I Questions regarding traveling speed in time and gravity as a force

[Grasshopper]

I don’t understand how there CAN’T be time curvature since clocks do not tick at the same rate relative to altitude. I mean, when you look straight down at a curved surface, the only reason you know it’s curved is because points of perspective appear further away from each other or closer to each other than they would be if the surface were flat. Likewise, if you compare ticks of a clock where time curves to a clock where time doesn’t curve, I imagine the time between ticks on the curved time clock will be greater the closer you get to the surface, while a “flat time” clock would have the same interval between ticks (relative to the clocks you compare with, which are at different altitudes). Drawing little dots (ticks) that get further and further away from each other on one end and closer and closer to each other intuitively feels to me like an exact analogue of one dimensional curvature. If I wanted to draw a 1D curvature, that is exactly how I’d draw it. Like this:

If you are running holding a leaking bottle, the drops will be further apart when you’re running fast, and closer when you’re running slow. Which means there’s an acceleration, and if you plot that on a time vs space graph, you’re plotting a curve. The analogy only goes so far, but the plot points resemble a curve to me.
That’s kind of what curved spatial surfaces do when you look down at them: the distance between points you’d use to gauge how far away something is different than if the surface were flat. Since clocks in a gravitational field do something analogous to what a curved surface does, I don’t see what is wrong with saying time is curved.
And as for the spacetime thing, in SR it is already clear that one man’s time is another man’s spacetime (much like one man’s electric field is another man’s combination of a magnetic field and electric field). Since even in flat spacetime you have the mixing of space and time depending on reference frame, the notion of curved spacetime — including curved time for reasons above — doesn’t seem problematic to me.

 

[anuttarasammyak]

I understand "mixing time and space" as being displaced in the spatial axis, which means changing the coordinates of both time and space. But not the bending of the axis. The axis are there just to measure the passing of time and distance. I don't understand what the heck it means to bend the line that is just used to count seconds.

Starting from comparison of IFRs in SR, you would share interest with me in observing rotating system that is like a merry-go-round. The larger the radius, the faster the piece moves tangent so #1 Slower time goes. the pieces go out if not bound toward the center. #2 Force (centrifugal force) appears. Perimeter rope of length $2\pi r$ is not enough due to Lorentz contraction of the parts. #3 Perimeter / radius > 2$\pi$. These amazing features in rotating system as function of radius are almost what you would expect in "bending of the axis", aren't they ?

Honestly saying, rotating system is a fake of bending spacetime because we can make it global IFR just by stopping rotation. Gravity does them in real way that we can cancel them only locally but not globally.

 

[A.T.]

I don’t understand how there CAN’T be time curvature since clocks do not tick at the same rate relative to altitude.

The problem lies with the term "curvature". It can mean different things. The cone surface in the video below has no intrinsic curvature (that's why we can roll it out flat without distortion). But it shows variable distances along the time dimension depending on altitude, which implies varying time dilation and gravity (apple falls down).

Of course if you draw the global picture, there will be intrinsic curvature (we cannot roll it out flat without distortion), because there is no way to connect the local cones without introducing it:

 

[ersmith]

@ersmith
You painted two examples: one with a stationary clock, where it only moves throught the time axis (parallel to the time axis, perpendicular to space axis) and the other with a moving clock, having an angle between both axis. So, the moving clock will tick slower than the stationary clock, because part of that spacetime limit is now into the spatial axis and not only in the time axis.
I understand "mixing time and space" as being displaced in the spatial axis, which means changing the coordinates of both time and space. But not the bending of the axis. The axis are there just to measure the passing of time and distance. I don't understand what the heck it means to bend the line that is just used to count seconds.

The line that "just counts seconds" in one coordinate system (e.g. that in which the clock is stationary) counts both seconds and meters in another coordinate system (e.g. one in which the same clock is considered "moving").

Now consider the case where a clock is considered to be "accelerating". In that case the relationship between space and time is continuously changing. For example, suppose an observer is inside an accelerating spaceship with two very accurate clocks. She drops one and uses the other to record the ticks of the falling clock (adjusting for light speed travel time, of course). Each tick of the falling clock is a microsecond, so if it takes 1 second to drop to the floor of the spaceship cabin then the observer will record a million ticks. These ticks all take place at different points in the cabin, so (relative to the observer's preferred reference frame, where she is considered stationary) they each have a different point in both space and time, whereas the clock the observer continues to hold only change in time (relative to the observer). If we plot a spacetime diagram in the observer's reference frame, the held clock follows a "straight" line in spacetime (along the observer's time axis) and the falling clock follows a "bent" or "curved" line as it continuously changes speed.

Now let's look at it from the perspective of the falling clock. In its preferred reference frame it remains stationary, and its time axis is straight, but the observer, rocket, and held clock are all accelerating relative to it and the held clock's spacetime diagram is a curved line. This is also the point of view that an outside observer would have.

So which one is "truly" accelerating? Well, we can attach an accelerometer to both clocks and see that it's the held one which is accelerating (the rocket is pushing on it) and the falling one is not. Put another way, the held clock is experiencing a force, the falling one is not.

Now consider the same experiment on the surface of the Earth. Again, we would find that the falling clock experiences no forces, whereas a held clock is continuously experiencing a 1g acceleration. This is why gravity is considered a fictional force in GR; the clock that's "falling" is not actually experiencing any force (at least not so far as we can measure) whereas a clock that's supposedly "at rest" on the surface of the Earth is experiencing a force, and you can physically measure that with a scale or other force measuring device.

