Spacetime curvature observer and/or coordinate dependent?

[Passionflower]

Spacetime curvature observer and/or coordinate dependent?

In another topic several people suggested that spacetime curvature is not absolute, it apparently depends on the observer and/or coordinate system. Apparently if someone goes fast (whatever that might mean in relativity) curvature is more in the time direction while if someone goes slow (again whatever that might mean) curvature is more in the spatial direction.

[sarcasm on]Seems we got a way to determine absolute motion as we simply have to check the curvature for the two components to see how fast we are going[/sarcasm off]

Any takers?

 

[Jonathan Scott]

Spacetime curvature observer and/or coordinate dependent?

In another topic several people suggested that spacetime curvature is not absolute, it apparently depends on the observer and/or coordinate system. Apparently if someone goes fast (whatever that might mean in relativity) curvature is more in the time direction while if someone goes slow (again whatever that might mean) curvature is more in the spatial direction.

[sarcasm on]Seems we got a way to determine absolute motion as we simply have to check the curvature for the two components to see how fast we are going[/sarcasm off]

Any takers?

That sort of curvature is related to acceleration, and one person's accelerating frame of reference is another's free fall, so it does depend on the frame of reference.

However, "going fast" is never a correct description from one's own point of view, so it's not quite clear what you mean.

If you are describing different test objects moving through the same static fields, the curvature doesn't change but the effect changes with velocity, and it's the other way round from what you say, in that the curvature of space only affects moving things (as should be obvious).

In a weak approximation, using an isotropic coordinate system (with one or more static sources), relativistic coordinate momentum per energy (not quite the same as velocity because the coordinate speed of light changes) changes at a rate g (1 + v²/c²) if the effective Newtonian field is g.

If the test object is moving tangentially to the field, the same rule applies to the velocity, hence the effect that light is deflected by twice the Newtonian acceleration.

 

[Mentz114]

Spacetime curvature observer and/or coordinate dependent?

In another topic several people suggested that spacetime curvature is not absolute, it apparently depends on the observer and/or coordinate system.

The components of the Riemann tensor will change value in a local frame according to
<br /> R_{abcd} = \Lambda^\alpha_a \Lambda^\beta_b \Lambda^\gamma_c \Lambda^\delta_d R_{\alpha\beta\gamma\delta} <br />
where a,b,c,d are local frame indexes and α,β,γ,δ are coordinate indexes. The tetrad \Lambda is defined so that
<br /> \eta_{ab} = \Lambda^\alpha_a \Lambda^\beta_b g_{\alpha\beta}<br />
where η is the Minkowski metric.

Apparently if someone goes fast (whatever that might mean in relativity) curvature is more in the time direction while if someone goes slow (again whatever that might mean) curvature is more in the spatial direction.

Nobody said that.
In the case of the Schwarzschild vacuum the velocities are relative to the source.

[sarcasm on]Seems we got a way to determine absolute motion as we simply have to check the curvature for the two components to see how fast we are going[/sarcasm off]

What curvature and what components ? Velocities are relative to the source of the field.

 

[Passionflower]

The components of the Riemann tensor will change value in a local frame according to
<br /> R_{abcd} = \Lambda^\alpha_a \Lambda^\beta_b \Lambda^\gamma_c \Lambda^\delta_d R_{\alpha\beta\gamma\delta} <br />
where a,b,c,d are local frame indexes and α,β,γ,δ are coordinate indexes. The tetrad \Lambda is defined so that
<br /> \eta_{ab} = \Lambda^\alpha_a \Lambda^\beta_b g_{\alpha\beta}<br />
where η is the Minkowski metric.

Yes the components change but curvature is absolute.

Do you think we should talk about coordinate effects when we describe curvature or do you think that is wrong or misleading to say the least?

 

[Mentz114]

Another example of curvature being dependent on an observer is the Langevin observer in the Born chart, where the 3D submanifold is curved ( non-Euclidean). That curvature depends on the angular velocity.

If the Minkowski metric is written in spherical polar coords it acquires non-zero components of the Riemann tensor ( I can't remember how many - perhaps only one).

Why did you phrase your question in the way that you did if you didn't want to talk about coordinate effects ?

When I refer to curvature I mean the Riemann tensor. Do you think the only important number is the invariant Ricci scalar ?

In answer to your question I would say that observed effects of curvature are observer dependent. For instance the Hagihara observer sees very different tidal effects compared to a Gull-Painleve observer, even though they are both moving in the same Schwarzschild spacetime.

 

[Passionflower]

In answer to your question I would say that observed effects of curvature are observer dependent.

So then tell me how does an observer observe the effects of

A) spatial curvature
B) temporal curvature

?

 

[PAllen]

If the Minkowski metric is written in spherical polar coords it acquires non-zero components of the Riemann tensor ( I can't remember how many - perhaps only one).

Hmm. How can a tensor (all components, everywhere) vanish in one coordinate system but not in another? Seems to violate the tensor transform law...

 

[Dale]

Spacetime curvature observer and/or coordinate dependent?

