Mass equivalence and instantaneous gravitational field

[GregAshmore]

Here is Max Born explaining the crash of a train, in which the train is regarded to be at rest:

We may choose a coordinate system rigidly attached to the train, in which case, at the moment of collision, the whole world makes a jerk relative to this system; everywhere we get a very strong gravitational field parallel to the original motion and this field causes the destruction of the train. Why does the church tower in the neighboring village not tumble down, too? ... The answer is this: the church tower does not fall down because, during the retardation, its relative position to the distant cosmic masses is not changed at all. The jerk, which as seen from the train, the whole world experiences, affects all bodies equally from the most distant stars to the church tower. All these bodies fall freely in the gravitational field which is present during the retardation.

There are two peculiar features of this gravitational field:

1. Causation. The field appears coincidentally with the collision of the train with an obstacle. If a passenger had pulled the emergency cord, a somewhat weaker field would have been generated. The field disappears in an even more remarkable manner, becoming weaker as each piece of wreckage comes to rest. It is difficult to imagine physical connections between these causes and events.

2. Propagation. The field appears everywhere in the universe at the same instant. If instantaneous force of gravity at a distance is a weakness of Newton's theory, then instantaneous acceleration of gravity at a distance must be a weakness of Einstein's theory.

With this in mind, it seems to me that the "resting train" view of the event is not as "real" as the "resting earth" view.

Comments?

 

[bcrowell]

In GR, the gravitational field g is not observable. The Earth's field is not observable, for example, because a free-falling observer can't detect it. The violations of causality you're talking about don't occur because there is something special about the gravitational field in Born's example. The violations of causality occur because you attribute an independent physical existence to a gravitational field, when actually it doesn't have any.

Mathematically, observables in general relativity have to be tensors. One of the properties of a tensor is that if it's zero in one coordinate system, it's zero in every system. If it's nonzero in one coordinate system, it's nonzero in every system. The gravitational field g isn't a tensor.

 

[GregAshmore]

In GR, the gravitational field g is not observable. The Earth's field is not observable, for example, because a free-falling observer can't detect it. The violations of causality you're talking about don't occur because there is something special about the gravitational field in Born's example. The violations of causality occur because you attribute an independent physical existence to a gravitational field, when actually it doesn't have any.

Mathematically, observables in general relativity have to be tensors. One of the properties of a tensor is that if it's zero in one coordinate system, it's zero in every system. If it's nonzero in one coordinate system, it's nonzero in every system. The gravitational field g isn't a tensor.

I'm in the process of learning tensors, so I'm not in a position to respond with confidence to that portion of your comments. However, I will observe that the qualitative explanation of the train wreck has a gravitational field present when considered with the train at rest, but no gravitational field when considered with the Earth at rest. Thus one and the same event has a gravitational field, or not, depending on the coordinate system to which it is transformed.

As to the existence of the gravitational field, it seems to me that it does have an existence--is physically present--when the event is evaluated with reference to the resting train. It is true that the gravitational field cannot be directly observed, just as the wave field of the photon cannot be observed in the open space between its source (e.g. the sun) and its destination (e.g. the earth). Nevertheless, the effect of the gravitational field--the motion of the Earth and stars--can be observed. If this were not so, then the gravitational field would be just as irrelevant as Lorentz's ether.

 

[bcrowell]

I'm in the process of learning tensors, so I'm not in a position to respond with confidence to that portion of your comments. However, I will observe that the qualitative explanation of the train wreck has a gravitational field present when considered with the train at rest, but no gravitational field when considered with the Earth at rest. Thus one and the same event has a gravitational field, or not, depending on the coordinate system to which it is transformed.

Right.

As to the existence of the gravitational field, it seems to me that it does have an existence--is physically present--when the event is evaluated with reference to the resting train. It is true that the gravitational field cannot be directly observed, just as the wave field of the photon cannot be observed in the open space between its source (e.g. the sun) and its destination (e.g. the earth). Nevertheless, the effect of the gravitational field--the motion of the Earth and stars--can be observed.

GR correctly predicts the motion of the Earth without needing to invoke anything like a gravitational field.

If this were not so, then the gravitational field would be just as irrelevant as Lorentz's ether.

Yes, it is just as irrelevant as Lorentz's ether.

 

[GregAshmore]

GR correctly predicts the motion of the Earth without needing to invoke anything like a gravitational field.

I am unable to correlate this statement with what I read in Einstein, Born, Hawking, and Wald. GR is first and foremost a field theory. In it, gravity is a field, just as electromagnetism is a field. There are so many statements which explicitly refer to the gravitational field as an integral part of GR that I can't begin to list them.

Have I misunderstood you?

 

[bcrowell]

I am unable to correlate this statement with what I read in Einstein, Born, Hawking, and Wald. GR is first and foremost a field theory. In it, gravity is a field, just as electromagnetism is a field. There are so many statements which explicitly refer to the gravitational field as an integral part of GR that I can't begin to list them.

Have I misunderstood you?

You might want to try looking at a few quotes like that and seeing what physical quantity they're actually referring to. Relativists will often speak loosely of "the gravitational field," meaning the characteristics of spacetime caused by gravity. What is not observable in GR is the gravitational field g, the thing that has units of acceleration and that we think of as equaling 9.8 m/s2 on the Earth's surface. Dip into one of those books where you see the phrase "gravitational field," and I think you'll probably see symbols for things like the metric and the curvature tensors -- no g.