So saying that a dropped apple falls down to the Earth is not really accurate. It's more accurate to say that the Earth's surface accelerates upwards to meet it. This is where our intuition about space and time becomes difficult, because naturally we assume that if the Earth's surface is everywhere accelerating then the Earth must be growing (or shrinking). That would only be true if spacetime were flat around the Earth. The fact that the Earth isn't changing in size shows that spacetime around the Earth is curved.

 

[ffp]

@robphy Oh, sorry I wasn't talking about the worldline, but about the time axis.

No, it is not

I was talking about the light cone. While I don't know much about relativity, I know what a light cone is. And it doesn't exist in real world. You don't see cones for every object/particle around. It is just a tool to demonstrate what I said in the post you quoted. If you think I am wrong, please explain why. That might be a good game-changing information.

Not time. Spacetime. You have been told this more than once already.

You have also already been told that asking for experimental "proof" of a theory is pointless. Theories are models. It is impossible to prove a model correct. The best you can do is to subject it to more and more stringent experimental tests, and see if it continues passing those tests. So far, GR has passed every experimental test that has been done.

Do you think there is much difference in stating that gravity comes from time or spacetime? I still find it an odd statement. I can understand space being curved. I can understand time being curved. I can understand both having a relation. I can't understand the dimension of time (even that it's spacetime, it still have differentiation of space and time, they are not the same) affecting the displacement of objects in space (I'm hitting that key because I saw that explanation several times, including in books that were recommended here!).

Yes, GR has passed all the tests. Those proved time dilation, curved space (or spacetime if you will), constant speed of light, etc. However nothing related to gravity and its origins, as far as I know. What you think about that?

Not in the theoretical model of GR, no.

Many physicists believe that GR might not be a fundamental theory, and that some more fundamental theory of gravity (the usual term is "quantum gravity") might treat gravity as a "fundamental force" like the other three interactions in our current Standard Model of particle physics. Usually such theories are assumed to be a theory that somehow unifies all of the fundamental interactions. But no such theory has yet passed any experimental tests. And if you want to discuss such theories, the place to do it is in the Beyond the Standard Model forum, not the relativity forum.In GR, the spacetime geometry is caused by the distribution of matter and energy in the universe.

That is very intersting! This more fundamental theory disregard gravity being a product of spacetime curvature? I will take a look at them and in the respective forums.
Also, this means that not all scientific community agree about the explanation of gravity provided by GR?

@Grasshopper Great post. I don't have an issue with "cruved time" either, as long as it works as you said, by having different flow of time rates, and not affecting objects in space.

@anuttarasammyak You said that time goes slower for points farther from the the center as they rotate faster. There is length contraction because of that speed difference. Centrifugal force appears. I don't get the perimeter/radius >2pi. Nor how this is analog to bending the axis.

@ersmith The "line that count seconds" I was talking about is the time axis.
The Earth surface going up thing is another issue of GR that I can't understand. Why having a spacetime curved around Earth stops its expansion? Also, about the accelerometer, they works similar to a spring inside the housing attached to a small part, it's the acceleration of the inside in relation to the housing that is measured. It reads 0 in freefall because the housing and the device inside are both accelerating equally, while on Earth you are holding the housing.

 

[jbriggs444]

I was talking about the light cone. While I don't know much about relativity, I know what a light cone is. And it doesn't exist in real world. You don't see cones for every object/particle around. It is just a tool to demonstrate what I said in the post you quoted. If you think I am wrong, please explain why. That might be a good game-changing information.

A light cone is "real" in the sense that many physicists would use the term. It is something that is "invariant". We call a thing invariant if it holds independent of the frame of reference or coordinate system that you adopt.

Whether event a is on the light-cone of event b is an invariant fact of the matter. It does not depend on one's point of view. This is enough to make it "real" in our eyes.

We can do physical measurements to determine where the light cone for a particular event is. Measurements that give the same result regardless of one's choice of reference frame. That is as real as it gets.

Contrast this with the question of whether two events occur at the same time. The answer to this question is not invariant. It depends on an arbitrary choice of synchronization convention. There is no "real" answer to this question.

 

[anuttarasammyak]

I don't get the perimeter/radius >2pi. Nor how this is analog to bending the axis.

Rotating system people measure the perimeter for r=1 m with measure and find perimeter is more than 2$\pi$ = 6.28.. m.

I tried to understand what you mean by saying "bending the axis". If your bending axis has nothing to do with phenomena of time pace change, emergent force or geometry change, I have nothing more to do for now, please forget what I said.

 

[PeterDonis]

I don’t understand how there CAN’T be time curvature

Time isn't a surface (or more generally a manifold), so "time curvature" makes no sense. Spacetime is what is curved in the presence of gravitating masses.

 

[PeterDonis]

I was talking about the light cone.

Doesn't change my response at all.

And it doesn't exist in real world.

"Exist in the real world" is not a scientific concept to begin with, so it's off topic here. This is a physics forum, not a philosophy forum.

Our model of spacetime, including light cones, makes very accurate predictions that have passed thousands of experimental tests. That's the most you can ask of a scientific model.

Do you think there is much difference in stating that gravity comes from time or spacetime?

Nobody has said either of those things. What we have said is that gravity comes from curvature of spacetime.

I can understand space being curved.

Spacetime being curved is the same kind of thing. Spacetime is a geometric manifold, just like the "space" you are imagining, and the same geometric concepts apply to it.

I can understand time being curved.

I don't see how, since, as I said in post #68 just now, time isn't a surface (or more generally a manifold), so saying it is curved makes no sense.

