Experimental measurement of Einstein's Field Equations

[JustinLevy]

Experimental "measurement" of Einstein's Field Equations

The question is essentially:
What if we took a strictly experimentalist view, and considered a phenomenological model for gravity that is a "generalization" of EFE:
[tex]R_{\mu \nu} - C_1\ g_{\mu \nu}\,R + C_2\ g_{\mu \nu} \Lambda = C_3\ T_{\mu \nu}[/tex]
for some constants C1,C2,C3.

Can we experimentally measure C1, C2, C3?
If so, how?

--------
Some background:
This started from a discussion with another student today regarding solving Einstein's field equations. It is our understanding that we can obtain a solution to EFE in vacuum by:
1) choosing a coordinate map, and using the metric components in this map
2) imposing some symmetry for a problem we're interested in, by constraining the form of the metric components
3) choose some boundary condition for the components at infinity
4) solve EFE in vacuum
5) use relation to Newtonian limit to determine constants from solving the differential equations

First, is this an adequate method?
Now onto the meat of the question.

Now suppose we were to solve instead, a "generalized" form of gravity with:
[tex]R_{\mu \nu} - C_1 g_{\mu \nu}\,R + C_2 g_{\mu \nu} \Lambda = C_3 T_{\mu \nu}[/tex]
for some constants c1,c2,c3.

If the constants are all non-zero, but [itex]\Lambda=0[/itex], then we get the SAME vacuum equations as EFE. In particular, we still get Schwarzschild and Kerr solutions.

So we'd still get the same orbital motion. We'd still get the same gravitational red-shift. In fact, we'd match all of the solar system tests of GR. Correct?

It appears that the only difference would be the field equations IN matter or when the cosmological constant is non-zero. So let's look at the cosmological constant next.

If we are interested in "Lambda-Vacuum" solutions,
[tex]R_{\mu \nu} - C_1 g_{\mu \nu}\,R + C_2 g_{\mu \nu} \Lambda = 0[/tex]
taking the trace and using [itex]g_{\mu \nu} g^{\mu \nu}=4[/itex] gives
[tex]R - 4 C_1 R + 4 C_2 \Lambda = 0[/tex]
[tex]R + 4 \frac{C_2}{1 -4 C_1} \Lambda = 0[/tex]

Where as, EFE give:
[tex]R - 4 \Lambda = 0[/tex]
So as long as
[tex]4 \frac{C_2}{1 -4 C_1} = -4[/tex]
[tex]C_2 = 4 C_1-1[/tex]
We can even satisfy the Lambda vacuum solutions.

Therefore, to have any hope of measuring the constants, we need non-zero stress-energy tensor to distinguish these theories experimentally. One such example would be the highly successful Friedman equations / FLRW metric and the expanding universe.

Are measurements of the expanding universe enough to allow one to "experimentally measure" the constants in Einstein's Field Equations?

I was hoping it was. But the other student pointed out that in reality the universe is not homogeneous, it is only on the larger scales that it appears so and thus justifies the assumption. So in some sense this too could be, in principle, written as a vacuum solution (with many spherical holes / boundaries) which contain mass ... and look like an effective EFE on larger scales? So by that argument, it still cannot restrict the constants?Comments?

 

[Mentz114]

Hi Justin,
where does that 'generalization' come from ? Does it extremise the Einstein-Hilbert action ?
I'm not clear about exprimental determination of the constants. Presumably you can come up with a model of some physical scenario, then make predictions and compare with data. Or would you be adjusting the constants to fit the data ?

This started from a discussion with another student today regarding solving Einstein's field equations. It is our understanding that we can obtain a solution to EFE in vacuum by:
1) choosing a coordinate map, and using the metric components in this map
2) imposing some symmetry for a problem we're interested in, by constraining the form of the metric components
3) choose some boundary condition for the components at infinity
4) solve EFE in vacuum
5) use relation to Newtonian limit to determine constants from solving the differential equations

First, is this an adequate method?