 

[GregAshmore]

You might want to try looking at a few quotes like that and seeing what physical quantity they're actually referring to. Relativists will often speak loosely of "the gravitational field," meaning the characteristics of spacetime caused by gravity. What is not observable in GR is the gravitational field g, the thing that has units of acceleration and that we think of as equaling 9.8 m/s2 on the Earth's surface. Dip into one of those books where you see the phrase "gravitational field," and I think you'll probably see symbols for things like the metric and the curvature tensors -- no g.

I have no problem with substituting "the characteristics of spacetime caused by gravity" in place of "field", if you wish.

Either way, when the train wreck is analyzed with the train at rest, there is a change to spacetime which extends instantaneously to the limits of the universe. The two-part question is, What causes this change, and how does this change become effective instantaneously?

 

[pervect]

It's somewhat sloppy wording to say that there is a "change in spacetime" when you experience a trainwreck, I think.

What happens is you are describing space-time with several different coordinate systems, an inertial coordinate system before the train-wreck, and accelerating coordinate system during the train-wreck, and another inertial coordinate system after the train wreck.

But the "change" in space-time caused by changing your coordinates is in no way a physical change. It's just a change in the labels, the coordinates, you use to describe it.

Without tensors, I'm not sure there is a lot more that can be said. But you can change your labels on space-time as fast as you like. There just isn't much physical significance to it.

When do your labels change? What notion of simultaneity do you use when you change your labels? THe answer to these are also obvious - you can do whatever you want. As long as you are self consistent.

By the way, there are some issues about how, exactly, you consistently "splice" together the coordinate systems before and after the train wreck (because their notions of simultaneity are different), but it would probalby be a digression to get into that.

 

[GregAshmore]

It's somewhat sloppy wording to say that there is a "change in spacetime" when you experience a trainwreck, I think.

What happens is you are describing space-time with several different coordinate systems, an inertial coordinate system before the train-wreck, and accelerating coordinate system during the train-wreck, and another inertial coordinate system after the train wreck.

But the "change" in space-time caused by changing your coordinates is in no way a physical change. It's just a change in the labels, the coordinates, you use to describe it.

Without tensors, I'm not sure there is a lot more that can be said. But you can change your labels on space-time as fast as you like. There just isn't much physical significance to it.

Actually, there is one coordinate system throughout the event, and it is fixed to the train. Nowhere in this narrative is there a transformation of coordinates.

The train wreck is certainly a physical event. The appearance of a gravitational field (or, if you prefer, a change in spacetime which is consistent with the effect of gravity) is the unseen physical event which Born proposes as the cause of the apparent event.

More precisely, this combination is proposed as the cause of the destruction: the newly-born gravitational field (which affects the entire universe, including the train) plus the "retarding forces" (which affect the train only).

I am asking what caused the appearance of the gravitational field, and how is it that the field affected the entire universe instantaneously.

 

[bcrowell]

I have no problem with substituting "the characteristics of spacetime caused by gravity" in place of "field", if you wish.

Either way, when the train wreck is analyzed with the train at rest, there is a change to spacetime which extends instantaneously to the limits of the universe. The two-part question is, What causes this change, and how does this change become effective instantaneously?

No, the difference between "the characteristics of spacetime caused by gravity" and "field" is not just a change of terminology. When those books use "field" loosely, they mean "the observable characteristics of spacetime caused by gravity." The change you're referring to is not an observable change. What causes this change? What causes it is the choice of an accelerating coordinate system.

Actually, there is one coordinate system throughout the event, and it is fixed to the train. Nowhere in this narrative is there a transformation of coordinates.

You can think of a noninertial frame as a frame that switches from one inertial frame to another inertial frame. That way of thinking about it is particularly natural here because the train's frame *is* a valid inertial frame (in the Newtonian sense) both before and after the crash.

 

[pervect]

Maybe we do need to get into the nature of the coordinate system you are using, and also some of the self-consistency issues, which aren't immediately obvious. I think, on due consideration, that they are relevant to the question you're trying to ask.Let's spell out in more detail how "the" coordinate system of the moving train is defined. You're starting out by assuming that a coordinate system exists, but when you analyze the situation carefully, it becomes less clear. For instance, consider the following quote from MTW's textbook "Gravitation".

IT is very easy to put together the words "the coordinate system of an accelerated observer", but it is much harder to find a concept that these words might refer to. The most useful comment on these words is that, if taken seriously, they are self contradictory.

So, let's talk about what you might mean by "the" coordinate system of the train. It seems a bit counterproductive to me to insist that you define it. So I'll offer some definitions. If they aren't what you had in mind, you'll need to spell out in more detail what your assumptions are, and how you avoid the problem I'll mention.

So I'll start by assuming that we initially have some inertial coordinate system, a "moving coordinate system", that co-moves with the train. In the real world it probably wouldn't be inertial and the space-time wouldn't be flat, we'd have issues with gravity and rotation and what-not, but I think the extra complexity will just make the problem harder to discuss.

And I'll assume we have another inertial coordinate system, one that is stationary.

Because space-time is being assumed to be flat (no gravity), both coordinate systems obey the Lorentz transform.

Now, I'll assume that we want the train coordinate system, before the crash, to assign coordinates to events identically to the manner that the moving inertial coordinate system did/does, and we wish the train coordinate system, after the crash to assign coordinates to events identically to the manner that the stationary system does. In between, during the crash, we'll have to use some as-yet-to-be defined system of coordinates.

This set of assumptions seems innocuous, but it already has the seeds of the problem.

The moving coordinate system has a different notion of simultaneity than the stationary coordinate system. When we draw the space-time diagram (Figure 6.2 in the text), if we draw the line representing the points where t=-a in the moving coordinate system, before the crash, and we draw the line representing the points where t=+b in the stationary coordinate system after the crash, we find that the lines cross.