(even that it's spacetime, it still have differentiation of space and time, they are not the same)

It is true that spacelike and timelike vectors are different kinds of things in spacetime; but those are not the same as "space" and "time". "Space" and "time" are coordinate-dependent, and as you have already been told, coordinate-dependent concepts are the wrong thing to focus on if you want to understand physics. The right things to focus on are invariants; whether a particular vector is spacelike or timelike (or null, aka lightlike) is an invariant.

GR has passed all the tests. Those proved time dilation, curved space (or spacetime if you will), constant speed of light, etc. However nothing related to gravity and its origins, as far as I know.

I have no idea what you're talking about. GR is a theory of gravity. All of the experimental tests of GR involve gravity.

This more fundamental theory disregard gravity being a product of spacetime curvature?

No. It says that gravity being a product of spacetime curvature is an approximation valid in a particular range of circumstances (roughly, when spacetime curvature is small enough that quantum gravity effects can be ignored--which turns out to be a pretty wide range; our current best guess is that quantum gravity effects don't become significant until spacetime curvature reaches the Planck scale, which corresponds to an energy density about 94 orders of magnitude larger than the energy densities in our solar system). This is no different from Newtonian gravity being an approximation to General Relativity valid in a particular range of circumstances (when gravity is weak and all speeds are slow compared to the speed of light).

this means that not all scientific community agree about the explanation of gravity provided by GR?

No. It just means that the scientific community is considering the possibility that GR might not be a fundamental theory. GR will still be a valid approximation within its domain no matter how that comes out, which means that within that domain, GR's explanation of gravity using spacetime curvature will still be valid, since it will still make accurate predictions. Just as the Newtonian model of gravity as a force still works just fine if you're trying to predict the trajectory of a baseball, let's say, here on Earth.

 

[ersmith]

The "line that count seconds" I was talking about is the time axis.

There is no unique time axis. There are infinitely many possible choices for a time axis. There's no point in talking about "time" as a thing independent of "space", because one observer's time axis is skewed into space in another observer's coordinate system. This is really important: you won't understand relativity if you don't get that space and time are not independent. That's why saying something like "gravity is caused by curvature of time" is nonsense: "time" isn't a well defined thing, you always have to consider space along with it. It's also why light cones are so important: all observers, regardless of how they divide spacetime into "space" and "time", can agree on the contents of light cones and hence on what things are "past", "future", and "elsewhere". Note my choice of words: "present" does not have a unique meaning in relativity (due to the relativity of simultaneity) and "past" and "future" are only defined relative to a light cone.

The Earth surface going up thing is another issue of GR that I can't understand. Why having a spacetime curved around Earth stops its expansion?

OK, this one is hard. At some level this question is like "why in Newtonian physics do things move in a straight line unless acted on by a force"? The answer is, that's just the way things work. Similarly, in GR objects not acted on by a force move in a "straight line" in curved spacetime. In the absence of any forces everything on the Earth's surface would follow geodesics in spacetime, i.e. would keep falling towards the center of the Earth (forming a black hole).

Similarly for a "falling" apple. It's just following the nearest thing to a straight line that exists in local spacetime, namely a path that leads towards the center of the Earth in the future. It'll follow that path until acted upon by a force, e.g. contact forces from the surface of the Earth (or from your hand, if you catch it). Note that its path goes through both space and time.

 

[Grasshopper]

Time isn't a surface (or more generally a manifold), so "time curvature" makes no sense. Spacetime is what is curved in the presence of gravitating masses.

But isn’t one observer’s time another observer’s time and space? (Edit — I guess that would make the entire thing spacetime curvature anyway, wouldn’t it?)

 

[Grasshopper]

@ffp

I take it you have no problem with “An object remains in uniform motion unless acted by an outside force,” right?

Well, if spacetime is curved, then “uniform motion” would be along geodesics rather than straight lines, would it not, and those geodesics near Earth lead toward the surface. I can’t really see the logical problem here.

 

[jbriggs444]

But isn’t one observer’s time another observer’s time and space?

[Trying to present this at as simple a level as possible -- because that matches my level of understanding]

Yes. The coordinate delta for the separation between two events may have a non-zero time component and zero spatial components according to one coordinate system. Meanwhile, the coordinate delta based on another coordinate system may have a [larger] time delta along with non-zero spatial deltas.

So the separation might be (4,0,0,0) in one coordinate system and (5,3,0,0) in another.

"Curvature" as the term is used in this context is the intrinsic curvature of a [sub-]space. It has nothing to do with wavy world-lines. It has to do with things like spheres where the surface area does not match up nicely with the radius. Or triangles where the angles do not add to 180 degrees.

The usual introduction to intrinsic curvature is by way of analogy. One considers the surface of a sphere. For instance, the surface of the Earth. And one considers a triangle from north pole, down to the equator, westward about 10,000 kilometers (1/4 of the way around the earth) and then back up to the north pole. Each angle on this triangle is 90 degrees. The sum is 270 degrees, not 180 degrees. This reflects the fact that the surface of the Earth is curved. This is intrinsic curvature. One cannot flatten out the surface of the Earth without wrinkling it or tearing it.

By contrast, we could consider a roll of paper with a triangle drawn on it. The sides of the triangle would appear to be curved. But the angles at the corners would still add to 180 degrees. And if you unrolled the paper, you'd have a nice flat ordinary triangle. That would be extrinsic curvature.

"intrinsic curvature" is a property of the space. [There is a conceptual leap that can be made here. One can consider a two dimensional surface with the same topology as the surface of the Earth without demanding that it be embedded in a three dimensional parent space. But that is a hard leap to make, so I will not belabor that point].

"extrinsic curvature" has to do with the way we represent the space -- how we "embed" it as a sub-space in a flat parent space. Like rolling up a 2 dimensional map to fit into a three dimensional tube.

