Logic of GR as mathematically derived

[grav-universe]

Uncapable of working through the tensors of the EFE's or visualizing the concepts upon which they are based as expressed in that manner, I have begun reverse engineering GR in an attempt to determine the basic concepts upon which it is based to find the foundation of its mathematical logic and reason. I want to see them and relate them one by one. At this point, however, I am stuck, still requiring some bits of information in order to work it out fully, so I will post what I have so far so that perhaps the members here can help to fill in the blanks. The values for time dilation and length contraction are what I am trying to find overall, depending upon some initial condition for the coordinate system used, but by bypassing the EFE's or if they are basically the same thing, just by taking each part of what can be deduced logically and putting those together algebraicly. We can start with some basic assumptions,

1. SR is valid locally

2. The speed of light is measured at c locally

3. Shells of identical attributes are spherical according to an observer at infinity

4. Mass is invariant

Add to this list as many as possible. I might be taking some minor assumptions for granted that may be important.

Then we have a list of assumptions about the field and the motion of a particle through the field, such as

A. Gravitational flux strength - The field of flux of gravity flows outward from the center. As the flux lines cross a shell, its strength is inversely proportional to the area of the surface of the shell, the number of flux lines being constant, but the strength diminishing with the number per area. A distant observer measures the area to be 4 pi r^2, and with a local tangent contraction of L_t, the local observer will measure the surface area to be (4 pi r^2) / L_t^2. In the radial direction, each flux line can be visualized as a beam of particles crossing the surface, and the strength is proportional to the frequency at which they cross, which is inversely proportional to the local time dilation z at that surface. The strength of gravity, the locally measured acceleration, then, is a' = F [L_t^2 / (4 pi r^2)] [1 / z], where F is some constant. By taking the low gravity approximation where L_t = z = 1 and a = - G M / r^2, we find F = - 4 pi G M, so that gives us a' = - G M L_t^2 / (r^2 z). Hopefully this accurately describes the field of flux and I have explained it well enough.

B. Energy conservation - Not so much conservation though, really, more a relation, but for a photon falling radially through the field, it works out to

f_r z_r = f_s z_s

That is, the frequency at any point in the field is the same according to a distant observer, with each successive pulse traveling through the field in the same time between any two points, so the rate of reception at r must be the same as the rate of emission at s according to the distant observer. The locally measured frequency, then, is just inversely proportional to the local time dilation z. Relating this to the energy of a photon, we have

(h f_r) z_r = (h f_s) z_s

E_r z_r = E_s z_s

And if we relate this to a massive particle as it passes each location in the same way as we did for a photon, we have

[m c^2 / sqrt(1 - (v'_r/c)^2)] z_r = [m c^2 / sqrt(1 - (v'_s/c)^2)] z_s

z_r / sqrt(1 - (v'_r/c)^2) = z_s / sqrt(1 - (v'_s/c)^2)

z / sqrt(1 - (v'/c)^2) = K

where K is a constant depending upon the conditions of radial freefall. For a massive particle falling from infinity, for instance, with initial conditions z = 1 and v' = 0, then K = 1.

C. Conservation of momentum - Here we just assume the locally measured angular momentum is constant for all shells, so that m v'_t r' is constant. r' would be the inferred distance to the origin, what the distant observer says would be measured if a length contracted ruler at r were extended with the same radial contraction L all the way to the center. For less ambiguity, however, we can restate that to say m v'_t (r / L) is a constant. Since m is invariant, we can reduce that to read P = v'_t r / L = constant.

Please add to this list.

Okay, so now to define invariants. Here I am defining invariants as quantities that remain the same regardless of the coordinate system. If we transform one GR coordinate system to another, we are changing the positions of the shells. However, the locally measured acceleration at that shell will remain the same, so a' is an invariant in this sense. The time dilation z at that shell will also remain the same, so z is an invariant. L and L_t change as the positions of shells and the ends of a local ruler change according to a distant observer, so those are not invariants. We have already assumed m to be an invariant, although it may be possible to keep it invaraint as an initial condition anyway while letting other factors vary. v' is the locally measured radial speed of a particle which will be invariant upon freefalling from some other shell no matter how we re-position them, while v that the distant observer measures is v = z L v', dependent upon L, so of course is not invariant.

Let's get started. Classically, the locally measured acceleration would be a' = d(v'^2) / (2 dr'). With local SR, with change in speed decreasing as one approaches c, however, it can be demonstrated that a' = d(v^2) / [2 dr' (1 - (v'/c)^2)]. While the former equation would still be the instantaneous coordinate acceleration that a local observer would measure, if we define a' instead only as the acceleration of a particle falling from rest at r, the latter equation finds a' as defined as such regardless of the initial measured speed of the particle. So from assumption B, for a particle falling from infinity with K = 1, that becomes

a' = d(v'^2) / [2 dr' (1 - (v'/c)^2)]

a' = c^2 d(1 - z^2) L / (2 dr z^2)

a' = c^2 [-2 dz z] L / (2 dr z^2)

a' = - c^2 dz L / (z dr)

Or likewise, we can find it with

a' = d(v') / [dt' (1 - (v'/c)^2)]

a' = c d(sqrt(1 - z^2)) / [z dt z^2]

a' = c [- dz z / sqrt(1 - z^2)] / [z^3 (dr / v)]

a' = - c dz (z L v') / [sqrt(1 - z^2) z^2 dr]

a' = - c^2 dz L / (z dr)

We can now relate assumptions A and B as

a' = - c^2 dz L / (z dr) = - G M L_t^2 / (r^2 z)

c^2 (dz / dr) L = G M L_t^2 / r^2

This is an important step, I think. It relates the time dilation and radial and tangent length contractions independent of the coordinate system involved. z is a function of r here, though, so let's arbitrarily set z = sqrt(1 - 2 G M / (r c^2)) as a coordinate choice, although many are possible. Then we get

c^2 [2 G M / (2 r^2 c^2 sqrt(1 - 2 G M / r))] L = G M L_t^2 / r^2

L / sqrt(1 - 2 G M / r) = L_t^2

L = L_t^2 sqrt(1 - 2 G M / r)

But we can only have one coordinate choice and the rest must be derived, so I'm not sure how to go further to derive one of the other two and then the last. Other coordinate choices will give other relations for other coordinate systems. This one is for Schwarzschild, of course, but since we are only allowed one coordinate choice, I am at a standstill. For example, we could perhaps set z = L as a coordinate choice and find the relation to L_t, but then we don't know what z and L are, only that they are equal. And other coordinate choices also give z = L with different L_t, such as z = L = 1 / sqrt(1 + 2 G M / r) for instance, which is another valid GR coordinate system. We might set L_t = 1 but are still left with just a relation between z and L, and we can't set L_t = 1 and z = L as two coordinate choices as far as I can see. I need some other piece of information. Any ideas?

 

[Dale]

Seems to me like you are putting a lot of work in here. Why not put that effort into learning tensors instead?

 

[grav-universe]

Seems to me like you are putting a lot of work in here. Why not put that effort into learning tensors instead?

Yes, lol. I've considered it. I've watch all of Leonard Susskind's lectures on GR, lots of tensor math, but I think they confused me more than resolved anything, other than the simple algebraic representations. Too many notations and they don't follow algebraic rules. I don't want to get them mixed up. And I like to visualize, so regardless, I would still like to break down the overall math into each of the basic features and build back up from there from whatever principles and foundations that can be determined.

 

[pervect]

Unfortunately, points 1 and 2 are the only really good ones about GR. The rest is already starting to drift off and become shaky - how shaky they get depends on where you go with them. It's not a disaster yet, but neither is it a firm foundation to build on.

One issue of many.

If you decide you don't understand tensors and can't learn them, you can't understand the stress energy tensor. As a result you're most likely going to try to visualize everything in terms of "mass". Unfortunately, mass is 1 number, and it takes 10 or so numbers to represent the stress-energy tensor. So you're basically not going to actually get GR. You're going to get some pseudo-theory that you call GR in your own mind that's some chimerical offshoot of Newton's theory (where mass does cause gravity).

 

[PeterDonis]

Uncapable of working through the tensors of the EFE's or visualizing the concepts upon which they are based as expressed in that manner, I have begun reverse engineering GR in an attempt to determine the basic concepts upon which it is based to find the foundation of its mathematical logic and reason.