If the velocity of the train is v, and we draw our space-time diagram in the coordinates of the station frame, then we can write:

t_moving = constant -> gamma * (t_station - (v/c^2)*x_station) = constant

or t_station = v/c^2 * x_station + constant

This line has a positive slope on the space-time diagram, and will interesect the line of constant time in the station frame, the line t_station = +b (which has zero slope), because it's not parallel, and any two non-parallel lines intersect.

This implies that the SAME event has two different time coordinates -a, and +b.

This is generally regarded as nonsensical behavior. The textbook solution to this problem is to limit the size of "the" coordinate system of an accelerated observer to the region of space-time where the lines of constant time do not cross. In this standard textbook approach, the accelerated observer does have a local coordinate system that has the desired properties I mentioned above, but said coordinate system does not cover all of space-time.

 

[Dale]

I am asking what caused the appearance of the gravitational field

The curved coordinate system used.

how is it that the field affected the entire universe instantaneously.

Because the curves in the coordinate system covered the entire universe.

 

[GregAshmore]

It seems to me that the essential feature of Born's example is being overlooked, namely, that the train is regarded at rest throughout this narrative. The Earth and the distant stars are moving uniformly, then suddenly decelerate until they are also at rest. The problem for an observer on the stationary train is to provide a physical explanation for two things: the sudden deceleration of the cosmos, and the restriction of destruction to the train only. The deceleration of the cosmos is explained by the temporary appearance of a gravitational field (both Born and Einstein use 'field'); the destruction of the train only is attributed to the restraining force which acts on the train only. All well and good. Now the question is, what caused the gravitation, and how is it that the gravitation was instantaneously effective throughout the cosmos?

I have Wald's book, but the math is unintelligible to me, having no prior exposure to tensors. I do not have MTW. MTW is a bit pricy, but I don't mind that. Some also recommend Schutz. I'm an engineer. In college I did well in calculus and partial differential equations. I took a graduate course in linear algebra and did well. With this background, which books would you suggest for self study in GR?

 

[pervect]

I would suggest first studying special relativity if you haven't already, with a text that includes the 4-vector approach, such as Taylor & Wheeler's "Space time physics".

Then I'd suggest graduate level electromagnetism (if you like math, Jackson, Classical Electrodynamics is the standard, hard, textbook though Griffiths might be an easier read), and picking up some basics of tensors from their use in electromagnetism and how electromagnetism is treated via special relativity.

With a knowledge of special relativity using 4-vectors, and of electromagnetism using tensors and 4-vectors, you should be in good shape to tackle GR.

Then pick the GR textbook of your choice- I'm afraid I'm not familiar with all the modern choices. I've seen at least one poster apparently confused by Schutz, for whatever it's worth, but I can't say they wouldn't be just as confused if they had studied some other book.

With a graduate level course in linear algebra, you should be familiar with vector spaces, and their duals. A vector space just has the properties of linearity, allowing multiplication by scalars, and the adding of vectors. The dual of a vector space is another vector space of the same dimension, which is the linear map of vectors of the vector space to scalars.

If you have a good understanding of dual vectors, how the metric defines the length of a vector, the dot product of vectors, and a natural mapping of vectors to their duals, the rest of tensors is pretty straightforwards. Tensors are just a map from M vectors and N dual vectors to a scalar. That's really all there is to it.

MTW has one of the best sections on accelerated observers - though it does require understanding tensors and the particular notation that MTW uses for them to follow their discussion of accelerated observers.

In any case, unless you're really disgustingly rich, I'd suggest checking a book out from a local university library before buying it.

 

[Dale]

Now the question is, what caused the gravitation, and how is it that the gravitation was instantaneously effective throughout the cosmos?

This question has already been asked and answered a couple of times during the course of this thread. Why don't you ask a new question that goes to your confusion about the answer. Repeating the same question can only get you the same answer again.

Do you understand how curved coordinates lead to gravitational fields in flat spacetime? E.g. are you familiar with Rindler coordinates?

 

[bcrowell]

I would suggest first studying special relativity if you haven't already, with a text that includes the 4-vector approach, such as Taylor & Wheeler's "Space time physics".

I would second pervect's suggestion of Taylor and Wheeler as a starting point, but I don't think the next step after that should be graduate-level physics texts. I would suggest going on after that to Exploring Black Holes, by the same authors. If you want to learn more GR after that, the next place to go would be undergrad texts like Hartle, or my own free book at lightandmatter.com.

 

[pervect]

bcrowell's suggestion might be better than mine. Once upon a time there weren't any undergraduate level treatments of GR - there were only popularizations and graduate level texts. This seems to be changing.. If you want to go the full distance, though, I'd recommend learning at least some graduate level SR before tackling graduate level GR. It's better to learn tensors in the context of a theory that you're familiar with rather than learning tensors along with an unfamiliar theory.

 

[GregAshmore]

This question has already been asked and answered a couple of times during the course of this thread. Why don't you ask a new question that goes to your confusion about the answer. Repeating the same question can only get you the same answer again.

Do you understand how curved coordinates lead to gravitational fields in flat spacetime? E.g. are you familiar with Rindler coordinates?

My understanding is that the curvature of spacetime is the result of the local mass-energy distribution. Therefore, curved coordinates are an effect of gravitation, not a cause of gravitation.

That principal is at the root of my question. The reason I have asked it so many times is that none of the answers I have received have dealt with the cause of the acceleration of the cosmos during the train wreck.

I'll restate my question in terms of spacetime curvature. Perhaps that will help.

