron-lose-energy/581791#581791 – HolgerFiedler Sep 24 '20 at 19:47
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Why do particles have to follow the spacetime curvature like how trains must follow the track? Why can't the metaphorical train just like cut through the center of the Earth? – ray lin Oct 18 '20 at 18:09
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@raylin particles are excitation in the fields, which are embedded into spacetime itself. If spacetime is curved, and this embeds the fields, then the excitation exists inside the curved fields, whose curvature they must follow. – Árpád Szendrei Oct 18 '20 at 19:11
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Actually, for non-relativistic velocities, it's the curvature of time (being a part of the curved spacetime) that changes the velocity of a massive object near the Earth. Because time runs slower the closer you are to Earth, the velocity of a massive object will change to maximize the time passed in the frame of the object (called the proper time).
Usually, the change of a massive object's velocity is shown by taking a stretched sheet of flexible rubber and then putting a massive object in the middle, because of which the rubber sheet will bend and is said to represent the curvature of space. See, for example, this demonstration. It seems that a small marble put on this sheet will change its velocity because the sheet is curved. But all this happens only because the real gravity is pulling on the marble (and on the heavy object that curves the sheet), and that's what makes the marble move (because the curvature of time, as part of the curved spacetime). So the false impression is given that the curvature of space is the reason.
For objects moving at relativistic velocities, it's a combination of both the curvature of space and the curvature of time which changes the velocity. In the case of light, it's only the curvature of space which makes the light move in a geodesic path, a path with the smallest distance in the curved space. Light doesn't move through time. Time stands still for a photon.
In flat space (where we can apply special relativity contrary to general relativity) we can compare this with the case of special relativity in which an on an object always moves through spacetime with the same velocity, the speed of light. Non-moving objects only travel through time. Objects moving with a non-zero velocity move both through space and time. Massless objects move only through space. But the object moves always with the speed of light through the flat spacetime with the velocity of light.
How cause mass the curvature of space. I think it's just a fact of Nature, which can't be explained (not even in the context of a not yet found quantum gravity, which makes one doubt if it even exists). In general, space(time) without mass is flat, so I guess space(time) with mass has to be different from flat space, i.e. curved. Mass and space can't exist without each other.
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So the concentric nature of the curvature of spacetime is what causes acceleration from another frame of reference (earth)? – spaceface Aug 18 '20 at 00:19
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@spaceface If a massive object like the earth has the form of a sphere, then the curvature of spacetime is the same in all directions. The curvature can't be concentric. Curvature can be equal on concentric circles around the earth but the curvature itself can't be concentric. Curvature varies from place to place. It's the local curvature, as seen by a stationary observer on earth (who finds himself in a non-inertial frame), that gives us the impression that an object accelerates. In the inertial frame of an observer falling freely together with the object, the object is at rest, – Deschele Schilder Aug 18 '20 at 01:02
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@spaceface or has a constant velocity, the observer doesn't see the object accelerate. You could say that it is us who are accelerating towards the object (though we don't move wrt to the center of the earth) and so for us, it seems the object accelerates towards us. It may seem strange that we accelerate without changing our position (we stay at the same distance from the center), but that's accounted for in General Relativity. I've read a question on this site, of why this is the case but I'll have to find it. When I've found it (I can't recall how the question was asked) – Deschele Schilder Aug 18 '20 at 01:13
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@spaceface I'll give you the link. But to answer your question: it is not the concentric Nature of curvature which accelerates an object to earth, but the curvature (as seen by a stationary person on earth) at the position of the object. And as said in my answer, for slow-moving objects it's mainly the curvature of the time part of the curved spacetime that causes acceleration (from the point of view of an observer standing on earth). When the velocity of the object increases the space part (of the spacetime curvature) shows more influence on the acceleration – Deschele Schilder Aug 18 '20 at 01:32
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@spaceface (again, from the perspective of a person standing on earth) while the influence of the time part decreases until the object reaches the speed of light (when the object is a massless photon, for example) and then it's only the space part of the curvature that influences the acceleration. The time part plays no role anymore (which is why time doesn't proceed for photons). GR is difficult to get a grip on. I can remember how I was struggling to get some understanding. But in time it becomes clearer. Nobody knows by the way what space actually is.. – Deschele Schilder Aug 18 '20 at 01:35
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@ descheleschilder - Setting aside relativistic speeds for now, let us consider just normal or non-relativistic free falling objects. I'm just trying to understand the principal of how someone standing on surface of earth is non-inertial, or accelerating, and how that results in a free-falling object actually accelerating at 1G (9.81m/s/s). Afterall, the free falling object is going hit the ground with ever increasing force (F=ma), right? – spaceface Aug 18 '20 at 03:56