In other words, you are attempting to reverse engineer the basic concepts of GR by discarding one of the basic concepts of GR.

The values for time dilation and length contraction are what I am trying to find overall

Then you aren't trying to find the basic concepts of GR in general; you are trying to understand particular derived concepts that only appear in certain particular solutions. There are many spacetimes in GR that have no useful concept of "length contraction" or "time dilation".

1. SR is valid locally

This is generally true in GR, yes.

2. The speed of light is measured at c locally

This is also generally true in GR, yes.

3. Shells of identical attributes are spherical according to an observer at infinity

This is only true in certain particular solutions.

4. Mass is invariant

This is also true only in certain particular solutions. In fact, "mass" can only be defined in certain particular solutions.

Add to this list as many as possible. I might be taking some minor assumptions for granted that may be important.

The assumptions you are making, that restrict you to only those particular solutions that meet the above requirements, are not what I would call "minor". That's not to say that these solutions aren't interesting; they are. But you should be clear that what you are doing is not "GR in general". It is only one particular application of GR.

A. Gravitational flux strength - The field of flux of gravity flows outward from the center...

All of this is also true only in certain particular solutions; but at this point I'm taking it as given that you are only interested in those solutions, and that you understand that "GR" in general is much broader than what you are considering.

A distant observer measures the area to be 4 pi r^2, and with a local tangent contraction of L_t, the local observer will measure the surface area to be (4 pi r^2) / L_t^2.

This is wrong. The are is 4 pi r^2, period. That is the definition of the "r" coordinate. What you are calling the "local tangent contraction"--I assume you mean that the coefficient of dr^2 in the metric is not 1, but 1/(1 - 2m/r)--doesn't change the physical area of a 2-sphere at radial coordinate r. All it changes is the physical length between two 2-spheres with radius r and r + dr, compared to the difference in area between those two 2-spheres. The areas are 4 pi r^2 and 4 pi (r + dr)^2; but the physical distance between the two 2-spheres is not dr, but dr/sqrt(1 - 2m/r). (So "tangent contraction" is a bad name; if there's any effect at all, it's radial, not tangential.)

In the radial direction, each flux line can be visualized as a beam of particles crossing the surface

This is not part of the standard interpretation of gravity around a spherically symmetric body, and it also doesn't seem necessary to the rest of your own model; why are you including it? (Strictly speaking, the "field of flux lines" view itself isn't really part of the standard interpretation either, but it's an obvious enough interpretation; however, that interpretation doesn't require any further talk about beams of particles.)

The strength of gravity, the locally measured acceleration, then, is a' = F [L_t^2 / (4 pi r^2)] [1 / z], where F is some constant. By taking the low gravity approximation where L_t = z = 1 and a = - G M / r^2, we find F = - 4 pi G M, so that gives us a' = - G M L_t^2 / (r^2 z). Hopefully this accurately describes the field of flux and I have explained it well enough.

I can't be sure whether or not this formula corresponds to the correct GR one. The correct GR formula, for the particular solution you are talking about, is (in conventional units)

$$a = \frac{GM}{r^2 \sqrt{1 - 2 GM / c^2 r}}$$

Note that this formula is not approximate; it's exact. I don't see how the formula you wrote corresponds to it, but maybe I'm missing something.

B. Energy conservation
...
The locally measured frequency, then, is just inversely proportional to the local time dilation z.

This looks OK, provided the emitter and receiver are both at rest in the static field. What you are calling "local time dilation" is only valid for objects at rest in the static field in any case.

And if we relate this to a massive particle as it passes each location in the same way as we did for a photon
...
z / sqrt(1 - (v'/c)^2) = K

What you are calling K here is usually called "energy at infinity", and it is indeed a constant of the motion for a freely falling object. This part looks fine, with the same proviso as above that what you are calling z (another often used name for it is the "gravitational potential") assumes an "observer" who is at rest in the static field.

C. Conservation of momentum - Here we just assume the locally measured angular momentum is constant for all shells, so that m v'_t r' is constant.

Did you mean "conservation of angular momentum"? Also, what is "m" here? Is it supposed to be the mass of the gravitating body? If so, that body's angular momentum has to be zero in the set of particular solutions that meet your requirements above. A rotating body is not spherically symmetric, and your assumptions above include spherical symmetry.

If you are just trying to express the angular momentum of an object in a free-fall orbit around the gravitating body, you are correct that that is also a constant of the motion. You might want to check out the Wikipedia page on geodesics in Schwarzschild spacetime (which is the particular solution you are considering here):

http://en.wikipedia.org/wiki/Schwarzschild_geodesics

r' would be the inferred distance to the origin, what the distant observer says would be measured if a length contracted ruler at r were extended with the same radial contraction L all the way to the center.

The actual "radial contraction" is itself a function of r, so it changes as you go in towards the center. So the actual physical distance to the origin has to be calculated by integrating $dr / \sqrt{1 - 2 G M / c^2 r}$ from r = 0 to whatever radius you are interested in.

You should note, though, that the standard definition of the angular momentum of an object in a free-fall orbit uses the radial *coordinate* r, *not* the "inferred distance to the origin". That's because the radial coordinate is a local quantity, but the inferred distance is not; it requires evaluating an integral, as I just said.

If we transform one GR coordinate system to another, we are changing the positions of the shells.

Not necessarily; at least, you need to more precisely define what "changing the positions of the shells" means.

However, the locally measured acceleration at that shell will remain the same, so a' is an invariant in this sense. The time dilation z at that shell will also remain the same, so z is an invariant.

How are you determining which shell in coordinate chart B corresponds to which shell in coordinate chart A? (This ties into what "changing the position of the shells" means.) If you are *defining* "the same shell" as "has the same a' and z", then what, exactly, is the "position" of the shell?

L and L_t change as the positions of shells and the ends of a local ruler change according to a distant observer, so those are not invariants.

See above comments about angular momentum. Properly defined, the angular momentum of a gravitating body *is* an invariant; and so is the angular momentum of an object in a free-fall orbit around the body.

We have already assumed m to be an invariant

You actually don't have to assume this; you can prove it based on other assumptions.

v' is the locally measured radial speed of a particle

Locally measured *by an observer who is static in the field*.

which will be invariant upon freefalling from some other shell no matter how we re-position them

Again, if you are using the invariance of v' as part of the definition of "the same shell", then what does "repositioning" the shell mean?

while v that the distant observer measures is v = z L v', dependent upon L, so of course is not invariant.

First, see comments above about L. Second, how does the distant observer "measure" v?

Classically, the locally measured acceleration would be a' = d(v'^2) / (2 dr').

Why is the factor of 2 there?

With local SR, with change in speed decreasing as one approaches c, however, it can be demonstrated that a' = d(v^2) / [2 dr' (1 - (v'/c)^2)].

I think you are confusing coordinate acceleration and proper acceleration here, but I'm not sure. My understanding above was that a' was supposed to be proper acceleration, but now you seem to be saying that a' is coordinate acceleration. It can't be both. At this point I can't really go further, since I'm not sure how you're interpreting the quantities in your formulas.

 

[grav-universe]

Unfortunately, points 1 and 2 are the only really good ones about GR. The rest is already starting to drift off and become shaky - how shaky they get depends on where you go with them. It's not a disaster yet, but neither is it a firm foundation to build on.

One issue of many.

If you decide you don't understand tensors and can't learn them, you can't understand the stress energy tensor. As a result you're most likely going to try to visualize everything in terms of "mass". Unfortunately, mass is 1 number, and it takes 10 or so numbers to represent the stress-energy tensor. So you're basically not going to actually get GR. You're going to get some pseudo-theory that you call GR in your own mind that's some chimerical offshoot of Newton's theory (where mass does cause gravity).

Right. I would like to know what each of those components represent algebraicly so I can break them down and put them together in a way that I can visualize better.

 

[grav-universe]

In other words, you are attempting to reverse engineer the basic concepts of GR by discarding one of the basic concepts of GR.

Not discarding it, just wanting to break it down into its basic components.

Then you aren't trying to find the basic concepts of GR in general; you are trying to understand particular derived concepts that only appear in certain particular solutions. There are many spacetimes in GR that have no useful concept of "length contraction" or "time dilation".

I haven't heard of those. They sound interesting but probably wouldn't be useful in terms of the way I am trying to do this.