I'll also simplify the example. Instead of having the train collide with an obstacle, I'll have the brakes applied at full force.

The train is at rest (and remains at rest throughout).

The cosmos (the earth, the sun, and the distant stars) is moving at constant speed. Spacetime is essentially flat. (The curvature due to the masses of the Earth and sun do not figure in the episode, so they can be ignored.)

The brakes are applied. Coincidentally with the application of the brakes, the cosmos decelerates. What is the physical cause of this deceleration?

The immediate cause is a [newly formed] distortion of spacetime, such that the cosmos freely falls. The direction of the fall is opposite to the original direction of travel, so the speed of the cosmos relative to the train is reduced. The train cannot freely fall in this curvature, due to the force of the brakes.

When the cosmos reaches zero speed relative to the train, the distortion of spacetime disappears. In the [once again] flat spacetime, the cosmos remains motionless with respect to the train.

The distortion of spacetime is the immediate cause of the deceleration, but it is not the underlying cause. Spacetime does not distort of itself; it is distorted by the presence of mass-energy.

The question is: What causes the distortion of spacetime, and how is it that the distortion is present instantly throughout the cosmos?

The answer cannot be that the selection of coordinates causes the appearance of the distortion, for there is only one coordinate system under discussion. That coordinate system was selected before the event in question was even thought of. Without a change in the selection of the coordinates, the selection of coordinates cannot explain the distortion of spacetime.

Indeed, it can be argued that, for the observers on the train, there is no other coordinate system. After all, isn't that the whole point of relativity? Any observer may consider himself to be at rest, and is able, in that resting position, to correctly explain (and predict) the behavior of nature, using the same laws as any other observer.

With this in mind, I can address one of your earlier replies.

So, let's talk about what you might mean by "the" coordinate system of the train. It seems a bit counterproductive to me to insist that you define it. So I'll offer some definitions. If they aren't what you had in mind, you'll need to spell out in more detail what your assumptions are, and how you avoid the problem I'll mention.

So I'll start by assuming that we initially have some inertial coordinate system, a "moving coordinate system", that co-moves with the train. In the real world it probably wouldn't be inertial and the space-time wouldn't be flat, we'd have issues with gravity and rotation and what-not, but I think the extra complexity will just make the problem harder to discuss.

And I'll assume we have another inertial coordinate system, one that is stationary.

When I speak of "the" coordinate system of the train, I mean the one and only coordinate system relative to which all events in this narrative take place. That coordinate system is by definition at rest. The spacetime location of every other object in the cosmos is expressed in these coordinates, and no other coordinates. That is one of the two essential conditions of the principle of relativity.

The other essential condition is that the same mathematical laws which are used in the train coordinate system may be used by an observer resting in any other coordinate system. That observer will consider himself to be at rest throughout the narrative, as well.

 

[GregAshmore]

bcrowell's suggestion might be better than mine. Once upon a time there weren't any undergraduate level treatments of GR - there were only popularizations and graduate level texts. This seems to be changing.. If you want to go the full distance, though, I'd recommend learning at least some graduate level SR before tackling graduate level GR. It's better to learn tensors in the context of a theory that you're familiar with rather than learning tensors along with an unfamiliar theory.

Thanks to both of you for your suggestions. I'll review them and make a decision.

I think my question in this thread is qualitative, and can be answered without recourse to the details of the equation of GR.

 

[bcrowell]

My understanding is that the curvature of spacetime is the result of the local mass-energy distribution. Therefore, curved coordinates are an effect of gravitation, not a cause of gravitation.

What DaleSpam refers to as a curved coordinate system is not the same as curved spacetime. Curvature of spacetime is defined in such a way that its existence or nonexistence doesn't depend on the choice of coordinates.

I'll restate my question in terms of spacetime curvature. Perhaps that will help. [...]

The cosmos (the earth, the sun, and the distant stars) is moving at constant speed. Spacetime is essentially flat. (The curvature due to the masses of the Earth and sun do not figure in the episode, so they can be ignored.)

That's fine. We ignore the curvature due to the Earth and sun. Therefore we're talking about flat spacetime.

The immediate cause is a [newly formed] distortion of spacetime, such that the cosmos freely falls. The direction of the fall is opposite to the original direction of travel, so the speed of the cosmos relative to the train is reduced. The train cannot freely fall in this curvature, due to the force of the brakes.

When the cosmos reaches zero speed relative to the train, the distortion of spacetime disappears. In the [once again] flat spacetime, the cosmos remains motionless with respect to the train.

There is no curvature of spacetime in your example. The spacetime curvature (again, neglecting the Earth and sun's gravity) is zero before, during, and after the train's deceleration.

The distortion of spacetime is the immediate cause of the deceleration, but it is not the underlying cause.

No, the cause of the stars' acceleration is the choice of a coordinate system.

Spacetime does not distort of itself; it is distorted by the presence of mass-energy.

Right.

The answer cannot be that the selection of coordinates causes the appearance of the distortion, for there is only one coordinate system under discussion. That coordinate system was selected before the event in question was even thought of. Without a change in the selection of the coordinates, the selection of coordinates cannot explain the distortion of spacetime.

When we refer to a coordinate system, we mean a coordinate system that covers an entire region of spacetime, which means that it covers a certain amount of time. A non-inertial coordinate system does not have to be thought of as a change from one inertial system to another inertial system. The coordinate system tied to the train is non-inertial.

Indeed, it can be argued that, for the observers on the train, there is no other coordinate system. After all, isn't that the whole point of relativity? Any observer may consider himself to be at rest, and is able, in that resting position, to correctly explain (and predict) the behavior of nature, using the same laws as any other observer.