This how Schwarzschild got the spherically symmetric solution. But it's not that easy usually. One can define a symmetry ( say axial with a reflection symmetry in a plane ) and then choose a coordinate chart that fits as well as possible. But some symmetries ( eg reflection ) have to be introduced algebraically ( ie in the metric coefficients ) and then one hopes that the Einstein tensor will resemble some real collection of matter and energy.

In the case of FLRW, the isotropy and homogeneity conditions make it a one-dimensional problem, but it's a miracle ( well, nearly) that the the Einstein tensor has the perfect fluid form.

Have you seen this excellent short article ?

http://en.wikipedia.org/w/index.php?title=Exact_solutions_in_general_relativity&oldid=45119659

What we'd all like is some method for writing down an EMT and then generating the metric, but that is nigh impossible. From what I've gathered getting exact solutions is a mixture of craftiness, maths, experience and a slice of luck.

Have you seen the Lennells space-time, which models a disc of matter rotating around a black-hole ? The maths involved in getting that is truly frightening.

(arXiv:1003.1453v1 [nlin.SI] 7 Mar 2010)

 

[JustinLevy]

where does that 'generalization' come from ? Does it extremise the Einstein-Hilbert action ?

That 'generalization' is just made up for the purpose of this discussion. Sort of like imagine, in classical electromagnetics, Coulombs law was generalized to:
[tex]F = \frac{q_1 q_2}{4 \pi \epsilon_0 r^{C1}}[/tex]
it is easy to imagine setting up an experiment to determine C1 to some precision. But the problem here is along these lines...

I'm not clear about exprimental determination of the constants. Presumably you can come up with a model of some physical scenario, then make predictions and compare with data.

The problem is that it appears all the solar system tests of GR will be passed regardless of what values of the "generalization" constants are! This seems really strange to me.

How could we, in principle, measure the constants to show they match with Einstein's Field Equations?

 

[Altabeh]

You're actually chipping away at the EFE by forcing it to look like

1) It's going to violate the energy-momentum conservation law,

[tex]T^{\mu\nu}_{;\nu}=G^{\mu\nu}_{;\nu}=0,[/tex]

This is clear from the form of your proposed equation: To have this resolved you must put [tex]C_1=\frac{1}{2}[/tex] and this is a necessity for the above law to hold for general field equations.

2- The constant [tex]C_2[/tex] is meaningless there in case the Lambda has entered the EFE. The Lambda factor if not constant must have this property to vanish under the covarient differentiation i.e.

[tex]\Lambda_{,\nu}=\Gamma^{\alpha}_{\alpha\nu}\Lambda.[/tex]

This is a crucial requirement for the conservation law to hold. So there is only one constant left and, of course, it has a reason behind it to simply be looked at differently. This constant does sound capable of making the EFE have solutions that only differ in numerical and statistical estimates with the observational results! This is because putting such constant into equation doesn't at all put a stringent requirement on the field equations and is more of a "coupling constant" in terms of today's physics! It can be fune tunned to make the static solutions of EFE compatible with the information gathered from all over the universe! But nevertheless this is really unimportant sometimes because you can fit the data on the material distribution in the universe within the matter tensor. It seems more to be a scaling factor for energy of the universe than a reasonable constant that can "really" do something.

The only interesting point about [tex]C_3[/tex] is that it could cause an extraordinary increase in the curvature of spacetime and on tiny scales of an atom this should be important from the standpoint of a larger effect of gravity as a force in comparison to the other three forces. But the main problem of a modified EFE of this kind is that it cannot reduce to the Newton's theory for weak static gravitational fields. So let alone such a makeup of the right-hand side of the EFE.

AB

 

[JustinLevy]

Okay, I agree with your comment about C2.
Assuming the Cosmological constant could only be measured gravitationally, there is no independent means to obtain it experimentally, and thus C2 can just be absorbed into the cosmological constant itself.