This is generally true in GR, yes.

This is also generally true in GR, yes.

This is only true in certain particular solutions.

That's interesting. I suppose a distant observer could infer non-spherical shells, but I am looking for symmetry with static observers and a non-rotating mass.

This is also true only in certain particular solutions. In fact, "mass" can only be defined in certain particular solutions.

That's interesting also. I thought mass was considered an invariant. I am including the mass equivalent of any energy present as part of the mass also, anything that contributes to the field. You state this here, but below you state "You actually don't have to assume this; you can prove it based on other assumptions." referring to the invariance of mass, so I'm not sure exactly what you are saying.

The assumptions you are making, that restrict you to only those particular solutions that meet the above requirements, are not what I would call "minor". That's not to say that these solutions aren't interesting; they are. But you should be clear that what you are doing is not "GR in general". It is only one particular application of GR.

Would the last two be considered "conditions" then? An additional condition would be that we are only considering a non-rotating mass. Also, I suppose another assumption would be

5. The equivalence principle holds

All of this is also true only in certain particular solutions; but at this point I'm taking it as given that you are only interested in those solutions, and that you understand that "GR" in general is much broader than what you are considering.

This is wrong. The are is 4 pi r^2, period. That is the definition of the "r" coordinate. What you are calling the "local tangent contraction"--I assume you mean that the coefficient of dr^2 in the metric is not 1, but 1/(1 - 2m/r)--doesn't change the physical area of a 2-sphere at radial coordinate r. All it changes is the physical length between two 2-spheres with radius r and r + dr, compared to the difference in area between those two 2-spheres. The areas are 4 pi r^2 and 4 pi (r + dr)^2; but the physical distance between the two 2-spheres is not dr, but dr/sqrt(1 - 2m/r). (So "tangent contraction" is a bad name; if there's any effect at all, it's radial, not tangential.)

This is not part of the standard interpretation of gravity around a spherically symmetric body, and it also doesn't seem necessary to the rest of your own model; why are you including it? (Strictly speaking, the "field of flux lines" view itself isn't really part of the standard interpretation either, but it's an obvious enough interpretation; however, that interpretation doesn't require any further talk about beams of particles.)

I can't be sure whether or not this formula corresponds to the correct GR one. The correct GR formula, for the particular solution you are talking about, is (in conventional units)

$$a = \frac{GM}{r^2 \sqrt{1 - 2 GM / c^2 r}}$$

Note that this formula is not approximate; it's exact. I don't see how the formula you wrote corresponds to it, but maybe I'm missing something.

I didn't literally mean particles, more just points in the flux lines that cross the surface per unit time. As for the surface area, the distant observer would measure 4 pi r^2, but the rulers of the local static observers at the surface would be contracted by L_t in the tangent direction along the surface, so they would measure (4 pi r^2) / L_t^2. The local acceleration they measure is

a' = G M L_t^2 / (z r^2)

This solution can be found by reverse engineering the result at the end of the post, the result of which I verified by applying the radial and tangent length contractions and time dilation for three different coordinate systems and it works out correctly for each. What you wrote is the local acceleration in Schwarzschild, which with L_t = 1 and z = sqrt(1 - 2 G M / (r c^2)), works out as you have, right.

This looks OK, provided the emitter and receiver are both at rest in the static field. What you are calling "local time dilation" is only valid for objects at rest in the static field in any case.

What you are calling K here is usually called "energy at infinity", and it is indeed a constant of the motion for a freely falling object. This part looks fine, with the same proviso as above that what you are calling z (another often used name for it is the "gravitational potential") assumes an "observer" who is at rest in the static field..

Right, by local, I mean as measured by static observers.

Did you mean "conservation of angular momentum"? Also, what is "m" here? Is it supposed to be the mass of the gravitating body? If so, that body's angular momentum has to be zero in the set of particular solutions that meet your requirements above. A rotating body is not spherically symmetric, and your assumptions above include spherical symmetry.

If you are just trying to express the angular momentum of an object in a free-fall orbit around the gravitating body, you are correct that that is also a constant of the motion. You might want to check out the Wikipedia page on geodesics in Schwarzschild spacetime (which is the particular solution you are considering here):

http://en.wikipedia.org/wiki/Schwarzschild_geodesics

Right, that's what I meant, the conservation of angular momentum. M is the mass of the gravitating body, energy equivalent included, and m is the mass of the particle.

The actual "radial contraction" is itself a function of r, so it changes as you go in towards the center. So the actual physical distance to the origin has to be calculated by integrating $dr / \sqrt{1 - 2 G M / c^2 r}$ from r = 0 to whatever radius you are interested in.

You should note, though, that the standard definition of the angular momentum of an object in a free-fall orbit uses the radial *coordinate* r, *not* the "inferred distance to the origin". That's because the radial coordinate is a local quantity, but the inferred distance is not; it requires evaluating an integral, as I just said.

Right, that would be the ruler distance as I call it, the distance measured by rulers laid end to end radially between two points in the field. What I was referring to there I call the inferred distance, which is just r / L. I found that conserved quantity by reverse engineering the metric solution in another thread for a freefalling particle.

Not necessarily; at least, you need to more precisely define what "changing the positions of the shells" means.

Changing the positions simply means defining some coordinate transformation, r1 = F(r), such as transforming to Eddington isotropic coordinates.

How are you determining which shell in coordinate chart B corresponds to which shell in coordinate chart A? (This ties into what "changing the position of the shells" means.) If you are *defining* "the same shell" as "has the same a' and z", then what, exactly, is the "position" of the shell?

We define shell A at r as having a' and z as invariants. Whatever new position r1 we change it to through coordinate transformations, it must still have the invariants a' and z.

Again, if you are using the invariance of v' as part of the definition of "the same shell", then what does "repositioning" the shell mean?

We can change the position of the shell, but the local static observer must still measure v' for the same particle passing there, so regardless of the coordinate transformation.

First, see comments above about L. Second, how does the distant observer "measure" v?

The distant observer infers that the local static observer's radial ruler is contracted by L and the local static observer's clock is ticking at a rate of z, as compared to the distant observer's own ruler and clock. The local observer measures a speed of v', measured over infinitesimal distance and time, so the distant observer infers

v' = dr' / dt' = (dr / L) / (z dt) = (dr / dt) / (z L) = v / z L

Why is the factor of 2 there?

That is just the standard equation for determining coordinate acceleration, whereas v_final^2 - v_original^2 = 2 a d, or in this case, d(v'^2) = 2 a' dr'. That 2 gets me sometimes.

I think you are confusing coordinate acceleration and proper acceleration here, but I'm not sure. My understanding above was that a' was supposed to be proper acceleration, but now you seem to be saying that a' is coordinate acceleration. It can't be both. At this point I can't really go further, since I'm not sure how you're interpreting the quantities in your formulas.

a' is the acceleration of a particle dropped from rest at r as the local static observer there would measure it. The first equation gives the coordinate acceleration starting from any speed v'. Call it a'_v'. For an original speed greater than zero, such as with a particle freefalling past a static observer at v' > 0, the first equation will only give the coordinate acceleration a'_v' for that speed, which is measured smaller than a' with greater v' in SR. To find a', just that for a particle falling from rest there with v' = 0 initially, we would need to divide by (1 - (v'/c)^2), so a'_v' = a' (1 - (v'/c)^2), as determined in another thread.

 

[PeterDonis]

Not discarding it, just wanting to break it down into its basic components.

But the way you break it down into components is frame-dependent. The individual components don't always have the same physical meaning. If you restrict yourself to a particular scenario, and pick your frame appropriately, you might be able to assign a reasonably intuitive physical meaning to the tensor components; but any visualization you come up with based on that will be limited to that particular scenario. As long as you're OK with that, yes, we should be able to "break down" at least some of the tensor components for a particular scenario.

I am looking for symmetry with static observers and a non-rotating mass.

Ok, that narrows it down to a particular scenario. Basically, you have a spherically symmetric, static, non-rotating gravitating body with some radius r that is significantly greater than 2M, where M is its mass. That ensures that there are no questions about whether or not any horizon is present.

One note: so far you only seem to be considering the vacuum region exterior to the gravitating body. In the interior of the body, things work a bit differently because there is matter present.

That's interesting also. I thought mass was considered an invariant.