Yes, this is correct. The relevant law of nature here is the Einstein field equations. The field equations relate curvature to mass-energy. By taking the gravitational effects of the earth, etc., to be negligible, you're saying that you're talking about flat spacetime (zero curvature), which is a solution of the field equations if you take the Earth's mass-energy to be negligible. By describing that flat spacetime in the coordinate system of the train, you're choosing to describe flat spacetime in unusual coordinates. However, the field equations work regardless of the coordinate system you choose, so they are still satisfied in your coordinates. The curvature is still zero, so it still matches the zero mass-energy that is present.

 

[Dale]

My understanding is that the curvature of spacetime is the result of the local mass-energy distribution. Therefore, curved coordinates are an effect of gravitation, not a cause of gravitation.

There is a difference between the curvature of spacetime, which is a coordinate-independent (tensor) geometric quantity, and curved coordinates, which are obviously coordinate dependent. In curved spacetime you cannot use globally straight coordinates, but in flat spacetime it is still possible to use curved coordinates.

To understand the difference consider a piece of paper and a balloon, both are 2D surfaces, one is curved in a coordinate independent sense and the other is flat in a coordinate independent sense. You can draw curved lines on the flat piece of paper, but you cannot draw globally straight lines on the balloon. In your example the spacetime is essentially flat, but the train-centered coordinates are curved. It is like drawing curved lines on the piece of paper.

In Newtonian physics you can also have curved coordinates, which produce fictitious forces. For example if you have your coordinates curved in the shape of a parabola then you have a uniform fictitious force. In GR the equivalence principle equates these coordinate dependent fictitious forces with gravity. The coordinate independent curvature is related to tidal forces.

That principal is at the root of my question. The reason I have asked it so many times is that none of the answers I have received have dealt with the cause of the acceleration of the cosmos during the train wreck.

Yes, they have. The cause is the curved coordinates.

I'll restate my question in terms of spacetime curvature. Perhaps that will help.

I'll also simplify the example. Instead of having the train collide with an obstacle, I'll have the brakes applied at full force.

The train is at rest (and remains at rest throughout).

The cosmos (the earth, the sun, and the distant stars) is moving at constant speed. Spacetime is essentially flat. (The curvature due to the masses of the Earth and sun do not figure in the episode, so they can be ignored.)

This is correct, the spacetime is essentially flat (coordinate independent) in this scenario.

The brakes are applied. Coincidentally with the application of the brakes, the cosmos decelerates. What is the physical cause of this deceleration?

The curvature of the coordinates.

The cause is a [newly formed] distortion of spacetime, such that the cosmos freely falls. The direction of the fall is opposite to the original direction of travel, so the speed of the cosmos relative to the train is reduced. The train cannot freely fall in this curvature, due to the force of the brakes.

When the cosmos reaches zero speed relative to the train, the distortion of spacetime disappears. In the [once again] flat spacetime, the cosmos remains motionless with respect to the train.

The spacetime remains flat the entire time (i.e. there are no tidal forces involved). It is only the coordinates which are curved.

The question is: What causes the distortion of spacetime, and how is it that the distortion is present instantly throughout the cosmos?

Because you have specified the curved coordinates instantly throughout the cosmos.

The answer cannot be that the selection of coordinates causes the appearance of the distortion, for there is only one coordinate system under discussion. That coordinate system was selected before the event in question was even thought of. Without a change in the selection of the coordinates, the selection of coordinates cannot explain the distortion of spacetime.

Indeed, it can be argued that, for the observers on the train, there is no other coordinate system. After all, isn't that the whole point of relativity? Any observer may consider himself to be at rest, and is able, in that resting position, to correctly explain (and predict) the behavior of nature, using the same laws as any other observer.

Sure, as long as he uses the correct expression for the metric in his coordinate system. The metric captures the coordinate-independent curvature, and the expression for the metric in a given coordinate system also reflects the curving of the coordinates.

EDIT: I see bcrowell was a LOT faster than I was, that is what happens when I watch a tv show in the middle of posting!

 

[GregAshmore]

What DaleSpam refers to as a curved coordinate system is not the same as curved spacetime. Curvature of spacetime is defined in such a way that its existence or nonexistence doesn't depend on the choice of coordinates.

okay

That's fine. We ignore the curvature due to the Earth and sun. Therefore we're talking about flat spacetime.There is no curvature of spacetime in your example. The spacetime curvature (again, neglecting the Earth and sun's gravity) is zero before, during, and after the train's deceleration.

I don't know how to square this statement with Born: "All these bodies [the cosmos] fall freely in the gravitational field which is present during the retardation." Bodies fall freely in a gravitational field because spacetime is curved. Therefore, there is curvature of spacetime during the acceleration of the cosmos.

Born hints that the mass of the distant stars is responsible for the gravitation in this case.
Einstein is more explicit. In this quote he says that the mass of the distant stars induces a gravitational field. Considered in the context of this quote, perhaps the meaning of my question is clarified.