As for C1, I understand what you are trying to argue, but as far as contact with experiment, I do not see how that mathematical requirement can be measured experimentally. In GR, mathematically it was shown by Noether, that we can't in general define an energy or momentum in a volume of space which is conserved. So what does it even mean physically to demand that it is locally conserved? Are you suggesting we can experimentally measure this "violation" of energy/momentum conservation?

If there is an experiment which could measure this, that would solve C1, but I don't see how to do it.
(EDIT: Oops, previous sentence originally said "solve C2". I meant C1 here in continuation of the previous paragraph discussion.)

So there is only one constant left and, of course, it has a reason behind it to simply be looked at differently. This constant does sound capable of making the EFE have solutions that only differ in numerical and statistical estimates with the observational results! This is because putting such constant into equation doesn't at all put a stringent requirement on the field equations and is more of a "coupling constant" in terms of today's physics! It can be fune tunned to make the static solutions of EFE compatible with the information gathered from all over the universe! But nevertheless this is really unimportant sometimes because you can fit the data on the material distribution in the universe within the matter tensor. It seems more to be a scaling factor for energy of the universe than a reasonable constant that can "really" do something.

I'm not understanding this. Since the solar system experimental tests are based on the vacuum solutions, aren't they completely independent of C3?

To not be independent of C3, somehow step 5 of "solving the vacuum equations" above needs to be wrong. Somehow those constants would have to depend on C3. But since the theory gives us differential equations for the metric, then necessarily there will be unknown integration constants that must be fit. Comparing to the Newtonian limit is a good way to get these, but it appears impossible in principle to obtain them from the differential equations themselves. You'd need something more than the differential equations to restrict the solutions.

 

[Mentz114]

I agree with Altabeh, what he says is pretty self evident. As for the no divergence condition, I suppose you could let that go if you had matter sinks, where mass and energy were destroyed, or matter sources, where it's created. But physical laws are based on the conservation of energy so that wouldn't be a popular move.

GR is very hard to meddle with. The EFE as they stand guarantee a certain level of connection with reality. Tinkering with this is like messing with the Euler-Lagrange equations.

Are you familar with PPN formalism ? Maybe you got the idea from that.

 

[JustinLevy]

I agree with Altabeh, what he says is pretty self evident. As for the no divergence condition, I suppose you could let that go if you had matter sinks, where mass and energy were destroyed, or matter sources, where it's created. But physical laws are based on the conservation of energy so that wouldn't be a popular move.

I agree it is an attractive line of argument, but like I said: Since GR already doesn't have global energy or momentum conservation ... how can we experimentally test to show that C1 has to have a particular form?

As I said above, if there is an experiment which could measure this, that would solve C1, but I don't see how to do it.

Please give an example if you see one.

Or alternatively: How do you know that the divergence free-ness of Tuv isn't an additional restriction that limits the physical solutions to the Einstein field equations (sort of like how many people consider solutions violating the null energy condition in EFE to be non-physical solutions)?

---

Regarding divergence.
Since
[tex]G_{ab} = R_{ab} - {1 \over 2}g_{ab}\,R + g_{ab} \Lambda[/tex]
and
[tex]G_{ab;c} = g_{ab;c} = 0[/tex]
then doesn't that mean:
[tex](G_{ab}\,g^{ab})_{;c} = G_{ab}\,{g^{ab}}_{;c} + G_{ab;c}\,g^{ab} = 0[/tex]
[tex](G_{ab}\,g^{ab})_{;c} =(R - {1 \over 2}4\,R)_{;c}=0[/tex]
therefore
[tex]R_{;c} = 0, (g^{ab}R)_{;c} = 0[/tex]?
Which would mean C1 can be any value and still satisfy the divergence condition?

EDIT:
Ooops. I was saying C2, when I meant C1. Fixed now.

 

[Altabeh]

As for C1, I understand what you are trying to argue, but as far as contact with experiment, I do not see how that mathematical requirement can be measured experimentally.