In another topic several people suggested that spacetime curvature is not absolute, it apparently depends on the observer and/or coordinate system. Apparently if someone goes fast (whatever that might mean in relativity) curvature is more in the time direction while if someone goes slow (again whatever that might mean) curvature is more in the spatial direction.

[sarcasm on]Seems we got a way to determine absolute motion as we simply have to check the curvature for the two components to see how fast we are going[/sarcasm off]

Any takers?

There are a bunch of different ways that you might use the word curvature. Several of them are tensor of higher order than 1, so I don't know if you want to call those absolute or not. One of them is a scalar field on spacetime, so that is clearly absolute.

Then one other thing that you could talk about would be the curvature of a worldline. I think that this is the curvature referred to in the other discussion. That curvature is a scalar (the magnitude of the 4-acceleration), but it also depends on the worldline. So I don't know if you want to call that absolute or not.

 

[Mentz114]

Hmm. How can a tensor (all components, everywhere) vanish in one coordinate system but not in another? Seems to violate the tensor transform law...

I just checked and I'm referring to an embedded spherical surface, which has one non-zero Riemann component. But probably not relevant to this topic.

So then tell me how does an observer observe the effects of

A) spatial curvature
B) temporal curvature

?

Off topic. It belongs in the other thread. Why don't you answer the points I made in post#5 ?

 

[PAllen]

I just checked and I'm referring to an embedded spherical surface, which has one non-zero Riemann component. But probably not relevant to this topic.

Right, that's no problem. A 2-sphere can be embedded in Euclidean 3-space, for example.

 

[Passionflower]

Off topic. It belongs in the other thread. Why don't you answer the points I made in post#5 ?

All I want is to address statements that bodies are affected differently by space and time curvature depending on their velocity. These statements I consider wrong and very misleading.

However it seems that quite a few folks on this forum are actively supporting these statements.

So now I wonder why I am even bothering...

 

[Mentz114]

All I want is to address statements that bodies are affected differently by space and time curvature depending on their velocity. These statements I consider wrong and very misleading.

However it seems that quite a few folks on this forum are actively supporting these statements.

The geodesic equations show clearly that coordinate acceleration depends on velocity. They contain the Christoffel symbols which, along with their derivatives appear in the curvature tensors.

Your question about measuring temporal curvature brought to mind gravitational time dilation, which depends only on the time parts of the metric. The acceleration needed to remain at constant r in the Schwarzschild vacuum is the Newtonian value divided by the square root of g₀₀. But I'm not sure if that is relevant either.

Anyhow it is late in my timezone so I'm clocking off.

 

[Jonathan Scott]

All I want is to address statements that bodies are affected differently by space and time curvature depending on their velocity. These statements I consider wrong and very misleading.

However it seems that quite a few folks on this forum are actively supporting these statements.

So now I wonder why I am even bothering...

If you move in a straight line in local space which is curved (as described by the metric) relative to the coordinate system, then that obviously adds to the acceleration relative to the coordinate system. Just as in ordinary mechanics, if the linear curvature is 1/r, then the resulting acceleration is v²/r. In this case, r = c²/g, so this causes an acceleration of g times v²/c² (for motion perpendicular to the direction of the field).

 

[Jonathan Scott]

All I want is to address statements that bodies are affected differently by space and time curvature depending on their velocity. These statements I consider wrong and very misleading.

However it seems that quite a few folks on this forum are actively supporting these statements.

So now I wonder why I am even bothering...

If you're bothered by the implications for the principle of equivalence of different accelerations for different velocities, that's not actually a problem if you think about it more carefully.

The principle of equivalence holds within a local coordinate system which is small enough to approximate SR. If you consider a local box at rest in the field and view it from an isotropic coordinate system, it appears to be curved, along with local space, towards the source. A light ray crossing the box is twice as curved (at 2g). This means that relative to a local horizontal line across the box, which follows the curvature of space, the light actually appears to be accelerating downwards at only 1g, the same as any material object at rest or moving at any other speed. It's all consistent.

 

[Naty1]

All I want is to address statements that bodies are affected differently by space and time curvature depending on their velocity. These statements I consider wrong and very misleading.

I agree based on prior explanations from a number of people here in the forums:

The curvature or flatness of space (as opposed to spacetime) IS a coordinate dependent idea.

Overall, can be a confusing issue due in large part to language...

From my [physics forums] notes of posts:

[one perspective]

Any situation involving a rapidly moving massive body's gravitational effect and a 'stationary' observer can be transformed to an equivalent question about the interaction between a rapidly moving observer and a 'stationary' massive body. So all observations relating to a rapidly moving massive body can be answered as if the body is stationary...as if all measures are local. Local measures trump distant measures.

https://www.physicsforums.com/showthread.php?p=3661242&posted=1#post3661242 And a further perspective:

For a single electron [ or black hole] as an example, the rest energy density of the electron is the only thing that causes spacetime curvature. The kinetic energy is frame-dependent, just as the velocity is; in the electron's rest frame it is zero, and we can predict all physical observables, like whether the electron forms a black hole, by solving the EFE in the electron's rest frame.
This means that no matter how fast you see a particle whizzing by, it will never become a black hole.