For two-dimensional surfaces, curvature is easy. You have [locally] saddle shapes with negative curvature or spherical shapes with positive curvature. So one number is enough to specify the [local] curvature of the space. When you get into three or four dimensions, curvature involves more components. We call it a tensor.

For one dimension -- lines, there is no such thing as intrinsic curvature. The thread on a spool has no such notion. Sure, it may be extrinsically wound up on the spool. But intrinsically, there is no difference between a straight thread and a thread on a spool. All a blind bug on the thread can determine is how far he has crawled so far.

 

[Grasshopper]

[Trying to present this at as simple a level as possible -- because that matches my level of understanding]

Yes. The coordinate delta for the separation between two events may have a non-zero time component and zero spatial components according to one coordinate system. Meanwhile, the coordinate delta based on another coordinate system may have a [larger] time delta along with non-zero spatial deltas.

So the separation might be (4,0,0,0) in one coordinate system and (5,3,0,0) in another.

"Curvature" as the term is used in this context is the intrinsic curvature of a [sub-]space. It has nothing to do with wavy world-lines. It has to do with things like spheres where the surface area does not match up nicely with the radius. Or triangles where the angles do not add to 180 degrees.

The usual introduction to intrinsic curvature is by way of analogy. One considers the surface of a sphere. For instance, the surface of the Earth. And one considers a triangle from north pole, down to the equator, westward about 10,000 kilometers (1/4 of the way around the earth) and then back up to the north pole. Each angle on this triangle is 90 degrees. The sum is 270 degrees, not 180 degrees. This reflects the fact that the surface of the Earth is curved. This is intrinsic curvature. One cannot flatten out the surface of the Earth without wrinkling it or tearing it.

By contrast, we could consider a roll of paper with a triangle drawn on it. The sides of the triangle would appear to be curved. But the angles at the corners would still add to 180 degrees. And if you unrolled the paper, you'd have a nice flat ordinary triangle. That would be extrinsic curvature.

"intrinsic curvature" is a property of the space. [There is a conceptual leap that can be made here. One can consider a two dimensional surface with the same topology as the surface of the Earth without demanding that it be embedded in a three dimensional parent space. But that is a hard leap to make, so I will not belabor that point].

"extrinsic curvature" has to do with the way we represent the space -- how we "embed" it as a sub-space in a flat parent space. Like rolling up a 2 dimensional map to fit into a three dimensional tube.

For two-dimensional surfaces, curvature is easy. You have [locally] saddle shapes with negative curvature or spherical shapes with positive curvature. So one number is enough to specify the [local] curvature of the space. When you get into three or four dimensions, curvature involves more components. We call it a tensor.

For one dimension -- lines, there is no such thing as intrinsic curvature. The thread on a spool has no such notion. Sure, it may be extrinsically wound up on the spool. But intrinsically, there is no difference between a straight thread and a thread on a spool. All a blind bug on the thread can determine is how far he has crawled so far.

Right, that makes sense in terms of the limitations of one dimension. However, what about a two dimensional surface in which an observer is limited to traveling on one dimension. If unit distances for him change in length, is that not an example of curvature or distortion? Or is that something such an observer can never measure?

On the other hand, you could say he just has a weird coordinate system. But, if the two dimensional surface is one such that distance on the surface is not $z = \sqrt{x^2 + y^2}$ , wouldn’t the 2D surface be curved, and if so, why wouldn’t it affect distance measurement in the one dimension on which he is confined?

 

[jbriggs444]

Right, that makes sense in terms of the limitations of one dimension. However, what about a two dimensional surface in which an observer is limited to traveling on one dimension. If unit distances for him change in length, is that not an example of curvature or distortion? Or is that something such an observer can never measure?

A unit distance is a unit distance. It cannot change in length, by definition. If it did change length, how would you know, other than by comparing it to something else?

On the other hand, you could say he just has a weird coordinate system. But, if the two dimensional surface is one such that distance on the surface is not $z = \sqrt{x^2 + y^2}$ , wouldn’t the 2D surface be curved, and if so, why wouldn’t it affect distance measurement in the one dimension on which he is confined?

It has nothing to do with coordinate systems. Intrinsic curvature exists independently of coordinate systems. If you can trace three straight lines that meet in 90 degree corners, you have curvature regardless of whether you are using lines of latitude and longitude at one angle or at another.

What you could say is that he has a weird "metric". Consider, for instance, a flat map of the Earth using a Mercator projection. The distance between points on this map that you measure with your ruler on the paper will not match the correct distances that you pace out on the surface of the Earth. The shortest path line that you draw on the map with a straight-edge will not match the shortest path great circle that you would follow on the surface of the Earth.

A "metric" is a function that works pretty much like the table of distances that you can find in old maps and atlases. You find a row with your starting city on the left. You find a column with your destination city on the top. You read off the distance between those two cities from the cell where the row and column intersect.

The metric is not the same thing as a coordinate system.

A "coordinate system" for a two dimensional space is a mapping between pairs of coordinate values and points in the space. For a two dimensional space one has coordinates with two values each. For three dimensional space, one has coordinates with three values, etc.

One can think of the metric as giving the distance between a given coordinate pair e.g. between $(x_1,y_1)$ and $(x_2,y_2)$. Or as giving a distance between the points at those coordinates. The latter choice makes the metric independent of coordinates but makes it more difficult to write down explicitly.