This is a serious mistake you're making without hesitation! Experiments are all based on a regular notion that we all know a great deal about: "Conservation law" which you seem to ignore it under the context of "experimental evidence". I understand why such a musing could find a way to your mind and stay there. You're saying this just because the current GR is somehow incapable of creating a good framework for discussing most of things we face in the real world at large scales of the solar system or the universe but in response to the question "how do we get out of this mess" you propose that something wrong (basically the equation you're giving is nothing true with [tex]C_1\neq 1/2[/tex] and it won't be able to cover experimental observations ever) knocks spots off the original setup of GR! Actually if we hit a snag or obstacle in a street, we can't destroy the buildings around to find another path to the other side of that obstacle.

In GR, mathematically it was shown by Noether, that we can't in general define an energy or momentum in a volume of space which is conserved.

I suggest you to take serious reviews over this misleading argument that Noether says you can't define a conserved quantity in some region of space. Actually the conservation laws (I'm not sure about the violation of momentum conservation) only are very slightly violated in GR and it is highly unlikely to have them overally violated and no experiment or theory supports this!

So what does it even mean physically to demand that it is locally conserved? Are you suggesting we can experimentally measure this "violation" of energy/momentum conservation?

Look there is nothing absolute about things like conservation of energy and momentum and all we know is that they are conserved in any region of spacetime under time and space translations respectively. Even if we take for granted that the energy conservation is violated, experiments have not shown this to occur in large amounts thus not strong to confirm your theory with [tex]C_1\neq 1/2[/tex].

If there is an experiment which could measure this, that would solve C2, but I don't see how to do it.

.

I don't know if there is one. Maybe other people know!

I'm not understanding this. Since the solar system experimental tests are based on the vacuum solutions, aren't they completely independent of C3?

This is remarkably illogical! You are saying that you want to form a general EFE that leads us for almost any circumstances given, to the right solutions compatible with experiments. Then you're just talking about vacuum solutions and say this does not have anything to do with C_3. So what the heck is C_3 all about? Furthermore, who says they are all based on vacc solutions? If this is the case, then we don't have any material distribution in the universe so no curvature to measure and nothing special about the univserse! It's just like a bubble that expands as time elapses!

To not be independent of C3, somehow step 5 of "solving the vacuum equations" above needs to be wrong. Somehow those constants would have to depend on C3. But since the theory gives us differential equations for the metric, then necessarily there will be unknown integration constants that must be fit. Comparing to the Newtonian limit is a good way to get these, but it appears impossible in principle to obtain them from the differential equations themselves. You'd need something more than the differential equations to restrict the solutions.

The theory with [tex]C_3[/tex] is only consistent when [tex]C_1= 1/2[/tex]. If so, then we can say, ignoring 5th step as being impossible to be taken within the new theory, [tex]C_3[/tex] acts as a coupling constant. Otherwise you can't make use of Newton's theory where first there is a Lambda term and secondly C_3 is not equal to its real value in GR.

AB

 

[Altabeh]

Regarding divergence.

[tex]G_{ab;c} = g_{ab;c} = 0[/tex]

I suggest you to first study the mathematics of GR in such a way that I can really rely on what you say. It is divergence and must be

[tex]G_{new}^{ab}_{;b} = (R^{ab} - {1 \over 2}g^{ab}R + g^{ab} \Lambda)_{;b}=0,[/tex]

thus

[tex]G_{new}^{ab}_{;b}=G^{ab}_{;b}+ g^{ab}_{;b}\Lambda=0.[/tex]

AB

 

[JustinLevy]

I suggest you to take serious reviews over this misleading argument that Noether says you can't define a conserved quantity in some region of space. Actually the conservation laws (I'm not sure about the violation of momentum conservation) only are very slightly violated in GR and it is highly unlikely to have them overally violated and no experiment or theory supports this!