Those simple perspectives clinch the issue for me;there was no dissent in the thread(s).

Other [supporting] perspectives:

[I don't have the original thread(s)]

DrGreg:

"... let's restrict our attention to 2D spacetime, i.e. 1 space dimension and 1 time dimension, i.e. motion along a straight line. …
In the absence of gravitation, an inertial frame corresponds to a flat sheet of graph paper with a square grid. If we switch to a different inertial frame we "rotate" to a different square grid, but it is the same flat sheet of paper. (The words "rotation" and "square" here are relative to the Minkowski geometry of spacetime, which doesn't look quite like rotation to our Euclidean eyes, but nevertheless it preserves the Minkowski equivalents of "length" (spacetime interval) and "angle" (rapidity).)

If we switch to a non-inertial frame [an accelerated observer] but still in the absence of gravitation), we are now drawing a curved grid, but still on the same flat sheet of paper. Thus, relative to a non-inertial observer, an inertial object seems to follow a curved trajectory through spacetime, but this is due to the curvature of the grid lines, not the curvature of the paper which is still flat.

When we introduce gravitation, the paper itself becomes curved. (I am talking now of the sort of curvature that cannot be "flattened" without distortion. The curvature of a cylinder or cone doesn't count as "curvature" in this sense.) Now we find that it is impossible to draw a square grid to cover the whole of the curved surface. The best we can do is draw a grid that is approximately square over a small region, but which is forced to either curve or stretch or squash at larger distances. This grid defines a local inertial frame, where it is square, but that same frame cannot be inertial across the whole of spacetime.

So, to summarize, "spacetime curvature" refers to the curvature of the graph paper, regardless of observer, whereas visible curvature in space is related to the distorted, non-square grid lines drawn on the graph paper, and depends on the choice of observer..."DrGreg:
When we talk of curvature in spacetime (either curvature of a worldline, or curvature of spacetime itself) we don't mean the kind of curves that result from using a non-inertial coordinate system, i.e, non-square graph paper in my analogy.

Four acceleration:
[both from pervect I suspect]

In Minkowski coordinates in Special Relativity, 4-acceleration is just the coordinate derivative of 4-velocity with respect to arc-length (proper time), and the 4-velocity is the unit tangent vector of worldline. As the 4-velocity has a constant length its derivative must be orthogonal to it. The 4-acceleration is the curvature vector; orthogonal to the worldline and its length is the reciprocal of the worldline's "radius of curvature".

In non-inertial coordinates in GR, the 4-acceleration is defined as a covariant derivative. This takes into account (and removes) any curvature of spacetime or "apparent curvature" due to using a "non-square grid", and leaves us with curvature that is a property of the worldline itself, not the spacetime or the choice of coordinates. Then everything else I said in the last paragraph is still true in a coordinate-independent sense.

A geodesic has zero 4-acceleration and zero curvature.
Pervect:
If you draw a largish parallelogram on a curved surface, you'll find that the opposite sides might not necessarily be equal in length, even though the sides are parallel. Now, imagine such a parallelogram, but that one of the sides of this parallelogram is the time axis on a space-time diagram. What you'll see is something that looks pretty much like gravitational time dilation. One timelike worldline will be shorter than another, even though they are connected by parallel geodesics.

from another discussion:


The Stress energy Tensor

The "amount of gravity produced" by the object is not a function of its energy alone, it's a function of its stress-energy tensor [ SET] of which energy is only one component. In a frame in which the object is moving, there will be other non-zero components of the SET as well as the energy, and their effects will offset the apparent "effect" of the increase in energy, so the final result will be the same [gravitation] as it is for a frame in which the object is at rest.

[A way to think about this is that in the frame of the object, the 'other' mass is the one that is moving...so the moving object cannot have 'greater gravity...]

 

[Naty1]

Jonathan Scott post #2:

That sort of curvature is related to acceleration, and one person's accelerating frame of reference is another's free fall, so it does depend on the frame of reference.

is entirely correct, but that is not the 'spacetime curvature' as described in my prior post by Dr Greg. These 'acceleration effects' are the curved grids lines as described by Dr Greg, not the 'spacetime' graph paper.

 

[Naty1]

Dalespam:

That curvature is a scalar (the magnitude of the 4-acceleration), but it also depends on the worldline. So I don't know if you want to call that absolute or not.

This too is curved grid lines on gravitational spacetime...so as noted, you have to define to what type of curvature you refer. But for a given worldine seems like it is a scalar invarient.

 

[Naty1]

passionflower...I think your original question deserves a lot more discussion...It has taken me a long time to develop a perspective with help from several experts here who agree with you...this discussion has so far not caused me to change my mind.

post #15:

passionflower:
All I want is to address statements that bodies are affected differently by space and time curvature depending on their velocity. These statements I consider wrong and very misleading.

me:
I agree based on prior explanations from a number of people here in the forums:

I 'wanna' reneg on my agreement, but agree with passionflower's original implied statement:

Spacetime curvature IS NOT observer and/or coordinate dependent?

that GRAVITATIONAL spacetime curvature [curved graph paper as described by docAl] is invarient, but acceleration overlays [curved grid lines drawn on the curved graph paper] are
not.