With that said, yes: If the metric is other than $d=\sqrt{x^2+y^2}$, is still suitably smooth and is more than just a re-scaling then the curvature in the neighborhoods of at least some particular points will be non-zero. [I'm no expert on differential geometry, but I am pretty sure that holds]

If you change the metric and continue to identify a particular pair of points then yes, that changes the distance between those points, by definition. What is your point? And what does that have to do with confining a path to one dimension?

 

[ffp]

@jbriggs444 Yes, I understand the concept of light cones. My point was that it is an abstract mathematical tool. It is not real like an electron. The light cone was used in comparison of the statement that "the time axis is curved" meaning anything more than "time flow in different rates if the axis is curved".

Spacetime being curved is the same kind of thing. Spacetime is a geometric manifold, just like the "space" you are imagining, and the same geometric concepts apply to it.

Fair enough. Then I shall think of spacetime as similar to space but fundamentally related to time. Makes sense, if we can imagine space being curved, spacetime can be curved as well.

I don't see how, since, as I said in post #68 just now, time isn't a surface (or more generally a manifold), so saying it is curved makes no sense.

You hit the spot here. That is exactly my issue here. Time isn't a surface. It isn't spatial. So it can't be curved. I can understand "time being curved" as time flowing in different rates though. What you think?

I have no idea what you're talking about. GR is a theory of gravity. All of the experimental tests of GR involve gravity.

They involve, but they do not prove that an apple fall on Earth because of the curvature of spacetime, right? We just can explain things that we couldn't with Newtonian gravity.

No. It says that gravity being a product of spacetime curvature is an approximation valid in a particular range of circumstances (roughly, when spacetime curvature is small enough that quantum gravity effects can be ignored--which turns out to be a pretty wide range; our current best guess is that quantum gravity effects don't become significant until spacetime curvature reaches the Planck scale, which corresponds to an energy density about 94 orders of magnitude larger than the energy densities in our solar system). This is no different from Newtonian gravity being an approximation to General Relativity valid in a particular range of circumstances (when gravity is weak and all speeds are slow compared to the speed of light).

No. It just means that the scientific community is considering the possibility that GR might not be a fundamental theory. GR will still be a valid approximation within its domain no matter how that comes out, which means that within that domain, GR's explanation of gravity using spacetime curvature will still be valid, since it will still make accurate predictions. Just as the Newtonian model of gravity as a force still works just fine if you're trying to predict the trajectory of a baseball, let's say, here on Earth.

And GR not being a fundamental theory doesn't means that the explanation of why gravity happens is wrong? I know we will still use it, the same way we still use Newton's gravity, but we will know that the sourec of gravity is not the one stated.
The thing is, until GR we didn't know Newtonian gravity didn't worked in some circumstances, then we considered it true. Now we know that GR doesn't work in some circumstances, shouldn't we consider it to be incomplete? Again, not telling that we should throw GR away, just that the conceptual explanations, like why an apple fall down, are still controversial.
Also, wouldn't gravitons be incompatible with GR?

@ersmith About your second quote, It is ok to accept "that's just the way things work" when we have an intuitive perception of things. We accepted Newton's theory of masses attract masses because that's the way things works (even that it might be wrong). We can see an apple falling down.
Now, when someone says that Earth surface is constantly expanding and it stays the same because of spacetime curvature, that's just too much to accept, don't you think?
It might be just me, but I don't like accepting things that aren't obvious/intuitive without a good explanation. That's what I'm trying to get here. A good reason of why spacetime curvature makes the apple fall.

Your example of the apple is like saying "the future of the apple is on the surface of the Earth". I think I'll just have to accept that explanation.

@Grasshopper Yes. The thing i, while spacetime is one thing, we still can differentiate traveling in space and traveling in time. In this case, the traveling in time of a still object in space is affecting the traveling in space. As if we are converting the travel in time into traveling in space because spacetime is curved. While we "know" that for change traveling in time into space a force is required.

 

[ersmith]

They involve, but they do not prove that an apple fall on Earth because of the curvature of spacetime, right? We just can explain things that we couldn't with Newtonian gravity.

The path of an apple falling to Earth is a geodesic in the Schwarzschild solution to Einstein's field equations. In other words, GR certainly does predict that apples fall to Earth, and with the same acceleration (to within experimental error) as Newton predicted. You may not like the verbal explanations, but the math is clear.

@ersmith About your second quote, It is ok to accept "that's just the way things work" when we have an intuitive perception of things.

Our intuition is formed from a very specific set of circumstances, and is not generally a good guide to reality outside of those circumstances. For example, Newton's first law is not intuitive at all -- in our everyday experience bodies in motion tend to stop fairly quickly. We now know that's due to friction, air resistance, etc., but the law only seems "intuitive" to you because you learned it at a young age.

Ultimately there's no getting around learning the math if you want to understand how things work.

It might be just me, but I don't like accepting things that aren't obvious/intuitive without a good explanation. That's what I'm trying to get here. A good reason of why spacetime curvature makes the apple fall.

The apple follows a "straight line" (geodesic) in curved spacetime. The curvature of the spacetime is described by Einstein's field equations, and depends on the configuration of matter and energy in the spacetime.

Just as two people who start out walking due north on Earth will eventually collide with each other, an apple and a planet moving directly "ahead" in the time direction through spacetime will eventually collide with one another, because while they start out with parallel time axes, the spacetime they are moving through is curved. GR describes exactly how it is curved.

Really, I'd suggest you switch your intuition around. The "natural" state of bodies is free fall. The apple is moving normally, it's the ground that isn't. The surface of the Earth would naturally collapse into a black hole if it were not for the forces holding it up, and we can actually feel those forces (e.g. on the soles of our feet) and measure them with accelerometers.