If we can show that C1 has to be within certain bounds to match experiment, then this answers the question.

If we considered the divergence free-ness of Tuv as an additional restriction that limits the physical solutions to the "generalized gravity" (sort of like how many people consider solutions violating the null energy condition in EFE to be non-physical solutions)... could we still get an experimental value for C1 somehow?

This is remarkably illogical!
...
Furthermore, who says they are all based on vacc solutions?

Maybe there are solar system tests of GR that I do not know about. But the gravitational red shift is based on the Schwarzschild solution. The perihelion advance of mercury can be considered a "test mass" moving in the background defined by the Schwarzschild solution (ie. considering the sun a static spherical mass).

If there are solar system tests which are not explained by vacuum solutions, please do fill me in. Because, to me, the comment doesn't seem as illogical as you have clearly taken it as.

 

[Altabeh]

If we can show that C1 has to be within certain bounds to match experiment, then this answers the question.

If we considered the divergence free-ness of Tuv as an additional restriction that limits the physical solutions to the "generalized gravity" (sort of like how many people consider solutions violating the null energy condition in EFE to be non-physical solutions)... could we still get an experimental value for C1 somehow?

Yes but this is highly unlikely to happen in reality because you're by construction considering another number instead of 1/2 for [tex]C_1[/tex] which destroys the conservation law of GR!

Maybe there are solar system tests of GR that I do not know about. But the gravitational red shift is based on the Schwarzschild solution. The perihelion advance of mercury can be considered a "test mass" moving in the background defined by the Schwarzschild solution (ie. considering the sun a static spherical mass).

If there are solar system tests which are not explained by vacuum solutions, please do fill me in. Because, to me, the comment doesn't seem as illogical as you have clearly taken it as.

Look, You can find many things like Kerr black holes or FRW spacetimes that each has its own source of matter and gravitational field. They are not all based upon vacuum solutions and only googling this can provide you with many examples that I can't give right now! But see for example Wiki's entry on observational evidences of black holes http://en.wikipedia.org/wiki/Black_hole#Observational_evidence".

AB

 

[JustinLevy]

Yes but this is highly unlikely to happen in reality because you're by construction considering another number instead of 1/2 for [tex]C_1[/tex] which destroys the conservation law of GR!

If you are willing to argue away the global energy and momentum "non-conservation" in GR as negligible since it is small, then I don't understand why, if I am asking how we can measure C1, that it isn't a similar issue that the divergence of R is small. I'm not proposing an alternative to GR, or pushing fringe theories, so please calm down. I am sorry if I am frustrating you.

I am asking: Can we experimentally constrain these constants? Can we, in this sense, "measure" Einstein's Field Equations? (and hence the title of this thread)

I understand that you are saying yes. But is there an experiment showing conservation of energy or momentum strong enough to be sensitive to potential violations due to ricci curvature? This would constrain C1. I would like to know about any such experiments. Since the Ricci curvature is due solely to the local energy content, and due to how weak gravity is, I'm skeptical there are experiments that sensitive. Hopefully my skepticism is wrong through, as measuring Einstein's field equations would be an interesting experimental feat!

Look, You can find many things like Kerr black holes or FRW spacetimes that each has its own source of matter and gravitational field. They are not all based upon vacuum solutions and only googling this can provide you with many examples that I can't give right now! But see for example Wiki's entry on observational evidences of black holes http://en.wikipedia.org/wiki/Black_hole#Observational_evidence".

AB

I am not saying there are no non-vacuum solutions to GR. I am saying the solar system tests are all vacuum tests of GR. And furthermore, the only non-vacuum "testing" of GR that I know of, is the prediction of the universe expanding. But that is on large scales where we don't know the matter content (actually, we need to include all kinds of unseen content to match GR), so I don't think that can be used to constrain C3 either. Maybe I am wrong. If someone knows how to use this to constrain C3, I'd much appreciate if you could point the way.