How can you experts agree on such an issue when no one entity in the Einstein Field Equations can claim preeminence in describing 'gravity'...

Also, if no matter how fast you see a particle whizzing by, it will never become a black hole, how can it change 'gravitational curvature'? Such speed does not add to gravitational curvature yet you generally PERCEIVE the spacetime as being curved ...the gravitational spacetime curvature component [curved graph paper] must be considered frame independent.

 

[Mentz114]

Is spacetime curvature observer dependent ?

If we take 'spacetime curvature' to mean gravity then the answer must be 'yes'.

1. A freely falling observer feels no gravity, but a stationary one does. This is because the elements of curvature that depend on the first derivatives of the metric can be transformed away for a point on a geodesic worldline. See for instance Fermi coordinates here http://en.wikipedia.org/wiki/Fermi_coordinates or Riemann coordinates here http://users.monash.edu.au/~leo/research/papers/files/lcb96-01.pdf. The latter is a lot better than the Fermi page.

2. Tidal effects ( also gravity ) can be different for different observers in the same spacetime. The tidal tensor in the coordinate basis is defined as
<br /> T_{\mu\nu}=R_{\mu\rho\nu\sigma}U^\rho U^\sigma<br />
(which already contains an observer dependent vector U) and being a tensor can be transformed to a local frame ( which may be geodesic or not) by the usual tetrad
<br /> \hat{T}_{ab}=\Lambda^\mu_a \Lambda^\nu_b T_{\mu\nu} <br />
and since the tidal effects are components of T, they can change without violating the tensorial properties of T. I gave an example in an earlier post of two observers who see different tidal effects in the Schwarzschild vacuum.

There could be something wrong with my logic, in which case, please politely point it out.

 

[yuiop]

Naty1 quoting someone else:

Any situation involving a rapidly moving massive body's gravitational effect and a 'stationary' observer can be transformed to an equivalent question about the interaction between a rapidly moving observer and a 'stationary' massive body. So all observations relating to a rapidly moving massive body can be answered as if the body is stationary...as if all measures are local. Local measures trump distant measures.

This should probably be qualified with the bolded words as:

Any situation involving a rapidly moving massive body's gravitational effect and a 'stationary' inertial observer can be transformed to an equivalent question about the interaction between a rapidly moving inertial observer and a 'stationary' massive body. So all observations relating to a rapidly moving massive body can be answered as if the body is stationary...as if all measures are local. Local measures trump distant measures.

and there probably should be a few more qualifying statement in there. Anyway, without any qualification it is not always true. For example a stationary observer in the field of a rotating gravitational body is not equivalent to a moving observer in the field of a non rotating gravitational body.

 

[PeterDonis]

If we take 'spacetime curvature' to mean gravity then the answer must be 'yes'.

If by "gravity" you mean "particular effects of gravity", then yes. As you point out, particular effects of gravity on particular observers will always be dependent on the observer's 4-velocity.

However, one could also mean by "spacetime curvature" or "gravity" something that is *not* observer-dependent, such as: the metric, the Riemann curvature tensor, or other tensors derived from it such as the Weyl, Ricci, or Einstein tensors. Most of the time I see the term "spacetime curvature" in the literature, it seems to refer to one of these objects, usually the Riemann curvature tensor.

So it seems to me that this is a question of terminology, not physics. Whether or not "spacetime curvature" is observer-dependent depends on what you define "spacetime curvature" to mean. What this tells me is that the term "spacetime curvature" is not a good one, and should be avoided whenever one is trying to be precise; instead, one should directly talk about something with a precise meaning, so that it is obvious whether or not it is observer-dependent.

 

[Naty1]

Mentz..post #19:...I don't have the mathematical skills to agree or disagree...hopefully someone will do one or the other...

yuiop..post#20: good clarification. [I updated my notes !]

PeterDonis:#21:

...What this tells me is that the term "spacetime curvature" is not a good one...

That is the conclusion I reached in quoting previous threads above.

But even beyond that, it seems to me [and I don't claim a complete mathematical insight into GR] that GR has infirmities similar to those of quantum mechanics. In QM people have different physical interpretations of some of the mathematics...the collapse of the wave function, the meaning and relationship different models within string theory, and ADS/CFT correspondnce, and on and on; in GR is it seems a lot of people have similar difficulty because time and distance don't have precise meanings, because energy of the gravitational field isn't amenable to a precise interpretation, and because 'g' doesn't even have a specific description with the EFE...