In the particular scenarios where it can be defined--or at least a particular subset of them--yes, the mass is an invariant. But there are possible scenarios where the "mass" is changing with time, and others where no useful concept of "mass" can be defined at all.

I am including the mass equivalent of any energy present as part of the mass also, anything that contributes to the field.

Yes, that's correct; in the cases where a "mass" can be defined, it will include these contributions.

You state this here, but below you state "You actually don't have to assume this; you can prove it based on other assumptions." referring to the invariance of mass, so I'm not sure exactly what you are saying.

What I meant by this is that, in those cases where a mass can be defined, it can be computed based on other quantities; and in the subset of those cases where the mass is invariant (doesn't change with time), the fact that it is invariant can also be computed based on other quantities. The mass itself is never a fundamental quantity.

Would the last two be considered "conditions" then?

I suppose you could view them that way, yes. My point is simply that whatever analysis we are going to do here is not generally applicable; it's only applicable in a particular case.

An additional condition would be that we are only considering a non-rotating mass.

Understood.

Also, I suppose another assumption would be

5. The equivalence principle holds

This one is always true in GR as well, so it won't restrict you any.

I didn't literally mean particles, more just points in the flux lines that cross the surface per unit time.

Ok. "Particles" is probably not a good term, then, since it suggests an underlying mechanism that doesn't seem to be what you are proposing.

As for the surface area, the distant observer would measure 4 pi r^2, but the rulers of the local static observers at the surface would be contracted by L_t in the tangent direction along the surface, so they would measure (4 pi r^2) / L_t^2.

No, they wouldn't. Rulers are not "contracted" at all, and in any case whatever "contraction" is present doesn't affect the tangential part of the line element (I'm assuming we are using the Schwarzschild line element), it only affects the radial part. Please re-read what I said about the definition of the r coordinate in terms of the physical area of 2-spheres, and what immediately follows.

Right, that's what I meant, the conservation of angular momentum. M is the mass of the gravitating body, energy equivalent included, and m is the mass of the particle.

Ok, then if the body is non-rotating, its angular momentum is zero. I suppose that does count as "conserved", but you can't write down a formula for the angular momentum--you just know it's zero because you assumed a non-rotating body.

Changing the positions simply means defining some coordinate transformation, r1 = F(r), such as transforming to Eddington isotropic coordinates.

We define shell A at r as having a' and z as invariants. Whatever new position r1 we change it to through coordinate transformations, it must still have the invariants a' and z.

We can change the position of the shell, but the local static observer must still measure v' for the same particle passing there

All ok.

The distant observer infers that the local static observer's radial ruler is contracted by L and the local static observer's clock is ticking at a rate of z, as compared to the distant observer's own ruler and clock.

To be clear, this is one *interpretation* of what the metric coefficients at a given radius r are telling you, and you have to be very careful in drawing inferences from it.

The local observer measures a speed of v', measured over infinitesimal distance and time, so the distant observer infers

v' = dr' / dt' = (dr / L) / (z dt) = (dr / dt) / (z L) = v / z L

This is one of those inferences you have to be very careful of. This v' that you have derived is not a physical observable, and is not subject to the usual limitations on physical quantities. So it's not, IMO, a good thing to focus on if you're trying to understand how GR works in this particular scenario.

a' is the acceleration of a particle dropped from rest at r as the local static observer there would measure it.

Ok, so it's a coordinate acceleration. It just so happens that the same formula (with a sign change if one wants to be precise) also describes the proper acceleration of the static observer, i.e., what he actually measures on his accelerometer. So in this case, a coordinate acceleration does correlate directly to a physical observable. But that's not always true.

 

[pervect]

Right. I would like to know what each of those components represent algebraicly so I can break them down and put them together in a way that I can visualize better.

The components are energy, momentum, and pressure (also called stress).

http://en.wikipedia.org/w/index.php?title=Stress–energy_tensor&oldid=517465899 has an overview. Their diagram is probably the most helpful part of the article.

http://upload.wikimedia.org/wikipedia/commons/3/37/StressEnergyTensor.svg

Note that the stress-energy tensor is symmetric, so that T_01 is equal to T_10. The wiki diagram gives two names to some of the quantities, i.e. momentum density for T_10 is equal to the "energy flux" T_01. In spite of the differeng names, they're equal.

http://math.ucr.edu/home/baez/einstein/ also has a description of both the Stress energy tensor, and Einstein's equation.. Baez's explanation, based on the rate of volume change of a sphere of "coffee grounds" is the simplest one I'm aware of that's technically accurate.

To state Einstein's equation in simple English, we need to consider a round ball of test particles that are all initially at rest relative to each other. As we have seen, this is a sensible notion only in the limit where the ball is very small. If we start with such a ball of particles, it will, to second order in time, become an ellipsoid as time passes. This should not be too surprising, because any linear transformation applied to a ball gives an ellipsoid, and as the saying goes, ``everything is linear to first order''. Here we get a bit more: the relative velocity of the particles starts out being zero, so to first order in time the ball does not change shape at all: the change is a second-order effect.

 

[grav-universe]

But the way you break it down into components is frame-dependent. The individual components don't always have the same physical meaning. If you restrict yourself to a particular scenario, and pick your frame appropriately, you might be able to assign a reasonably intuitive physical meaning to the tensor components; but any visualization you come up with based on that will be limited to that particular scenario. As long as you're OK with that, yes, we should be able to "break down" at least some of the tensor components for a particular scenario.

Yay. :)

Ok, that narrows it down to a particular scenario. Basically, you have a spherically symmetric, static, non-rotating gravitating body with some radius r that is significantly greater than 2M, where M is its mass. That ensures that there are no questions about whether or not any horizon is present.

One note: so far you only seem to be considering the vacuum region exterior to the gravitating body. In the interior of the body, things work a bit differently because there is matter present.

Right. I am only considering the vacuum region outside of the body and beyond any singular horizon.

In the particular scenarios where it can be defined--or at least a particular subset of them--yes, the mass is an invariant. But there are possible scenarios where the "mass" is changing with time, and others where no useful concept of "mass" can be defined at all.

Yes, that's correct; in the cases where a "mass" can be defined, it will include these contributions.

What I meant by this is that, in those cases where a mass can be defined, it can be computed based on other quantities; and in the subset of those cases where the mass is invariant (doesn't change with time), the fact that it is invariant can also be computed based on other quantities. The mass itself is never a fundamental quantity.

Okay, thanks. The mass (and energy equivalent) is constant over time and is in proportion to the gravity, made constant (in the form GM) for any r while changing functions for other values only or the mass is simply presumed to be an invariant at all r to begin with.

No, they wouldn't. Rulers are not "contracted" at all, and in any case whatever "contraction" is present doesn't affect the tangential part of the line element (I'm assuming we are using the Schwarzschild line element), it only affects the radial part. Please re-read what I said about the definition of the r coordinate in terms of the physical area of 2-spheres, and what immediately follows.

I'm not sure what you mean here. I am just using the standard interpretation for gravitational time dilation and contraction of rulers. For instance, in SC, a photon traveling radially would be measured locally (by a static observer) at c, but a distant observer infers that it travels at z L c, where z is the time dilation and L is the radial length contraction, where z = L = sqrt(1 - 2 m / r), so the distant observer infers a speed of the photon of (1 - 2 m / r) c.

Also, I am not applying just SC except for examples, but these solutions, the equations found here, will apply to any arbitrary coordinate system. I generally use the three coordinate systems SC, GUC, and EIC to verify the results.

Ok, then if the body is non-rotating, its angular momentum is zero. I suppose that does count as "conserved", but you can't write down a formula for the angular momentum--you just know it's zero because you assumed a non-rotating body.

m v' r / L_t refers to the mass of the particle, not the gravitating body, but since we are considering mass to be an invariant and the angular momentum constant also, we can just drop m and consider that P = v' r / L_t is constant for any r.

To be clear, this is one *interpretation* of what the metric coefficients at a given radius r are telling you, and you have to be very careful in drawing inferences from it.

This is one of those inferences you have to be very careful of. This v' that you have derived is not a physical observable, and is not subject to the usual limitations on physical quantities. So it's not, IMO, a good thing to focus on if you're trying to understand how GR works in this particular scenario.