You declared the fields that were called for in the clock example also as merely fictitious, only because the field lines of actual gravitational fields are necessarily brought forth by mass; in the discussed examples no mass that could bring forth those fields was present. This can be elaborated upon in two ways. Firstly, it is not an a priori necessity that the particular concept of the Newtonian theory, according to which every gravitational field is conceived as being brought forth by mass, should be retained in the general theory of relativity. This question is interconnected with the circumstance pointed out previously, that the meaning of the field components is much less directly defined as in the Newtonian theory. Secondly, it cannot be maintained that there are no masses present, that can be attributed with bringing forth the fields. To be sure, the accelerated coordinate systems cannot be called upon as real causes for the field, an opinion that a jocular critic saw fit to attribute to me on one occasion. But all the stars that are in the universe, can be conceived as taking part in bringing forth the gravitational field; because during the accelerated phases of the coordinate system K' they are accelerated relative to the latter and thereby can induce a gravitational field, similar to how electric charges in accelerated motion can induce an electric field. Approximate integration of the gravitational equations has in fact yielded the result that induction effects must occur when masses are in accelerated motion. This consideration makes it clear that a complete clarification of the questions you have raised can only be attained if one envisions for the geometric-mechanical constitution of the Universe a representation that complies with the theory.

No, the cause of the stars' acceleration is the choice of a coordinate system.

The primary cause of the deceleration of the cosmos is the pull on the emergency stop chord. Is the selection of a new coordinate system somehow bound up in the act of pulling on the chord? (I am not being facetious. This is at the heart of my question.)

Edit:
Furthermore, your statement seems to directly contradict Einstein in the above quote:

To be sure, the accelerated coordinate systems cannot be called upon as real causes for the field, an opinion that a jocular critic saw fit to attribute to me on one occasion.

When we refer to a coordinate system, we mean a coordinate system that covers an entire region of spacetime, which means that it covers a certain amount of time. A non-inertial coordinate system does not have to be thought of as a change from one inertial system to another inertial system. The coordinate system tied to the train is non-inertial.

I take this to mean that we describe the motion of the cosmos in the constant velocity segments with SR, as SR is valid in these segments and is more convenient. During the acceleration segment, it is necessary to use the equation of GR to describe the motion of the cosmos [in a relativistic manner]. This choice is forced upon us by the reality of the acceleration and the gravitational field which produces it. We do not cause the acceleration by choosing to evaluate the cosmos with GR.

Yes, this is correct. The relevant law of nature here is the Einstein field equations. The field equations relate curvature to mass-energy. By taking the gravitational effects of the earth, etc., to be negligible, you're saying that you're talking about flat spacetime (zero curvature), which is a solution of the field equations if you take the Earth's mass-energy to be negligible. By describing that flat spacetime in the coordinate system of the train, you're choosing to describe flat spacetime in unusual coordinates. However, the field equations work regardless of the coordinate system you choose, so they are still satisfied in your coordinates. The curvature is still zero, so it still matches the zero mass-energy that is present.

See the discussion above regarding the mass of the cosmos.

Edit:
DaleSpam posted his response while I was writing this response to bcrowell. Most of this response will apply well enough to DaleSpam's comments. The following is worth emphasizing:

DaleSpam said:

Yes, they have. The cause is the curved coordinates.

The obvious question is, What caused the coordinates to become curved?

 

[GregAshmore]

I bought Taylor for SR and Hartle for GR. Hartle uses the four-vector, then moves on to tensors. Taylor uses neither. (I need to own them because I will take months to work through them, and will need to refer back to them often.)

 

[Dale]

The primary cause of the deceleration of the cosmos is the pull on the emergency stop chord. Is the selection of a new coordinate system somehow bound up in the act of pulling on the chord? (I am not being facetious. This is at the heart of my question.)

Yes, indirectly. Pulling on the cord causes the train to become non-inertial (i.e. an accelerometer attached to the train reads a non-zero value). In order to have a non-inertial object remain at rest you must use curved coordinates.

Consider, for example, a rotating turbine blade. An attached accelerometer reads a non-zero value. Therefore we know that we will have to use curved coordinates if we wish to have the turbine blade remain stationary. In this case, the coordinates will need to be curved in a helical fashion and this will yield gravitational forces pulling outwards on the turbine blade in this frame.

The obvious question is, What caused the coordinates to become curved?

Our completely arbitrary choice to define the coordinate system such that a non-inertial object was always at rest. We did not need to make that choice, we could just as easily have chosen an inertial reference frame where the train was not always at rest. Then our coordinates would not have been curved.

Did you read and understand my comments above about the baloon and the piece of paper? The line on the piece of paper that represents the train has a sharp bend in it. So if we draw a coordinate system adapted to that line then the coordinate system will nessecarily be bent also.

 

[GregAshmore]

Yes, indirectly. Pulling on the cord causes the train to become non-inertial (i.e. an accelerometer attached to the train reads a non-zero value). In order to have a non-inertial object remain at rest you must use curved coordinates.

Consider, for example, a rotating turbine blade. An attached accelerometer reads a non-zero value. Therefore we know that we will have to use curved coordinates if we wish to have the turbine blade remain stationary. In this case, the coordinates will need to be curved in a helical fashion and this will yield gravitational forces pulling outwards on the turbine blade in this frame.
.
.
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Our completely arbitrary choice to define the coordinate system such that a non-inertial object was always at rest. We did not need to make that choice, we could just as easily have chosen an inertial reference frame where the train was not always at rest. Then our coordinates would not have been curved.

Did you read and understand my comments above about the baloon and the piece of paper? The line on the piece of paper that represents the train has a sharp bend in it. So if we draw a coordinate system adapted to that line then the coordinate system will nessecarily be bent also.

Yes, I believe I have understood, though I did not go through the exercise of drawing the diagrams. Similar diagrams are drawn in the x-ct plane to transform between inertial systems. The axes are not perpendicular in those diagrams, but oblique. Yet, the axes are still straight lines. When acceleration is introduced, the axes must curve in order to correctly represent the transformation.

When the acceleration of the cosmos relative to the train is expressed using differentials only, instead of points in a particular coordinate system, the result is the curved spacetime of the GR field equation.