As an example of GR difficulties, I posted "Relativity of potential energy" over two years ago when I knew even less than now:

https://www.physicsforums.com/showthread.php?p=2549413#post2549413

..But whereas potential energy is independent of the frame of reference used, kinetic energy has different values in different frames of reference, and so does total energy.

from THE RIDDLE OF GRAVITATION,1992, Peter Bergmann (a student of Einstein)

and was, frankly, rocked by the answers...including this from BenCrowell:

In GR, the gravitational field g is not really a well-defined thing. By the equivalence principle, you can always pick any point and make g=0 there by adopting a free-falling frame of reference. There isn't even a gravitational field tensor. The closest thing to a gravitational field in GR is the Christoffel symbol, and it doesn't transform as a tensor.

so I've been concluding now for sometime that the quote {Feynman??}

"shut up and calculate"

applies to not only QM but GR.

 

[PAllen]

Following Peter Donis's suggestion to define terms in relation to mathematical objects, let's make sure at least that there is no disagreement on the following:

1) Curvature scalars ( e.g. Ricci, Kretschmann) are invariant and thus coordinate and observer independent. (The value of Kretschmann curvature invariant is that it is not identically zero for vacuum regions in GR. Thus it can give a scalar measure of vacuum curvature).

2) Curvature tensors (e.g. Riemann, Ricci, Weyl) are most properly viewed as intrinsic geometric features of the manifold, and can be studied with coordinate free (and thus, obviously, observer independent) techniques. However, their expression as components is obviously coordinate dependent. Different observers (following different world lines), performing measurements influenced by e.g. the tt component of such an object (expressed in the local frame of that observer) will make different measurements (because the tt components will be different).

3) In a geometry that has natural symmetries (e.g. spherical, axial), you can choose (but don't have to) make statements about motion in coordinate systems that manifest the symmetry. In this sense, you can give meaning to statements like: two observers in different states of motion in such a coordinate system are more or less affected (for some measurement) by different curvature or metric components.

Thus, I find myself sympathetic to both points of view in this thread. They are emphasizing different aspects of the above observations.

 

[PAllen]

Naty1 quoting someone else:


This should probably be qualified with the bolded words as:
and there probably should be a few more qualifying statement in there. Anyway, without any qualification it is not always true. For example a stationary observer in the field of a rotating gravitational body is not equivalent to a moving observer in the field of a non rotating gravitational body.

Actually, the context of the discussion Naty got that quote from was purely boosted non-rotating black hole versus stationary black hole. Thus, SC geometry in two different coordinate systems.

 

[Naty1]

PAllen:

I find myself sympathetic to both points of view in this thread. They are emphasizing different aspects of the above observations.

seems a better course based on this discussion...bummer...

I also found this description in my notes from an earlier discussion:

The Riemann tensor is observer independent quantity (being a true tensor), but the tidal forces you measure with rods and accelerometers are only exactly equal to the Riemann tensor for someone who is not accelerating, for someone who is accelerating there is a (usually very tiny) difference between the observer-independent Riemann tensor components and the actual measurements of the gravity gradient, also known as tidal forces.

I can't vouch for its precision, but in the context of an earlier discussion elsewhere it made a lot of sense. Can some of you math whizzes comment on whether, in general, an 'invarient' tensor [a true tensor] is so only in inertial conditions? If so, do we still call it 'invarient' because its so close, or not use that terminology.

I've read in these forums of a theory of gravity which gives the same predictions as GR but uses the torsion defect instead of curvature. Does such a 'torsion' viewpoint offer any insights into these differing views of 'cuvature'?

Pallen:

Curvature tensors (e.g. Riemann, Ricci, Weyl) are most properly viewed as intrinsic geometric features of the manifold, and can be studied with coordinate free (and thus, obviously, observer independent) techniques. However, their expression as components is obviously coordinate dependent...

Is this why the " stress energy [momentum] tensor" is sometimes referred to as a 'pseudo tnesor'?...I've understood just the component origins of that statement...how it arises in the Einstein Field Equations...

 

[Mentz114]

I can't vouch for its precision, but in the context of an earlier discussion elswhere it made a lot of sense. Can some of you math whizzes comment on whether, in general, an 'invariant' tensor [a true tensor] is so only in inertial conditions? If so, do we still call it 'invariant' because its so close, or not use that terminology.

The tensorial property is the invariance of scalar contractions. For instance a 4-vector is a rank-1 tensor. Suppose we have two 4-vectors U^\mu, V^\mu and we make the contraction l=U^\mu V_\mu, where V_\mu = g_{\mu\alpha}V^\alpha. Now apply a coordinate transformation \Lambda to the vectors. The contraction of the transformed vectors is (\Lambda^a_\mu U^\mu)( \Lambda_a^\mu V_\mu). Because \Lambda is not a tensor but a transformation matrix, we can rearrange the last expression to (\Lambda^a_\mu \Lambda_a^\mu) U^\mu V_\mu = U^\mu V_\mu.

What makes U,V tensors is that despite having their components changed by a transformation, their contraction remains the same. The same logic can be applied to any contraction of indexed objects. The objects in question are tensors iff their scalar contractions are invariant.