Right, it is not physically observable, only inferred by the particular coordinate system applied. And since coordinate systems can change, what the distant observer infers is not invariant. Kind of the point of changing coordinate systems I guess. :)

Also, to be clear, I actually wrote that backwards, sorry. Since I was showing what the distant observer infers, it should have been

v = dr / dt = (L dr') / (dt' / z) = z L (dr' / dt') = z L v'

The primed units refer to local measurements made by a static observer and the unprimed units are what the distant observer infers.

 

[grav-universe]

The components are energy, momentum, and pressure (also called stress).

http://en.wikipedia.org/w/index.php?title=Stress–energy_tensor&oldid=517465899 has an overview. Their diagram is probably the most helpful part of the article.

http://upload.wikimedia.org/wikipedia/commons/3/37/StressEnergyTensor.svg

Note that the stress-energy tensor is symmetric, so that T_01 is equal to T_10. The wiki diagram gives two names to some of the quantities, i.e. momentum density for T_10 is equal to the "energy flux" T_01. In spite of the differeng names, they're equal.

http://math.ucr.edu/home/baez/einstein/ also has a description of both the Stress energy tensor, and Einstein's equation.. Baez's explanation, based on the rate of volume change of a sphere of "coffee grounds" is the simplest one I'm aware of that's technically accurate.

Cool. That baez link may be just what I needed. Thank you. :)

 

[PeterDonis]

I am just using the standard interpretation for gravitational time dilation and contraction of rulers.

I'll agree that this is more or less the standard interpretation for time dilation (though not without some qualifications), but it isn't for contraction of rulers. Here is some food for thought:

(1) You are interpreting a change in the coefficient of dt^2 in the metric as "time dilation" and a change in the coefficient of dr^2 as "contraction of rulers". But in the case of length contraction and time dilation in SR, which is supposed to be what motivates the terminology, the metric coefficients *do not change*. So the supposed analogy with SR seems fishy to me, to say the least.

(2) In the case of time dilation, since the spacetime is static, we can set up a direct measurement by sending photons back and forth between static observers, and observing their frequencies. But that works precisely because the spacetime is static--i.e., it doesn't change with time. The spacetime *does* change with radius, so we can't draw an analogy between time dilation and length contraction and say that they must go together the way they do in flat spacetime; there is a key physical difference between measurements of time intervals and measurements of radial intervals, which isn't there in flat spacetime.

(3) Saying that the change in the coefficient of dr^2 represents "contraction of rulers as measured by the distant observer" assumes that the distant observer can make some kind of measurement that shows this. But if we actually try to construct such a measurement, it doesn't work out that way.

Here's one example: suppose I set up a mechanical linkage between two static observers, one at some finite radius r > 2m and one so far distant that we can take his "r" to be infinity (so all his metric coefficients are the same as for flat spacetime). I construct the linkage--say it's a very, very long metal rod--far away from the gravitating body and make marks at either end that I verify are separated by the same length, call it 1 meter. Then I lower one end of the rod to the observer at r; the other end stays with the distant observer. The rod is oriented radially.

Now the distant observer moves the rod by 1 meter, using the marks to verify the distance. The observer at r measures how far the marks at his end move, compared to the length of a meter stick that he has locally. (A technical point: we assume that the meter stick is rigid enough that the proper acceleration required to hold it static at r does not change its length measurably, compared to its length when in free fall. We also make the same assumption about the rod--acceleration doesn't affect its observed length.) What does he observe? What do you predict? (I'll save the answer for a separate post after you've had a chance to respond.)

Right, it is not physically observable, only inferred by the particular coordinate system applied. And since coordinate systems can change, what the distant observer infers is not invariant.

To me, this takes away the point of the distant observer inferring these things in the first place. If he's inferring something that will change if he changes coordinates, why is he interested? Why isn't he interested in things that are invariant under coordinate changes? The latter are what tell you about the physics.

Kind of the point of changing coordinate systems I guess. :)

To me, and I think to most relativity physicists, the point of changing coordinate systems is that some invariants are much easier to compute in one coordinate system, while others are much easier to compute in a different coordinate system. Since what you're computing is an invariant, you can compute it in any coordinate system you want and the answer will be the same, so why not do it in the one that's easiest?

 

[grav-universe]

Here's one example: suppose I set up a mechanical linkage between two static observers, one at some finite radius r > 2m and one so far distant that we can take his "r" to be infinity (so all his metric coefficients are the same as for flat spacetime). I construct the linkage--say it's a very, very long metal rod--far away from the gravitating body and make marks at either end that I verify are separated by the same length, call it 1 meter. Then I lower one end of the rod to the observer at r; the other end stays with the distant observer. The rod is oriented radially.

Now the distant observer moves the rod by 1 meter, using the marks to verify the distance. The observer at r measures how far the marks at his end move, compared to the length of a meter stick that he has locally. (A technical point: we assume that the meter stick is rigid enough that the proper acceleration required to hold it static at r does not change its length measurably, compared to its length when in free fall. We also make the same assumption about the rod--acceleration doesn't affect its observed length.) What does he observe? What do you predict? (I'll save the answer for a separate post after you've had a chance to respond.)

Well, if I read that correctly, the inferred length according to the distance observer will decrease, moving down one full meter at the top and a "contracted" meter at the bottom. The ruler length of the rod, d = int dr / L, however, will remain the same. And the observer at r will measure that the rod has moved one full meter as measured locally in the same way that the distant observer does because each meterstick is identical locally, only different as inferred, depending upon the particular coordinate system applied.

I'll agree that this is more or less the standard interpretation for time dilation (though not without some qualifications), but it isn't for contraction of rulers. Here is some food for thought:

(1) You are interpreting a change in the coefficient of dt^2 in the metric as "time dilation" and a change in the coefficient of dr^2 as "contraction of rulers". But in the case of length contraction and time dilation in SR, which is supposed to be what motivates the terminology, the metric coefficients *do not change*. So the supposed analogy with SR seems fishy to me, to say the least.

(2) In the case of time dilation, since the spacetime is static, we can set up a direct measurement by sending photons back and forth between static observers, and observing their frequencies. But that works precisely because the spacetime is static--i.e., it doesn't change with time. The spacetime *does* change with radius, so we can't draw an analogy between time dilation and length contraction and say that they must go together the way they do in flat spacetime; there is a key physical difference between measurements of time intervals and measurements of radial intervals, which isn't there in flat spacetime.

(3) Saying that the change in the coefficient of dr^2 represents "contraction of rulers as measured by the distant observer" assumes that the distant observer can make some kind of measurement that shows this. But if we actually try to construct such a measurement, it doesn't work out that way.



To me, this takes away the point of the distant observer inferring these things in the first place. If he's inferring something that will change if he changes coordinates, why is he interested? Why isn't he interested in things that are invariant under coordinate changes? The latter are what tell you about the physics.



To me, and I think to most relativity physicists, the point of changing coordinate systems is that some invariants are much easier to compute in one coordinate system, while others are much easier to compute in a different coordinate system. Since what you're computing is an invariant, you can compute it in any coordinate system you want and the answer will be the same, so why not do it in the one that's easiest?

I'll try to state more clearly what I am trying to do here. It is indeed invariants I am looking for, constants of motion and so forth that can be put together in some way to determine the variables z, L , and L_t. Variable properties that a distant observer measures don't matter since they can be changed, so generally only locally measured invariant properties is what I am originally looking for. Those variables, z, L, and L_t are identical to the co-efficients in the metric of an arbitrary coordinate system, right, so we might as well just think of them that way. I am trying to find invariants that determine how the coordinate dependent co-efficients relate to each other, whereafter we can arbitrarily make a single coordinate choice, say L_t = 1 or z = L or z = sqrt(1 - 2 m / r), perhaps, to find something along the lines of SC, or any other, and determine what the other co-efficients must be from that. I want to do this by finding individual properties for states of motion through a vacuum spacetime that can easily be visualized and accepted, and then combined afterward to find how the co-efficients relate in order to form any metric we wish for any arbitrary GR coordinate system.

 

[PeterDonis]

the observer at r will measure that the rod has moved one full meter as measured locally in the same way that the distant observer does because each meterstick is identical locally

This is correct.

only different as inferred, depending upon the particular coordinate system applied.

But if the actual measurement says the meter stick's length is unchanged, on what basis do you "infer" that it has changed?