 

[Grasshopper]

A unit distance is a unit distance. It cannot change in length, by definition. If it did change length, how would you know, other than by comparing it to something else?It has nothing to do with coordinate systems. Intrinsic curvature exists independently of coordinate systems. If you can trace three straight lines that meet in 90 degree corners, you have curvature regardless of whether you are using lines of latitude and longitude at one angle or at another.

What you could say is that he has a weird "metric". Consider, for instance, a flat map of the Earth using a Mercator projection. The distance between points on this map that you measure with your ruler on the paper will not match the correct distances that you pace out on the surface of the Earth. The shortest path line that you draw on the map with a straight-edge will not match the shortest path great circle that you would follow on the surface of the Earth.

A "metric" is a function that works pretty much like the table of distances that you can find in old maps and atlases. You find a row with your starting city on the left. You find a column with your destination city on the top. You read off the distance between those two cities from the cell where the row and column intersect.

The metric is not the same thing as a coordinate system.

A "coordinate system" for a two dimensional space is a mapping between pairs of coordinate values and points in the space. For a two dimensional space one has coordinates with two values each. For three dimensional space, one has coordinates with three values, etc.

One can think of the metric as giving the distance between a given coordinate pair e.g. between $(x_1,y_1)$ and $(x_2,y_2)$. Or as giving a distance between the points at those coordinates. The latter choice makes the metric independent of coordinates but makes it more difficult to write down explicitly.

With that said, yes: If the metric is other than $d=\sqrt{x^2+y^2}$, is still suitably smooth and is more than just a re-scaling then the curvature in the neighborhoods of at least some particular points will be non-zero. [I'm no expert on differential geometry, but I am pretty sure that holds]

If you change the metric and continue to identify a particular pair of points then yes, that changes the distance between those points, by definition. What is your point? And what does that have to do with confining a path to one dimension?

It’s just difficult for me to grasp how curvature in spacetime doesn’t imply curvature in all axes. Trying to work my head around that.

 

[ffp]

The path of an apple falling to Earth is a geodesic in the Schwarzschild solution to Einstein's field equations. In other words, GR certainly does predict that apples fall to Earth, and with the same acceleration (to within experimental error) as Newton predicted. You may not like the verbal explanations, but the math is clear.Our intuition is formed from a very specific set of circumstances, and is not generally a good guide to reality outside of those circumstances. For example, Newton's first law is not intuitive at all -- in our everyday experience bodies in motion tend to stop fairly quickly. We now know that's due to friction, air resistance, etc., but the law only seems "intuitive" to you because you learned it at a young age.

Ultimately there's no getting around learning the math if you want to understand how things work.

Newton's first law is intuitive. It's understandable at least. It is not easily perceived, though. It makes sense and you wouldn't need more than some minutes to understand that things tend to stay in motion. You could see a video of an experiment in vacuum or space and that's it. That is the "that's the way things are" that are easy to accept, because you can directly prove and experiment.
GR is making a statement that is completely unbelievable. And are proven only through math (I'm talking about gravity not being a force and the apple falls down due to spacetime curvature thing).

The apple follows a "straight line" (geodesic) in curved spacetime. The curvature of the spacetime is described by Einstein's field equations, and depends on the configuration of matter and energy in the spacetime.

Just as two people who start out walking due north on Earth will eventually collide with each other, an apple and a planet moving directly "ahead" in the time direction through spacetime will eventually collide with one another, because while they start out with parallel time axes, the spacetime they are moving through is curved. GR describes exactly how it is curved.

Really, I'd suggest you switch your intuition around. The "natural" state of bodies is free fall. The apple is moving normally, it's the ground that isn't. The surface of the Earth would naturally collapse into a black hole if it were not for the forces holding it up, and we can actually feel those forces (e.g. on the soles of our feet) and measure them with accelerometers.

Ok, going by the analogy of the planet surface. We have two coordinates in a plane: x and y, being x horizontal and y vertical. If we draw a straight line parallel to y and then we curve that plane into a sphere like our planet, the line will not be straight anymore. It will be bent into the y-axis making a curved line. Like airplanes do to travel the shortest distance. But this only happens because x and y are spatialy perpendicular. Actually, because there is an angle between them.
So, can we say that time coordinate is perpendicular to the 3 spatial ones?

 

[jbriggs444]

It’s just difficult for me to grasp how curvature in spacetime doesn’t imply curvature in all axes. Trying to work my head around that.

An axis, being a one-dimensional object cannot have intrinsic curvature.

 

[PeterDonis]

Edit — I guess that would make the entire thing spacetime curvature anyway, wouldn’t it?

Yes, indeed.

 

[PeterDonis]

Time isn't a surface. It isn't spatial. So it can't be curved.

That's right. Spacetime can be curved, but time can't.

They involve, but they do not prove that an apple fall on Earth because of the curvature of spacetime, right?

You have already been told multiple times now that no scientific model can be "proved". So continuing to ask about "prove" is pointless and will just get your thread closed since other members' time should not be wasted on pointless questions.

GR not being a fundamental theory doesn't means that the explanation of why gravity happens is wrong?

No. It just means it's an approximation.

Or, if you insist on classifying things based on "right" and "wrong", then all theories are wrong--including whatever more fundamental theory ends up having GR as an approximation to it, if that ever happens. To be "right", a scientific theory would have to be proved--and, as you have already been told multiple times, no scientific theory can be proved. So thinking of theories in terms of "right" and "wrong" is pointless. The thing to look at is how accurate the theory's predictions are, and over what range of circumstances.

until GR we didn't know Newtonian gravity didn't worked in some circumstances

No, that's not correct. We knew Newtonian gravity made some incorrect predictions before GR was discovered (for example, the precession of Mercury's perihelion, as actually observed, was different from what Newtonian gravity predicted). We just didn't have any better model until GR was discovered.