You must have come across the Ricci scalar which is the self-contraction of the Ricci tensor, R=R^\mu_\mu. Because R_{\mu\nu} is a tensor the contraction is invariant under coordinate tansformation.

(I am not a "math whizz", this is standard textbook stuff.)

 

[Naty1]

Mentz: is that a yes or no to this question:

Can some of you math whizzes comment on whether, in general, an 'invariant' tensor [a true tensor] is so only in inertial conditions?

I really can't tell...thanks.

What makes U,V tensors is that despite having their components changed by a transformation, their contraction remains the same. The same logic can be applied to any contraction of indexed objects. The objects in question are tensors iff their scalar contractions are invariant.

I've seen that enough to know it is 'standard'...but so far the real meaning of 'contraction'
has eluded me. Any good online source that comes to mind??

Are the scalar contractions of the Einstein Stress Energy Tensor invarient...?? I seem to recall several sources indicating it is a 'pseudo tensor'...

Pervect provided this quote elsewhere from MTW:

[
… nowhere has a precise definition of the term “gravitational field” been
given --- nor will one be given. Many different mathematical entities are
associated with gravitation; the metric, the Riemann curvature tensor, the curvature scalar … Each of these plays an important role in gravitation theory, and none is so much more central than the others that it deserves the name “gravitational field.”
/QUOTE]

I rely on that so I don't get too discouraged!

 

[Dale]

Can some of you math whizzes comment on whether, in general, an 'invarient' tensor [a true tensor] is so only in inertial conditions?

I am not sure this is a direct answer to your question, but as much as possible I try to use the word invariant only to refer to scalars (tensors of rank 0). It is OK to use the word to refer to tensors of higher rank as they can be considered to be geometric objects which do not change under a coordinate basis change, but it can cause confusion since the components of a higher rank tensor do change under a coordinate basis change. It is probably more correct to call them covariant, but then that can be confusing in the case of one or more contravariant indices. I don't think that the usage is standardized yet.

 

[Mentz114]

Can some of you math whizzes comment on whether, in general, an 'invariant' tensor [a true tensor] is so only in inertial conditions?

A tensor is always a tensor, whatever frame it is transformed to. Tensors are not invariant, but 'covariant'. An invariant would be something that has exactly the same value whatever frame it is calculated in, like R^μ_μ.

...but so far the real meaning of 'contraction' has eluded me. Any good online source that comes to mind??

What exactly do you mean by 'real meaning' ? The physical interpretation or the mechanics ?

Are the scalar contractions of the Einstein Stress Energy Tensor invarient...?? I seem to recall several sources indicating it is a 'pseudo tensor'...

It is a tensor. So the scalars formed by contracting with itself and contracting with another rank-2 tensor will be invariant.

This is OK
http://en.wikipedia.org/wiki/Tensor_contraction and a search for 'tensor contraction' throws up a lot of stuff.

 

[pervect]

If by "gravity" you mean "particular effects of gravity", then yes. As you point out, particular effects of gravity on particular observers will always be dependent on the observer's 4-velocity.

However, one could also mean by "spacetime curvature" or "gravity" something that is *not* observer-dependent, such as: the metric, the Riemann curvature tensor, or other tensors derived from it such as the Weyl, Ricci, or Einstein tensors. Most of the time I see the term "spacetime curvature" in the literature, it seems to refer to one of these objects, usually the Riemann curvature tensor.

So it seems to me that this is a question of terminology, not physics. Whether or not "spacetime curvature" is observer-dependent depends on what you define "spacetime curvature" to mean. What this tells me is that the term "spacetime curvature" is not a good one, and should be avoided whenever one is trying to be precise; instead, one should directly talk about something with a precise meaning, so that it is obvious whether or not it is observer-dependent.

It also depends on what means by "observer dependent". Certainly the numerical values of the components of a tensor can vary when you change frames - in fact, they must change, to satisfy the tensor transformation laws.

I agree that in talking with a reader with some expertise, it's better to be more specific. "Space-time curvature" could mean any of a number of things.

On the other hand, talking to a reader without any expertise, it's not only not especially helpful to be technically precise, it's probably actually conuter-productive. Unfamiliar words tend to scare the new readers - sometimes, it seems to scare them away from grasping simple basic concepts (like geodesic deviation) that they probably could grasp with an approach that was less formal and more "chatty".

As far as the original question about whether or not spatial curvature doubles the deflection of light, there are a number of different approaches that all seem to converge on the answer "yes, it does".

The best documented approach doesn't use tensors at all, but the PPN parameter beta, which can be and is described as the amount of spatial curvature induced by a unit rest mass, for instance in MTW's "Gravitation". (Note that here we see yet another possible meaning for the term spatial curvature via this usage.)

One can then state that deflection of light would be half of what GR predicts if the PPN parameter beta (the spatial curvature) was zero, instead of unity, the value that GR predicts.

THere are other approaches as well, for instance on can decompose the Riemann tensor using the Bel decomposition, and arrive at similar results regarding light deflection from the geodesic deviation equation - in this case that half of the geodesic deviation is due to the electrogravitc part of the tensor, and the other half is due to the topogravitic parts.