Let me put the question another way: coordinate quantities by themselves have no physical meaning; only invariants do. What invariant changes at finite r that allows the distant observer to conclude that the meter stick's length has changed? (Obviously such an invariant will have to affect everything at r, not just the meter stick, so the *relative* lengths of things as measured locally remain the same.)

I think your answer to this question is going to be "none"; but that means that the distant observer's "inference" that the length has changed has no physical meaning. Why, then, does he bother?

I'll try to state more clearly what I am trying to do here. It is indeed invariants I am looking for, constants of motion and so forth that can put together to determine the variables z, L , and L_t.

But why do you want to determine those variables? It's the invariants that tell you about the physics; if you already know those, why bother computing extra variables that aren't invariant, and therefore don't tell you about the physics?

Variable properties that a distant observer measures don't matter since they can be changed, so generally only locally measured invariant properties is what I am originally looking for. Those variables, z, L, and L_t

Why do you think those are "locally measured invariant properties"? You go on to say that they are identical to metric coefficients, but metric coefficients aren't invariants.

I am trying to find invariants that determine how the coordinate dependent co-efficients relate to each other, whereafter we can arbitrarily make a single coordinate choice, say L_t = 1 or z = L or z = sqrt(1 - 2 m / r), or any other, and determine what the other co-efficients must be from that.

This seems backwards to me, as above. The only reason for being interested in coordinate dependent coefficients at all is to find out about the physics, which means finding out about invariants. Why would you want to use invariants to find out about coordinate dependent coefficients?

 

[DiracPool]

I've watch all of Leonard Susskind's lectures on GR, lots of tensor math, but I think they confused me more than resolved anything, other than the simple algebraic representations. Too many notations and they don't follow algebraic rules.

I agree about Lenny. I watched a large number of those videos too, not just GR, but the others also. He is such a good entertainer that he makes me feel like I'm learning something when I'm not. It's sad because they ostensibly seem to offer so much hope...You will achieve the "theoretical minimum" just by letting Lenny entertain you. In retrospect, I didn't learn much but spent a lot of time watching him "abstractfucate." The most pointed frustration I had with his lectures is that, with the possible exception of a few in the classical mechanics series, he essentially gives no concrete examples in his arguments. Accordingly, there's no fully worked problems that may anchor to personal experience some of these abstract tangents he goes on for for hours. All it takes is one specific example here and there to really anchor the maze of concepts he weaves. But, sadly you aren't going to find many there. If you want a good show, though, tune in.

 

[grav-universe]

But if the actual measurement says the meter stick's length is unchanged, on what basis do you "infer" that it has changed?

Let me put the question another way: coordinate quantities by themselves have no physical meaning; only invariants do. What invariant changes at finite r that allows the distant observer to conclude that the meter stick's length has changed? (Obviously such an invariant will have to affect everything at r, not just the meter stick, so the *relative* lengths of things as measured locally remain the same.)

The inferred length is what the distant observer would find if he were to actually map it out on paper from r=0 onward, equally spacing r=1, r=2 r=3, etc., by applying his current coordinate system. He then marks where each end of the meterstick lies at some r, r1 and r2, say, and notices the distance between those two points are smaller than they would be for a meterstick further out on the map. That would be the inferred length contraction, equal to the dr component of the metric as compared to his local meterstick. All inferred measurements are taken from this map, such as the coordinate speed inferred by how a particle would move across the paper, for instance, during the time that his own clock ticks t=1, t=2, t=3, and so on, as compared to the time that passes locally at r for a static observer that coincides with the particle, since the time dilation is an invariant for that shell regardless of the coordinate system applied. All in all, it does not say anything physical since the coordinate system can be changed, no, but rather just a helpful visualization, just inferred according to the particular coordinate system applied and mapped out accordingly.

I think your answer to this question is going to be "none"; but that means that the distant observer's "inference" that the length has changed has no physical meaning. Why, then, does he bother?

Why do we bother working out the metric for different coordinate systems? It is the same difference. Some visualizations are better than others, even though the invariants stay the same, sometimes allowing us to see how those invariants relate more clearly through another perspective.

But why do you want to determine those variables? It's the invariants that tell you about the physics; if you already know those, why bother computing extra variables that aren't invariant, and therefore don't tell you about the physics?

Same reason. To be able to switch between arbitrary coordinate systems. It is the invariants and constants I am after, taken from reasonable foundations that can be visualized, in order to find the variables z, L , and L_t to be used in a metric of the form

ds^2 = c^2 dt^2 z^2 - dr^2 / L^2 - dθ^2 r^2 / L_t^2

given some original arbitrary coordinate choice that determines the variables.

Why do you think those are "locally measured invariant properties"? You go on to say that they are identical to metric coefficients, but metric coefficients aren't invariants.

The variables are identical to the co-efficients. We would use them with measurements inferred by a distant observer to find local measurements. It is the invariants I am after in order to find them, though, as what the distant observer infers doesn't tell us anything physical, right. z is an invariant, though, for a particular shell, so really I am just trying to find it's relationship with L and L_t in the metric, a relationship that would be true regardless of the coordinate system, but where the actual variables can be determined to find the full metric after some coordinate choice has been made.

This seems backwards to me, as above. The only reason for being interested in coordinate dependent coefficients at all is to find out about the physics, which means finding out about invariants. Why would you want to use invariants to find out about coordinate dependent coefficients?

To form a metric solution.

 

[grav-universe]

Let me ask this. This Wiki link has the derivation for the Schwarzschild solution. For the 3 lines contained in the link under "Using the field equations to find A(r) and B(r)" and shown below, what is each saying physically?

$$\rm{4 \dot{A} B^2 - 2 r \ddot{B} AB + r \dot{A} \dot{B}B + r \dot{B} ^2 A=0}$$

$$\rm{r \dot{A}B + 2 A^2 B - 2AB - r \dot{B} A=0}$$

$$\rm{- 2 r \ddot{B} AB + r \dot{A} \dot{B}B + r \dot{B} ^2 A - 4\dot{B} AB=0}$$

 

[HomogenousCow]

I can hardly imagine how one would go about developing GR without tensors...

 

[PeterDonis]

The inferred length is what the distant observer would find if he were to actually map it out on paper from r=0 onward, equally spacing r=1, r=2 r=3, etc., by applying his current coordinate system.

...

He then marks where each end of the meterstick lies at some r, r1 and r2, say, and notices the distance between those two points are smaller than they would be for a meterstick further out on the map.

If you are just using "inferred length" as a convenient visualization, as here...

All in all, it does not say anything physical since the coordinate system can be changed, no, but rather just a helpful visualization, just inferred according to the particular coordinate system applied and mapped out accordingly.

...then all this is ok as far as it goes, but as I think I've said before, you have to be very careful not to draw invalid inferences from it. Judging by what I see here on PF, it is extremely difficult to abide by that restriction. "The piece of paper doesn't physically exist" means what it says; you can't draw any conclusions about the physics from your visualization using the piece of paper. You can only construct the visualization using conclusions about the physics that you got from somewhere else.

z is an invariant, though, for a particular shell

Yes (for "shells" with r > 2m; but in this thread we are, I believe, restricting to that case anyway).

so really I am just trying to find it's relationship with L and L_t in the metric, a relationship that would be true regardless of the coordinate system

No, it wouldn't, because the metric coefficients depend on the coordinate system. L and L_t are metric coefficients; they only make sense in the particular coordinate system you are trying to use. That's part of what "the distant observer's inferred distance isn't a physical quantity" means.

I do understand better where you're trying to go with this from your comments. In general what you're doing makes sense; I'm just trying to give some cautions about its limitations.

 

[PeterDonis]

This Wiki link has the derivation for the Schwarzschild solution.

There's an important qualifier here (which the Wiki page doesn't make very clear, but that's Wikipedia for you): this is a derivation of the Schwarzschild solution in a particular coordinate chart.

For the 3 lines contained in the link under "Using the field equations to find A(r) and B(r)" and shown below, what is each saying physically?

That can't be answered because of the qualifier I gave above; each of these equations is specific to the particular coordinate chart being used.

In general, the Einstein Field Equation (which the equations on the Wiki page that you refer to are components of) tells how the curvature of spacetime is related to the presence (or absence, in this case) of stress-energy (energy, momentum, pressure, and stress are all components of stress-energy). So in general, the equations you refer to are basically constraints on how the curvature can vary given that the spacetime is vacuum. But the specific form depends on the coordinate chart you adopt.