Now we know that GR doesn't work in some circumstances

No, we don't. We do not have any experiments or observations that contradict GR. The concern about GR possibly not being a fundamental theory is entirely theoretical; it is not driven by any experimental data.

the conceptual explanations, like why an apple fall down, are still controversial.

No, they're not. There is no controversy at all about what the GR model says, or how accurate its predictions are, or over what range of circumstances it has been tested. The only open question is whether we will some day discover some more fundamental theory of gravity to which GR is an approximation. But even if that happens, it won't change anything about how accurate GR's predictions are or over what range of circumstances those predictions work.

wouldn't gravitons be incompatible with GR?

If they are ever discovered (i.e., if quantum aspects of gravity are ever observed in experiments), yes, that would be an indication that GR is not a fundamental theory of gravity, since it's not a quantum theory. But that hasn't happened yet, nor is it expected to happen any time soon.

 

[Grasshopper]

@Grasshopper Yes. The thing i, while spacetime is one thing, we still can differentiate traveling in space and traveling in time. In this case, the traveling in time of a still object in space is affecting the traveling in space. As if we are converting the travel in time into traveling in space because spacetime is curved. While we "know" that for change traveling in time into space a force is required.

Why should there be a force to travel through space? Or to “convert” from time to space? All you have to do to convert a time coordinate to a mix of time and space is have a uniform velocity with respect to the other observer, and that requires no force, because F = ma, not F = mv. (“Convert,” I mean it’s just a coordinate transformation)

I guess I don’t understand why there needs to be a force if the motion is a geodesic, which means it is uniform motion in which Newton’s first law holds. It is my understanding that such motion requires no force.

 

[A.T.]

While we "know" that for change traveling in time into space a force is required.

The more general rule, that applies in Newton's and Einstein's models, is that a force is required to deviate from a geodesic path in space-time.

 

[Nugatory]

Newton's first law is intuitive.

That seems a rather strong claim when we consider that the first law was only accepted a few hundred years back. Aristotlean mechanics was around for two millenia - and Aristotle was just codifying an intuition that's been taken for granted for longer than recorded history.

GR is making a statement that is completely unbelievable. And are proven only through math (I'm talking about gravity not being a force and the apple falls down due to spacetime curvature thing).

No, GR is proven through many experiments, not just math. You find it counterintuitive only because you've spent an entire lifetime in the very special and unusual conditions at the surface of a planet; if you had lived your life in freefall it would never occur to you to think of gravity as a force.

 

[Dale]

There is no light cone in our universe

There certainly is. Any flash of light creates an easily measurable physical light cone.

That's a very nice example of how you can explain something in our real world without using the math/geometry. And it's also an example of how the geometry is just a drawing that, sometimes, might help visualize or calculate things.

I completely disagree with this. I would like for you to think about and explicitly respond to the following challenge:

Do you agree that a standard table's legs are physically perpendicular to the tabletop? If so, why? Any reason that you can give to justify the geometry of the table can be extended to justify the geometry of spacetime. If you don't agree that a standard table's legs are physically perpendicular to the tabletop, then what is the physical difference between a folding table with the legs extended or folded? Geometry is part of the physical universe.

Pseudo-Riemannian geometry is not Euclidean geometry, but it is every bit as legitimate a geometry as Euclidean geometry is. And furthermore geometry in the real world as measured by real-world instruments follows the rules of pseudo-Riemannian geometry, not Euclidean geometry.

I was not asking if our brains were capable of understanding the theory of relativity, that is obvious. I was talking about we being able to trully comprehend and grasp what it means to live in a 4 dimensional universe

The two are the same. If you can understand the theory then you can understand "what it means to live in a 4 dimensional universe".

That's good to read. What I'm asking is experimental proof for the statements, which I see often when reading/watching about relativity, that gravity is not a fundamental force the way the other 3 are and that is, instead, caused by the curvature of time. I've seen this more than once.

But, I think you didn't say that. In this case, let's take a step back and I ask you: Is gravity a fundamental force like electromagnetism, and nuclear weak and strong forces? And what is the cause of gravity?

"Gravity" is a little ambiguous of a word, so let me be clear:

Gravity, meaning the whole phenomenon of gravitation and everything related to gravitation is a fundamental interaction, but it is different from the other fundamental forces in that it does not produce proper acceleration of test objects. This meaning of "gravity" is caused by stress-energy and any relevant boundary conditions, at least in globally hyperbolic spacetimes.

Gravity, meaning specifically the local force of gravity is a fictitious force that is determined by your choice of coordinates, not something physical, and this includes the curvature of time. This meaning of "gravity" has no cause.

The curvature of spacetime is something physical and that is tidal effects and their relativistic generalizations which are the only "real" (meaning frame invariant) aspect of gravity.

 

[PeterDonis]

I guess I don’t understand why there needs to be a force if the motion is a geodesic

There doesn't. An object whose worldline is a geodesic feels zero force.

 

[Grasshopper]

That seems a rather strong claim when we consider that the first law was only accepted a few hundred years back. Aristotlean mechanics was around for two millenia - and Aristotle was just codifying an intuition that's been taken for granted for longer than recorded history.
[…]

There are still people who refuse to accept Galilean relativity. Met some on a YouTube video comment section that got invaded by legitimate flat earthers (lol).

 

[ffp]

That's right. Spacetime can be curved, but time can't.