The curved-coordinate diagram is a representation of a single reality as seen from two different frames. Likewise, the field equation is a mathematical (as opposed to graphical) representation of a single reality as observed from two different frames.

Here I would stress that every observer is by definition at rest, observing from his stationary frame the movement of nature.

Each observer is observing reality. When the observer on the stationary train sees the cosmos freely falling, the cosmos is falling in a real gravitational field (or curved spacetime, if you prefer). The observer on the stationary Earth sees no gravitational field as he observes the decelerating train, because there really is no gravitational field present.

If we hear a report from both observers, we will hear two different accounts of the event. Both accounts are true accounts of the one real event.

If I choose to sit in the stationary train to observe the event, I do not thereby cause the gravitational field to come into being. The gravitational field is real in the stationary train, regardless of my choice of observation point.

This is why Einstein appeals to the masses of the distant stars to induce the gravitation seen in the stationary train, and why he rejects the suggestion that the acceleration of the coordinate system brings about the gravitational field.

Born, too, insists on the physical reality of gravitational effects from distant stars. This explains why a motionless planet is flattened at the poles. The mass of the stars, rotating around the planet, can be modeled as a hollow, thick-walled, rotating sphere. He cites Thirring as having calculated gravitational distortions which flatten the poles of the planet.

Born puts it this way:

We take as the basis of relativistic dynamics the assumption that distant masses as real causes must now replace what was previously taken as a fictitious cause of physical phenomena, absolute space.

We see that both Einstein and Born contend that distant masses are the real cause of the gravitational effects which are seen when accelerated motion is observed from a stationary frame.

I realize now that I can be more precise in the wording of my questions.

1. What is the causal connection between the application of the brake on the train and the induction of gravity by the distant stars?

2. How is it that the action on the train is communicated instantly to the distant stars, allowing the induction to begin immediately?

 

[Dale]

Yes, I believe I have understood, though I did not go through the exercise of drawing the diagrams. Similar diagrams are drawn in the x-ct plane to transform between inertial systems. The axes are not perpendicular in those diagrams, but oblique. Yet, the axes are still straight lines. When acceleration is introduced, the axes must curve in order to correctly represent the transformation.

Excellent. Sounds like you understand it.

When the acceleration of the cosmos relative to the train is expressed using differentials only, instead of points in a particular coordinate system, the result is the curved spacetime of the GR field equation.

Careful. The paper is still flat (i.e. spacetime is not curved) just the coordinate lines are curved. It is a small detail, but it is important to recognize coordinate-independent from coordinate-dependent facts.

The curved-coordinate diagram is a representation of a single reality as seen from two different frames. Likewise, the field equation is a mathematical (as opposed to graphical) representation of a single reality as observed from two different frames.

Here I would stress that every observer is by definition at rest, observing from his stationary frame the movement of nature.

Each observer is observing reality. When the observer on the stationary train sees the cosmos freely falling, the cosmos is falling in a real gravitational field (or curved spacetime, if you prefer). The observer on the stationary Earth sees no gravitational field as he observes the decelerating train, because there really is no gravitational field present.

If we hear a report from both observers, we will hear two different accounts of the event. Both accounts are true accounts of the one real event.

If I choose to sit in the stationary train to observe the event, I do not thereby cause the gravitational field to come into being. The gravitational field is real in the stationary train, regardless of my choice of observation point.

This is why Einstein appeals to the masses of the distant stars to induce the gravitation seen in the stationary train, and why he rejects the suggestion that the acceleration of the coordinate system brings about the gravitational field.

Born, too, insists on the physical reality of gravitational effects from distant stars. This explains why a motionless planet is flattened at the poles. The mass of the stars, rotating around the planet, can be modeled as a hollow, thick-walled, rotating sphere. He cites Thirring as having calculated gravitational distortions which flatten the poles of the planet.

Born puts it this way:


We see that both Einstein and Born contend that distant masses are the real cause of the gravitational effects which are seen when accelerated motion is observed from a stationary frame.

Born and Einstein are certainly much smarter than I am and may understand the meaning of "real" sufficiently for them to speak of such things, but I am not so smart and I have never been able to grasp the meaning of the word "real" in enough scientific rigor to make such distinctions. Nobody on these forums has been able to provide me a good scientific definition of the word "real". Some people here seem to think that things that are measurable must be real, and others seem to think that things which are coordinate-dependent cannot be real. These effects are measurable but coordinate-dependent, so the two groups disagree on their reality and I cannot decide.

I realize now that I can be more precise in the wording of my questions.

1. What is the causal connection between the application of the brake on the train and the induction of gravity by the distant stars?

2. How is it that the action on the train is communicated instantly to the distant stars, allowing the induction to begin immediately?

If you want to include the masses of the distant stars you certainly may. In that case you are dealing with a curved spacetime (the balloon instead of the paper) and the math becomes much more complicated as does the definition of the coordinates. I think that the extra complication is unnecessary and doesn't add any conceptual benefit, but if you want to use it all you have to do is to start with the usual FLRW metric and coordinates and change coordinates appropriately to keep the train at rest. The resulting expression for the metric will still be a solution to the Einstein field equations and will show the gravitational acceleration of the universe as expected.

 

[GregAshmore]

Careful. The paper is still flat (i.e. spacetime is not curved) just the coordinate lines are curved. It is a small detail, but it is important to recognize coordinate-independent from coordinate-dependent facts.

Understood that representation of a particular event in the field equation of GR does not necessarily result in a curved spacetime. There will be no curvature when the "special" case of an inertial motion (in the sense of Newton-Galileo) is modeled. However, my understanding is that accelerated motion will always be accompanied by a curved spacetime metric, as it is the curvature of the metric which produces the inertial motion (free fall) of a body in that region of spacetime.