 

[pervect]

The way distance works is this.

If you've got a curve, and you use standard definitions, everyone agrees on the length of the curve, the length of the curve is not observer dependent. Nor is the length of the curve coordinate dependent - it doesn't matter what your observer is, or what coordiates you choose, the curve has a length that's inddepenent of all that. So you'll never have a case of a distant observer saying a given curve has a different length than a nearby observer.

The length of a curve is observer independent , period.

But - we can't say the same about distance. Why is that? Well, the problem is in specifying the curve. If you have a curve, the length of it is observer independent. But the actual curve itself may not be.

That's the only tricky issue in measuring distanace. How do you specify this curve whose length you are to measure? Once you've managed to do that you've got the problem solved.

Generally, the curve is specified as the shortest curve connecting two points, where all points are on a surface of simultaneity. The surface of simultaneity CAN change with the observer, so in general distance (in SR) can depend on the observer. This is a consequence of the relativity of simultaneity.

But the ONLY reason this happens is when you have a disagreement on simultaneity. If you have a non-time varying gravitational field (like the Schwarzschild. case), and you're outside the event horizon, there is a shared notion of simultaneity for all static observers.

Thus, in the case of a static observer, IT DOESN"T MATTER where you are when you measure length. Whether you're "on site" or "at infinity" a meter, is a meter, is a meter. Thus it is sufficient to be able to measure the distance on-site with a ruler to determine, say, the height of a building, there's no need to spend a lot of time agonizing about your meter sticks changing length, and your clocks ticking at different rates, and all that. Your local clocks, and local rulers give you the answer you're looking for, the one and only answer.

And if the distant observer comes up with a different answer, he's simply wrong.

It's necessary that you an the distant observer share a common notion of simultaneity to make this statement - but as long as you restrict yourself to static observers, you hae agreement on length, just as you do in special relativity if you demand that everyone be at rest relative to one another by choosing some particular inertial frame.

 

[grav-universe]

Thank you everyone for your replies so far. :)

so really I am just trying to find it's relationship with L and L_t in the metric, a relationship that would be true regardless of the coordinate system

No, it wouldn't, because the metric coefficients depend on the coordinate system. L and L_t are metric coefficients; they only make sense in the particular coordinate system you are trying to use. That's part of what "the distant observer's inferred distance isn't a physical quantity" means.

By this I mean that I am looking for the relationships between z, L, and L_t that will be true regardless of the coordinate system, where upon making a coordinate choice for one, we will be able to find the other two. So far in the OP we found the relationship

(dz / dr) L = m L_t^2 / r^2

which should be true for any arbitrary coordinate system. For instance, in SC, we have

z = sqrt(1 - 2 m / r), L = sqrt(1 - 2 m / r), L_t = 1

dz / dr = (m / r^2) / sqrt(1 - 2 m / r)

[(m / r^2) / sqrt(1 - 2 m / r)] sqrt(1 - 2 m / r) = m (1)^2 / r^2

which works out. In GUC it is

z = 1 / sqrt(1 + 2 m / r), L = 1 / sqrt(1 + 2 m / r), L_t = 1 / (1 + 2 m / r)

dz / dr = (m / r^2) / (1 + 2 m / r)^(3/2)

[(m / r^2) / (1 + 2 m / r)^(3/2)] / sqrt(1 + 2 m / r) = m [1 / (1 + 2 m / r)]^2 / r^2

which also works out, and for EIC it becomes

z = (1 - m / (2 r)) / (1 + m / (2 r)), L = 1 / (1 + m / (2 r))^2, L_t = 1 / (1 + m / (2 r))^2

dz / dr = (m / r^2) / (1 + m / (2 r))^2

[((m / r^2) / (1 + m / (2 r))^2] / (1 + m / (2 r))^2 = m [1 / (1 + m / (2 r))^2]^2 / r^2

which works out as well. I'm still looking for another relationship that can be put together with this one so that the other two co-efficients can be found after making a coordinate choice for one.

 

[PeterDonis]

By this I mean that I am looking for the relationships between z, L, and L_t that will be true regardless of the coordinate system

How is that possible when L and L_t are coordinate system-dependent quantities? As you've defined them, they only *exist* in particular coordinate systems. They exist in SC and GUC, but that doesn't mean they exist in other charts.

 

[grav-universe]

How is that possible when L and L_t are coordinate system-dependent quantities? As you've defined them, they only *exist* in particular coordinate systems. They exist in SC and GUC, but that doesn't mean they exist in other charts.

When the metric is expressed in the same form as SC, for instance, which is what I want to do, only changing r to r1 as with GUC and EIC, then there is a particular z1, L1, and L1_t for the coordinate system of the new metric. Likewise if we know z1, we should be able to find L1 and L1_t from that through the relationships between them. The only difference is that I don't want to start with any particular solution such as SC. Rather, I want to start with the logical foundations and reasonable assumptions of individual components that lead up to any arbitrary coordinate system once certain coordinate choices have been made, but again, with the metric in the same form as SC. For starters, then, we could just find those foundations that lead up to SC, while defining the coordinate choices made for that, and then we should be able to find it in the same way for any similar arbitrary metric with some different coordinate choice.

 

[PeterDonis]

The only difference is that I don't want to start with any particular solution such as SC. Rather, I want to start with the logical foundations and reasonable assumptions of individual components that lead up to any arbitrary coordinate system once certain coordinate choices have been made, but again, with the metric in the same form as SC.

That right there is a huge restriction. You are saying you want a metric in the form

$$ds^2 = A(r) dt^2 + B(r) dr^2 + r^2 d\Omega^2$$

It's fairly straightforward (though a bit tedious in places) to show that the metric for any static, spherically symmetric spacetime can be written in this form. (MTW lays out the reasoning in some detail.) However, that doesn't mean this form is the best form for looking at all aspects of the physics. Essentially, you *are* starting with SC; any line element written in the above form is just a small variation on SC, and has all of the same restrictions as SC.

If all you want to study is static, spherically symmetric spacetimes (i.e., things like stars in stable equilibrium, *not* black holes), these restrictions aren't that much of an issue. But if you want to look at other kinds of spacetimes (e.g., black holes, or the FRW spacetimes used in cosmology), you won't get very far. We've already seen that the SC form of the line element doesn't work well at a BH horizon, because the spacetime at and inside the horizon is not static. An FRW spacetime is even worse: the universe is expanding, so the spacetime isn't static *anywhere*, and you can't even write the metric in the above form at all.

The general procedure of looking at the symmetries of a particular spacetime, and trying to match up coordinates with them, is fine as far as it goes; but there's no guarantee that it's going to give you a good handle on all the physics.

 

[grav-universe]

That right there is a huge restriction. You are saying you want a metric in the form

$$ds^2 = A(r) dt^2 + B(r) dr^2 + r^2 d\Omega^2$$

It's fairly straightforward (though a bit tedious in places) to show that the metric for any static, spherically symmetric spacetime can be written in this form. (MTW lays out the reasoning in some detail.) However, that doesn't mean this form is the best form for looking at all aspects of the physics. Essentially, you *are* starting with SC; any line element written in the above form is just a small variation on SC, and has all of the same restrictions as SC.

If all you want to study is static, spherically symmetric spacetimes (i.e., things like stars in stable equilibrium, *not* black holes), these restrictions aren't that much of an issue. But if you want to look at other kinds of spacetimes (e.g., black holes, or the FRW spacetimes used in cosmology), you won't get very far. We've already seen that the SC form of the line element doesn't work well at a BH horizon, because the spacetime at and inside the horizon is not static. An FRW spacetime is even worse: the universe is expanding, so the spacetime isn't static *anywhere*, and you can't even write the metric in the above form at all.

The general procedure of looking at the symmetries of a particular spacetime, and trying to match up coordinates with them, is fine as far as it goes; but there's no guarantee that it's going to give you a good handle on all the physics.

Right. That is all I want to find for now, a static external metric. Baby steps. :) That is the best I can visualize and work with at the moment, but perhaps later I could expand from what I can see of the metric here.

 

[PeterDonis]

Right. That is all I want to find for now, a static external metric.

Then you've already captured the two important local invariants in such a metric: the "time dilation factor" z (which could also be expressed as "gravitational potential"), and the "acceleration due to gravity" a. In the external vacuum region there are no other local invariants that I'm aware of.