The Bel decomposition requires a timelike vector field to define the decomposition - i.e. to define the split of space-time into space and time. The obvious choice for this vector field is the timelike killing vectors associated with a static metric.

 

[Mentz114]

Are the scalar contractions of the Einstein Stress Energy Tensor invariant...??

The trace is an invariant.

R^ab - (R/2) g^ab = κT^ab

-> R^a_b - (R/2) g^a_b = κT^a_b

-> R^a_a - (R/2) g^a_a = κT^a_a

-> R - 2 R = κT^a_a

-> T^a_a = -R/κ

using R = R^a_a and g^a_a = 4.

 

[PeterDonis]

Can some of you math whizzes comment on whether, in general, an 'invarient' tensor [a true tensor] is so only in inertial conditions?

I don't claim to be a math whiz, but I think the answer to this question is "no". Tensors are tensors in curved spacetime as well as in flat spacetime, in the presence of gravity as well as in its absence. They work the same way (with the one key proviso that in curved spacetime, you have to make sure you use covariant derivatives instead of just ordinary derivatives when writing tensor equations) and have the same invariance properties in both cases.

Is this why the " stress energy [momentum] tensor" is sometimes referred to as a 'pseudo tnesor'?

I think you are referring to this:

http://en.wikipedia.org/wiki/Stress–energy–momentum_pseudotensor

which is indeed a pseudo-tensor, but is *not* what appears on the RHS of the Einstein Field Equation; that is the "stress-energy tensor" (yes, the terminology can be confusing), which is a true tensor.

What makes the other thing a "pseudo-tensor" is that it tries to incorporate "energy stored in the gravitational field" as well as the ordinary stress-energy that appears on the RHS of the EFE; but there is no way to define "energy stored in the gravitational field" as a true tensor.

 

[PeterDonis]

I agree that in talking with a reader with some expertise, it's better to be more specific. "Space-time curvature" could mean any of a number of things.

On the other hand, talking to a reader without any expertise, it's not only not especially helpful to be technically precise, it's probably actually conuter-productive. Unfamiliar words tend to scare the new readers - sometimes, it seems to scare them away from grasping simple basic concepts (like geodesic deviation) that they probably could grasp with an approach that was less formal and more "chatty".

Yes, it's a judgment call which approach to use. What tends to flip me over into "need to use more specific, precise terminology" mode is when I find different people (one of whom may well be me) using the same word to refer to different things, and then arguing past each other.

Also, sometimes people will ask questions that really shouldn't be answered in the terms they were posed in, because those terms make implicit assumptions that can hinder understanding. But again, I agree it's a judgment call.

 

[Naty1]

hey, thanks a lot guys...especially:

PeterDonis...

Is this why the " stress energy [momentum] tensor" is sometimes referred to as a 'pseudo tnesor'? I think you are referring to this:

http://en.wikipedia.org/wiki/Stress%...m_pseudotensor

yes...your answer a big help, I could have read that for another ten years and never deciphered it on my own. I think that must have been where I did mislead myself.

Mentz:

...but so far the real meaning of 'contraction' has eluded me. Any good online source that comes to mind?? What exactly do you mean by 'real meaning' ? The physical interpretation or the mechanics ?

I'd be happy to start with either...physical would be an easier start...but the wikipedia explanation is 'above my paygrade"...I'll check out 'mathpages' online...

 

[PAllen]

Mentz:


I'd be happy to start with either...physical would be an easier start...but the wikipedia explanation is 'above my paygrade"...I'll check out 'mathpages' online...

The mechanics is straightforward. A contraction is formed from one or more tensors having (in total) a matched number of contravariant and covariant indexes; you pair these up (multiple ways to do it, each producing, in general, a different contraction), and sum over component products for all combinations matching on the chosen index pairs. Because the transform under diffeomorphism for contravariant indexes is the inverse of of that for covariant indexes, it is trivially guaranteed than any contraction is diffeomorphism invariant. I don't think there is a 'general' physical meaning to contractions. Because of the generality of the above principle, you can readily form contractions that don't have any reasonable physical meaning. The physical meaning of different contractions gets at the interpretation of GR as a physical theory rather than some arbitrary collection formulas from differential geometry.

[Edit: in the above, I am focusing contraction to a scalar not to a lower rank tensor. ]

 

[Mentz114]

Naty1, I haven't got much to add to PAllen's #35 except an example.

Raising an index is a contraction. This g^ab U_c is a rank-3 mixed tensor ( 3 distinct idexes), and if we contract it we get g^ab U_a or g^ab U_b. These are the same thing because the metric is symmetric and the result is actually U with a contravariant (raised) index

U^a = g^ab U_b = g^a0 U₀+ g^a1 U₁ +g^a2 U₂+g^a3 U₃

which is a linear combination of the covariant components of U.

 

[Naty1]

My attempt at a Summary:
[I’ve don’t think I’ve ever seen one so I thought I’d give it a try..]


[#19] Is spacetime curvature observer dependent ?