Born and Einstein are certainly much smarter than I am and may understand the meaning of "real" sufficiently for them to speak of such things, but I am not so smart and I have never been able to grasp the meaning of the word "real" in enough scientific rigor to make such distinctions. Nobody on these forums has been able to provide me a good scientific definition of the word "real". Some people here seem to think that things that are measurable must be real, and others seem to think that things which are coordinate-dependent cannot be real. These effects are measurable but coordinate-dependent, so the two groups disagree on their reality and I cannot decide.

Fair enough. I'm no Einstein, either. Even so, the goal of scientific endeavor is to properly characterize reality, or if you prefer, to distinguish between reality and fantasy. Einstein repeatedly used such terms to express the motive for his approach to the problem of accelerated motion. I daresay that if the gravitational effect observed from the stationary train is not "really real", Einstein would consider his work to be a failure. (This in spite of its successes in predicting physical events such as the precession of Mercury's orbit.)

If you want to include the masses of the distant stars you certainly may. In that case you are dealing with a curved spacetime (the balloon instead of the paper) and the math becomes much more complicated as does the definition of the coordinates. I think that the extra complication is unnecessary and doesn't add any conceptual benefit, but if you want to use it all you have to do is to start with the usual FLRW metric and coordinates and change coordinates appropriately to keep the train at rest. The resulting expression for the metric will still be a solution to the Einstein field equations and will show the gravitational acceleration of the universe as expected.

When the dynamics of this event are considered in terms of the equation of GR (leaving aside for the moment the question of reality), we start with a flat spacetime. What, then, is the causal relationship between the application of force to the wheels of the train and the resulting curvature of spacetime? And how is it that the curvature of spacetime appears instantly throughout the universe, rather than traveling as a gravitational wave?

 

[Dale]

Understood that representation of a particular event in the field equation of GR does not necessarily result in a curved spacetime. There will be no curvature when the "special" case of an inertial motion (in the sense of Newton-Galileo) is modeled. However, my understanding is that accelerated motion will always be accompanied by a curved spacetime metric, as it is the curvature of the metric which produces the inertial motion (free fall) of a body in that region of spacetime.

Wrt the metric the usage can be sloppy. Some people using the word "metric" refer to the metric tensor which is a coordinate-independent geometric quantity. Other people using the word "metric" refer to the expression for the line element which is a coordinate-dependent expression. Your response is correct using the second meaning, but I just wanted to let you know of the potential confusion in case you later get responses that use the first meaning.

Even so, the goal of scientific endeavor is to properly characterize reality, or if you prefer, to distinguish between reality and fantasy.

Unless you know the meaning of "reality" this definition of science is not very useful. I would rather say that the goal of science is to produce models that allow the accurate prediction of experimental results. If you can come up with a good definition of "reality" then we can use it to determine if an endeavor which produces accurate models also "properly characterizes reality".

When the dynamics of this event are considered in terms of the equation of GR (leaving aside for the moment the question of reality), we start with a flat spacetime. What, then, is the causal relationship between the application of force to the wheels of the train and the resulting curvature of spacetime? And how is it that the curvature of spacetime appears instantly throughout the universe, rather than traveling as a gravitational wave?

Again, spacetime is not significantly curved, just the coordinates. Also, it is not the application of force to the train that directly causes the curvature, it is the defining of your coordinate system such that a non-inertial object is at rest. You could use that coordinate system without applying force to the train, and you can apply force to the train without using that coordinate system. The application of force is only an indirect cause as I said above.

Here is how I would look at it:
1) In a coordinate-independent sense the worldlines of the train and the Earth are initially not parallel, the Earth's worldline remains straight (inertial) but the train's worldline curves (non-inertial) to become parallel to the earth's.
2) In one inertial reference frame the train is initially at rest and the Earth is moving, the force on the train is not balanced so it suddenly accelerates to match velocity with the earth.
3) In another inertial frame the Earth is initially at rest and the train is moving, the force on the train is not balanced so it suddenly decelerates to match velocity with the earth.
4) In a particular non-inertial frame the train is initially at rest and the Earth is moving, the force on the train is balanced with a transient gravitational force such that the train does not accelerate, however there is no opposing force on the Earth so the transient gravitational field brings it to rest.

 

[bcrowell]

Since GregAshmore is diving into the math, I suggest we adjourn this discussion for a while. GregAshmore, a good short-term goal would be to get to the point where you understand the contents of this WP article http://en.wikipedia.org/wiki/Rindler_coordinates . The next step would be to learn enough about tensors to understand why curvature tensors (such as the Riemann tensor) can't be made to be zero or nonzero by a change of coordinates, so that the spacetime represented by the Rindler coordinates is still flat. Hartle will also have a discussion of the equivalence principle, which should help you to understand why the Newtonian gravitational field g can't be a tensor, and can't be directly observable in GR.

 

[GregAshmore]

Agreed. I'll get busy on the math and check back in when I've done my homework.

Relativity is a tough subject, both conceptually and mathematically. I've had to learn many subjects on my own, both at work and for personal interest. Relativity is by far the most difficult of them all. (Although, I suspect that quantum theory may be just as difficult when I go beyond the qualitative discussion, into the technical details.)

I appreciate the time you three have taken to advance my understanding of relativity, and the direction you have given for further study.

 

[bcrowell]

Relativity is a tough subject, both conceptually and mathematically.

It's also a very beautiful subject, and fundamentally a simple one. Make sure to post on PF whenever you run into snags!