There is an invariant associated with the radial part of the metric, but it's not really a local one. To see what it is, go back to the thought experiment I mentioned earlier, with a long, radially oriented rod going between a distant observer with r -> infinity and a local observer at finite r. We saw that if the distant observer moves the rod radially by 1 meter, as seen by comparing a local meter stick to marks on the rod made while it was in empty space far from all gravitating bodies, the local observer also sees the rod move radially by 1 meter, as seen by comparing a local meter stick to marks on the rod at his end.

However, we could also ask a different question: what is the proper length of the rod between the two observers? I.e., how many marks, each separated by 1 meter (as seen locally), lie between them when the rod is oriented radially? This is where the effect of the dr^2 metric coefficient shows up; it makes the answer *larger* than it would be if spacetime were flat. Another way of expressing this is that the spatial geometry of a slice of constant Schwarzschild time is not Euclidean. In your terms, the geometry of the "flat sheet of paper" on which the distance covered by each coordinate interval dr is the same, is *different* from the geometry of the spatial slice where the distance covered by each coordinate interval dr changes with r. Going by the flat sheet of paper, we would expect r2 - r1 marks between the two observers; but going by the actual geometry of the spatial slice, using the SC chart, the number of marks is given by

$$s = \int_{r_1}^{r_2} \frac{dr}{\sqrt{1 - 2m / r}}$$

This turns out to be *larger* than r2 - r1, and that fact is an invariant; it doesn't depend on the particular coordinates you adopt. You can calculate it in GUC and get the same answer.

You can also calculate it in EIC, but there you have to be careful, because the labeling of the 2-spheres changes; you will have to find *different* values R1 and R2 for the limits of integration, to keep the physical invariants at each limit the same. But the metric coefficients will *not* be among those invariants.

One other thing to be careful of: you'll note that in defining this invariant s, I specified the geometry of a slice of constant *Schwarzschild* coordinate time. Charts with different slicings (such as Painleve or Eddington-Finkelstein or Kruskal) will have a *different* "proper distance" between r1 and r2, even if the r values themselves are identical. For example, in the Painleve chart, the proper distance in a slice of constant Painleve time between r1 and r2 *is* just r2 - r1; the spatial geometry of a slice of constant Painleve time *is* Euclidean. The slices of constant Painleve time are "cut at an angle" compared to the slices of constant Schwarzschild time. So even in a static spacetime, there are still subtleties.

 

[grav-universe]

Right, the proper length ds of a local radial ruler does not change with the coordinate system, so that is an invariant, and we have

ds = dr / L = dr1 / L1

Coordinate tangent length changes directly with r1 / r, since the circumference of a spherical shell varies directly with r1 / r according to a distant observer, while local observers measure the ruler distance around their shell the same, invariant, so

L1_t / L_t = r1 / r

L1_t / r1 = L_t / r

and z is an invariant, so

z1 = z

So far I haven't been able to use those to determine anything though.

 

[grav-universe]

So far I haven't been able to use those to determine anything though.

Well, actually, I suppose we could at least use them to verify that the relation

(dz / dr) L = m L_t^2 / r^2

will work for any arbitrary coordinate system of this metric form, transforming the variables z, L , and L_t to z1, L1, and L1_t with

(dz1 / dr1) L1 = m L1_t^2 / r1^2

dz1 (L1 / dr1) = m (L1_t / r1)^2

whereby dz1 = dz, L1 / dr1 = L / dr, and L1_t / r1 = L_t / r, so that we have

dz (L / dr) = m (L_t / r)^2

 

[grav-universe]

Okay well, the relations shown in the last couple of posts are coordinate independent invariants, so if we use those to find some relationship that works out for SC, then it will work for any coordinate system of the same metric form. So far I have happened upon the relationship

[d(L_t / r) / dr] L (r / L_t)^2 = - z

or d(L_t / r) (L / dr) / (L_t / r)^2 = - z

which is composed entirely of the invariants found in the last couple of posts as can be seen more clearly in the second equation. It works for SC, so it will work for any coordinate transformation from SC. Combining this with the relationship found in the OP, we can also gain

m d(L_t / r) = - dz z

which is also composed of invariants. In fact we can now see in the OP that those individual invariants shown there for a' are composed of these invariant factors also (and constants), such as

a' = - (G M) (L_t / r)^2 / z and

a' = - c^2 dz (L / dr) / z

We now have a second relationship to combine with the first, which by working through the last two for a', the first relationship was found to be

c^2 (dz / dr) L = G M L_t^2 / r^2

or dz (L / dr) = m (L_t / r)^2

so that if we make some arbitrary coordinate choice for one variable, z, L, or L_t in terms of r, we can now directly determine the other two from those two relationships, although it may still be difficult for some with the derivatives involved. I will continue to work through these some more to find more such relationships, but what I'm really looking for is the logical foundations behind them. I'm hoping I might be able to see something in one of them that looks like a reasonable assumption to base the particular resulting equation upon. Does anybody see anything about any of the first three equations shown in this post that looks like something can be determined about what logical foundation that equation might be based upon?

 

[grav-universe]

Here's a rather simple one I have found so far.

z^2 = 1 - 2 m (L_t / r)

So if we make a coordinate choice for z or L_t, we can easily find the other. Then we can find L with one of the other relationships found before, such as

L = m (L_t / r)^2 / (dz / dr)

I still need the logical foundation for it, though.

 

[grav-universe]

Here's something interesting. We can re-arrange that last one to get

(L_t / r) = (1 - z^2) / (2 m)

and combine it with

L = m (L_t / r)^2 / (dz / dr)

to get

L = (1 - z^2)^2 / (4 m (dz / dr))

which gives us the relationship between L and z, but if we then combine that with

a' = - c^2 (dz / dr) L / z

from the OP, we find

a' = - c^2 (1 - z^2)^2 / (4 m z)

which gives us a direct relationship between the local acceleration and time dilation, independent of r or any other factors other than the mass of the gravitating body M, since m = G M / c^2. So if one only knows the numerical time dilation at some shell (regardless of the value of r for that shell) and the mass of the gravitating body, they can find the local acceleration at that shell by applying this formula which is coordinate independent. Of course that also works vice versely, one can find the time dilation according to whatever the local acceleration is, rather than work with gravitational potential for a specific coordinate dependent r or whatever, but finding the solution for z in terms of a' in that manner appears to be long and messy.

 

[PeterDonis]

So if one only knows the local time dilation and the mass of the gravitating body M, they can find the local acceleration, independent of r.

This doesn't look right. I haven't worked through the details of your derivation, but just looking at the standard formula for the proper acceleration of a static observer, I don't see how knowing the time dilation z and the mass M, but not r, let's you find the acceleration:

$$a = \frac{M}{r^2 \sqrt{1 - 2M / r}} = \frac{M}{r^2 z}$$

So if I know z and M, I still can't find a without knowing r as well.

 

[grav-universe]

This doesn't look right. I haven't worked through the details of your derivation, but just looking at the standard formula for the proper acceleration of a static observer, I don't see how knowing the time dilation z and the mass M, but not r, let's you find the acceleration:

$$a = \frac{M}{r^2 \sqrt{1 - 2M / r}} = \frac{M}{r^2 z}$$

So if I know z and M, I still can't find a without knowing r as well.

Right. With this relationship you couldn't, simply because it includes r, but work through any specific numerical examples you like for the relationship in the last post with SC, GUC, EIC, or any other of the same form and it should work out (I haven't yet but I'm confident because I can see how it relates and that it is invariant). The local acceleration appears to be directly tied to the time dilation only (and M).

(By the way, just a quick mention that that would actually need to be a' = - G M L_t^2 / (r^2 z) to be coordinate independent, but of course in SC, L_t = 1 so amounts to the same thing for that coordinate system only.)

 

[PeterDonis]

Right. With this relationship you couldn't, simply because it includes r, but work through any specific numerical examples you like for the relationship in the last post with SC, GUC, EIC, or any other of the same form and it should work out (I haven't yet but I'm confident because I can see how it relates and that it is invariant). The local acceleration appears to be directly tied to the time dilation only (and M).

This doesn't make sense to me. The formula I posted shows that if you only know z and M, not r, you can't know a. In other words, given a fixed z and M, there are multiple possible values of a. So I don't see how it's possible for a to only be tied to z and M.