When we curve spacetime we are curving both space and time axis. While spacetime is a unique thing, we can still differentiate what is space and what is time. What does it means to bend that axis, then? Or it doesn't mean nothing and we have to only look at the plane and not the axis?

No. It just means it's an approximation.

Or, if you insist on classifying things based on "right" and "wrong", then all theories are wrong--including whatever more fundamental theory ends up having GR as an approximation to it, if that ever happens. To be "right", a scientific theory would have to be proved--and, as you have already been told multiple times, no scientific theory can be proved. So thinking of theories in terms of "right" and "wrong" is pointless. The thing to look at is how accurate the theory's predictions are, and over what range of circumstances.

Ok, I guess all of physics is true until a new, more complete theory comes up or for some reason is proven wrong. Just as Newton's theory is now considered "wrong"*, GR might be proved wrong with the development of physics, especially quantum physics.

* You might pardon my use of the word wrong. What I mean by wrong is the consensual view of the physics community. I know we are in a GR forum section. But I think that in any physics group, relativity is considered real. And by that I mean: if someone asks you what is the source of gravity you would answer the curvature of spacetime instead the pulling force created by masses, right?

No, we don't. We do not have any experiments or observations that contradict GR. The concern about GR possibly not being a fundamental theory is entirely theoretical; it is not driven by any experimental data.

I don't know exactly why GR is incompatible with quantum physics, that's why I asked before. But I thought it were some fundamental irreconcilable features of both. Like, if one is true the other can't be.

@Grasshopper Not to travel, to change direction or accelerate. Which is what happens when you change part of time traveling into space traveling.

There certainly is. Any flash of light creates an easily measurable physical light cone.

It is an abstract concept. You can't measure in reality because it is a reference for a point in time. And time is always running.

I completely disagree with this. I would like for you to think about and explicitly respond to the following challenge:

Do you agree that a standard table's legs are physically perpendicular to the tabletop? If so, why? Any reason that you can give to justify the geometry of the table can be extended to justify the geometry of spacetime. If you don't agree that a standard table's legs are physically perpendicular to the tabletop, then what is the physical difference between a folding table with the legs extended or folded? Geometry is part of the physical universe.

Pseudo-Riemannian geometry is not Euclidean geometry, but it is every bit as legitimate a geometry as Euclidean geometry is. And furthermore geometry in the real world as measured by real-world instruments follows the rules of pseudo-Riemannian geometry, not Euclidean geometry.

The table is physical and is inside space. I can measure its legs and the angle between them. Spacetime is not a spatial dimension. It's 3 spatial 1 time. How do you measure the angle between time and space?

Which leads me to the question I made before. The analogy of geodesics and the airplane routes. in a flat space a line is the shorter distance between 2 points. That becomes a curve when the flat space is now a spherical one. The flat and curved spaces have the same origin for both their axis and they are perpendicular. Does that means that in GR, space and time are perpendicular?

"Gravity" is a little ambiguous of a word, so let me be clear:

Gravity, meaning the whole phenomenon of gravitation and everything related to gravitation is a fundamental interaction, but it is different from the other fundamental forces in that it does not produce proper acceleration of test objects. This meaning of "gravity" is caused by stress-energy and any relevant boundary conditions, at least in globally hyperbolic spacetimes.

Gravity, meaning specifically the local force of gravity is a fictitious force that is determined by your choice of coordinates, not something physical, and this includes the curvature of time. This meaning of "gravity" has no cause.

The curvature of spacetime is something physical and that is tidal effects and their relativistic generalizations which are the only "real" (meaning frame invariant) aspect of gravity.

The theory of everything that lots of physics try to develop is, roughly speaking, the unification of the 4 fundamental forces. Considering GR don't treat gravity as a force, should it be unified too, or just the other 3?
From wikipedia "In physics, the fundamental interactions, also known as fundamental forces" so gravity is a fundamental force as much as electromagnetism. However it's a force caused by curvature of spacetime and that does not cause acceleration. Is that it?

EDIT: Another thought: In Newtonian gravity, the range of gravity is radial and infinite. Which means if the universe were completely emptied of masses and energy and two tennis balls would put several light-years apart, they would still be pulled together and one day they would touch. Does this applies to GR too, since in GR there is no force? Does the curvature of the tennis ball propagate light-years away to interact with the other ball?

 

[Dale]

It is an abstract concept. You can't measure in reality because it is a reference for a point in time. And time is always running.

Complete nonsense. This is how the GPS works, by physically measuring light cones. The intersection of four light cones identifies your spacetime event. If this geometry were purely abstract then GPS would be purely abstract too, and it isn't.

The table is physical and is inside space. I can measure its legs and the angle between them.

I can also measure the relative velocity (or rapidity) between two objects. This is the spacetime angle between them. This angle can be measured just like a spatial angle, so if the spatial angle is physical by virtue of being measurable, then so is the spacetime angle.

How do you measure the angle between time and space?

The easiest way is with a radar gun, but many other alternative devices exist. E.g. anemometers, speedometers, etc. Notice that we are not measuring the angle between time and space any more than you measure the angle between space and space for the table. For the table you are measuring the spatial angle between the tabletop and the leg. With a speedometer you are measuring the spacetime angle between your car and the road. These spacetime angles are every bit as physical and measurable as spatial angles.

That becomes a curve when the flat space is now a spherical one. The flat and curved spaces have the same origin for both their axis

This doesn't make any sense that I can see. A flat space cannot become a spherical space and I cannot even fathom in what sense you would say that they share an origin. They don't even naturally have origins to begin with.

Considering GR don't treat gravity as a force, should it be unified too, or just the other 3?

I think that is an open question at the moment.