None of the answers to the OP are precise…none can be unless an agreement is reached on exactly waht component of 'curvature' measurement will be used...and that would provide only a partial answer;the reason is given at the end of this post with some of the great insights insights along the way summarized here first.…

Originally Posted by Mentz114

If we take 'spacetime curvature' to mean gravity then the answer must be 'yes'.

PeterDonis adds:
If by "gravity" you mean "particular effects of gravity", then yes. As you point out, particular effects of gravity on particular observers will always be dependent on the observer's 4-velocity…. this is a question of terminology, not physics. Whether or not "spacetime curvature" is observer-dependent depends on what you define "spacetime curvature" to mean.

passionflower asked several questions that I really liked:

[1]

So then tell me how does an observer observe the effects of
A) spatial curvature
B) temporal curvature

and [2]

All I want is to address statements that bodies are affected differently by space and time curvature depending on their velocity. These statements I consider wrong and very misleading.

I’ll try some explanations; these, too will be imprecise, but hopefully again illustrate
The problem; feel free of course pick them apart if you don’t like them:

[1] Locally, everything stays the same...spacetime stays flat….but as a differential instead of infinitesimal observation is made, curvature becomes apparent. Time dilation enables the observer to make a comparison with a ‘distant’ observer to see their clocks run at different speeds; such an observer may also see a distant inertial observer apparently moving in an arc...with some acceleration, that is some curvature is apparent resulting from the acceleration of either. docAl might say: "yes, that's the effect of different grid lines but drawn on the original curved graph paper.…the original gravitational field. [This is another imprecise explanation.]

[2] Jonathan Scott provided insights to this one in posts #13,14; I want to try a complementary approach: I think he said different curvatures, accelerations and velocities are all related in curved spacetime.

I am reminded that ‘light follows null geodesics’ is an APPROXIMATION; a good one, but what really follow null geodesics are test particles...ideals with no mass, no energy, no disturbance of the initial gravitational field. So we see that photons of different energy follow slightly different paths. [I got this idea elsewhere from pervect.] So these ‘bodies’ are affected by energy….and you [OP] were worried about just speed affecting curvature’!

Next, suppose we have a particle with some mass at some velocity relative to an otherwise stationary gravitational field. Send a second particle at the same initial trajectory but, say, double the velocity of the first; or impart it with some spin [angular momentum] but at the original velocity: All the particles will follow different trajectories. In docAl's language, the original gravitational field maintains it's 'curved graph paper' shape, unless it’s evolving, but there is an overlay of different grid lines on that curved graph paper representing ‘spacetime curvature…...due to different energies; hence different paths result. The ‘gravitational curvature’ may remain invariant [depending on how we define that] but the effect of the now differently curved grid lines results in a different overall ‘curvature’.

We seem to have agreed that: An invariant would be something that has exactly the same value in whatever frame it is calculated; And Dalespam and Mentz seem to agree that: Invariant refers to scalars (tensors of rank 0), but Tensors are 'covariant' not invariant.” And Dalespam notes:

It also depends on what means by "observer dependent". Certainly the numerical values of the components of a tensor can vary when you change frames….[

but the Observers WILL agree on the total..] No controversy nor obstacles here.


Is that progress?? It sure helps me, but I don’t know about the OP…where is he??

Here comes the bad news:

So even if you all were to agree with all those descriptions, which would be a miracle, and the ones Pallen described in post #23, we are still faced with the problem of ascribing those descriptions of ‘grid lines’ and ’curved graph paper’ [or some other equivalents] to some mathematical component of the EFE.

That seems impossible because if my understanding is correct MTW lays out an absolute impass: [from my post #27, quote courtesy of pervect from elsewhere]

MTW

… nowhere has a precise definition of the term “gravitational field” been
given --- nor will one be given. Many different mathematical entities are
associated with gravitation; the metric, the Riemann curvature tensor, the
curvature scalar … Each of these plays an important role in gravitation
theory, and none is so much more central than the others that it deserves the name “gravitational field.”

I take gravitational field to be equivalent to ‘gravity’…or ‘spacetime curvature’ and so am stuck right here. Seems we have come as far as we can….


I do have a final question: Can someone give a sentence or three outline of the difference between ‘invariant’ and ‘covariant’: If I used the explanation above [invariant means an identical value while covariant means components may change among observers but the totals remains the same] would that be a start? Or is the essential difference something else?

my personal thanks to passionflower, the OP, for starting this...

 

[Mentz114]

I do have a final question: Can someone give a sentence or three outline of the difference between ‘invariant’ and ‘covariant’: If I used the explanation above [invariant means an identical value while covariant means components may change among observers but the totals remains the same] would that be a start?

I think you've got it. Instead of total use 'contraction'.

We can describe a mathematical expression as covariant, which means it's written in tensors. Doing physics covariantly means that we are not plagued with artifacts induced by coordinate basis changes.

An easy way to remember this is to think of the 4-velocity. The components are physically meaningful but frame dependent, while the contraction with itself is an invariant.