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Continued -- Now in the case of earthbound object, i think i can grasp that earth provides "push" against it, preventing it from free falling. And because it is constant push then it is considered acceleration? Perhaps somewhat akin to rotational acceleration at constant orbit.. I dont know. But that still does not explain to me how an object falls at accelerating speeds, even from earthbound objects frame of reference. So yeah.. I'm still confused. – spaceface Aug 18 '20 at 03:56
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@spaceface Suppose we find ourselves in the famous elevator accelerating through empty space. You can imagine that by this acceleration (which is why we find ourselves in the non-inertial frame of the elevator) we are pushed to the side of the lift opposite to the direction of the acceleration. Now we encounter a mass that's not accelerating (i.e. the mass finds itself in an inertial frame). According to us, the mass accelerates towards us. It feels like we are standing on the bottom of a huge mass. – Deschele Schilder Aug 18 '20 at 04:15
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@spaceface If the elevator's acceleration is 9,8 $\frac{m}{{sec}^2}$ than it feels like we are standing on earth. We see the mass fall freely towards us with the same acceleration. Now Einstein stated that there is (locally) no difference between the artificial gravity we experience in the elevator and a real gravity field: the equivalence principle. So because standing on earth is equivalent to being accelerated in an elevator, we see the mass fall freely. The mass experiences no force and finds itself in an inertial frame. – Deschele Schilder Aug 18 '20 at 04:22
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assume A is earthbound, and B is free falling. What is creating the differential in relative speed of B towards A? Since A is in a fixed position center of earth, and B is free falling towards it? – spaceface Aug 18 '20 at 04:29
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@spaceface You mean the difference, I guess. Like I said this difference is created because we are accelerating towards B. But in terms of curvature, B tries to maximize the time that passes for it. Now curvature of time means that time is running at different speeds, depending on how high you are above the earth. On the surface of the earth, time passes slower than at higher distances from the center of the earth. At infinity, time passes at it highest pace, as is always the case in empty space. When approaching infinity spacetime approaches zero curvature. It goes on... – Deschele Schilder Aug 18 '20 at 04:46
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@spaceface Because time passes at different rates at different altitudes, B will have to fall free If it ly to maximize the time that passes for it. If B stays put somehow somewhere above the atmosphere, the pace of time will be slower for B. So the time passed for B will not be maximized. Therefore it has to fall freely. Now you can ask yourself why time is curved. There are some nice layman's explanations, like Feynmann's example of an accelerating rocket with a periodically flashing light at the top (Google). If time is curved, space automatically is curved too. It's confusing, ain't it?! – Deschele Schilder Aug 18 '20 at 04:54
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As far as I know, it is not known why mass warps space (one of the biggest problems in physics, uniting general relativity with quantum mechanics). It is just a model, and all observations so far support this model. As for the second question, as far as I understand spacetime does not change an object's velocity, it just appears to change from an outside observer's viewpoint. That is, it is traveling straight and with constant speed through it's surrounding spacetime, but since this spacetime is curved locally, if you are watching from a distance (in a part of spacetime curved differently), it appears as if the object is accelerating or in a curved trajectory.
JustJoost
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If you notice in GR, it always mentions space time is curved, it does not mention space is curved. When a heavy object curves space time, not only space component is affected, the time component is also affected. We all travel through time, even an absolute stationary object travels/ is moving in time domain. When a object enters the curvature, the objects flat time domain enters to streched time domain, for an outside observer (outside curvature) this change appears as beginning of motion( like a slip on slippery surface or a rate of change -dt). As you move closer to the heavy object the more space time is curved hence the more rate of change you see.. hence the illusion of acceleration due to gravity appears. Let's consider this example, an object is moving 1 meter per second out side a curvature. Now the object enters a curvature where the space time is stretched to 2meter at the starting of curvature then 3 meter and then 4 meter and so on... please note that the stretched 2,3,4 meter are equivalent to 1 meter outside curvature, now along with the space, time is also stretched, ie at 2 meter is 2 seconds, at 3 meter, it is 3 seconds so on etc... Hence the 1m/s moving object after entering curvature appears to accelerate for an outside observer, hence the illusion of falling. We actually need to apply force to stop this falling object to the center of curvature.
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It is a postulate of general relativity that matter moves along the geodetics of curved space. Without this postulate it would not matter if you describe space with curved coordinates.
my2cts
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