Please note that Kerr black holes are vacuum solutions as well. So I do not see how observational evidence of black holes can possibly constrain C3. Maybe I am missing your point here.

 

[Altabeh]

If you are willing to argue away the global energy and momentum "non-conservation" in GR as negligible since it is small, then I don't understand why, if I am asking how we can measure C1, that it isn't a similar issue that the divergence of R is small. I'm not proposing an alternative to GR, or pushing fringe theories, so please calm down. I am sorry if I am frustrating you.

Why do you focus on the "non-conservation" of energy in GR locally? Did I say something like this in my other posts? When talking about slight violation of energy conservation in GR I mean that in a Cartesian-like coordinates the http://www.google.com/url?q=http://...section_link&resnum=1&ct=legacy&ved=0CAoQygQ" to find out what this [tex]r[/tex] is!)

You're not frustrating me, as I'm really calm and having my cup of tea!

I am asking: Can we experimentally constrain these constants? Can we, in this
sense, "measure" Einstein's Field Equations? (and hence the title of this thread)

If it comes to C_1, then No! But if that's about C_3 in case we ignore the Newtonian limit, Yes we can! At least I can assure you that if any discrepancy between experiments and theory exists, it is not because of C_1 being 1/2 but C_3 or maybe the matter tensor at heart! All vacuum solutions are compatible with experimental data gathered from all over the world and you cannot also close your eyes to the experimental errors due to thoroughly accountable reasons such as the lack of precision in the measurements.

You are going out on a limb here but the main problem is that putting constraints on C_1 by looking at the observational evidences is so ineffective because all you do is "fine tune" C_1 at each point of the spacetime and if the field was weak, then these coupling constants would be so much the same almost everywhere.

 

[Mentz114]

The thing that gets tested by our observations of solar system and astronomical dynamics is the Christoffel symbols. These are derived from the metric and have nothing to do with the EMT of the source, or whether it is a vacuum solution or not. So we are testing the metric, not whether it is a solution of the EFE. That's another question.
So I don't see how C1 affects anything except whether we believe the metric is a solution or not.

If one found found a spherically symmetric vacuum 'solution' with C1 <> 1/2, then we could test the two metrics against each other. In the event that the C1<>1/2 case was actually better, we would have to reject ALL of GR.

 

[JustinLevy]

If one found found a spherically symmetric vacuum 'solution' with C1 <> 1/2, then we could test the two metrics against each other. In the event that the C1<>1/2 case was actually better, we would have to reject ALL of GR.

That's the thing though. C1 can be any value, and it will not affect the vacuum solutions, since the vacuum solutions solve:
[tex]R^{\mu\nu}=0[/tex]
Regardless of C1.
So we wouldn't have to reject GR. In fact, as far as I know (and hence the question), there doesn't seem to be an easy way to measure C1. Altabeh's ideas along the lines of conservation of energy or momentum experiments seem like a decent path, but it sounds like the deviations due to C1<>1/2 would be too small to measure to put much constraint on C1.

If it comes to C_1, then No! But if that's about C_3 in case we ignore the Newtonian limit, Yes we can!

How? How do we get C_3 if we ignore the Newtonian limit?

Furthermore, it still seems strange to me that in the "normal approach" we need to use the Newtonian limit twice. First to set C3, and then again after solving the differential equations for the metric in order to set the integration constants. It's almost as if C_3 alone can't set the "coupling" constant and we still another equation.

Do we actually need more than the EFE, and the equation of geodesic motion, to solve for orbital motions around the sun? It seems like the answer is yes, using the above arguments. What am I missing? Is there another constraint that would help us determine the integration constants without having to refer to Newtonian limits again?

 

[atyy]

I'm not sure this is related, but following Altabeh's suggestion to look at energy-momentum conservation, Table 4 of http://relativity.livingreviews.org/Articles/lrr-2006-3/ gives limits on parameters which are labelled in Table 2 as describing violation of total momentum conservation.

 

[Altabeh]

That's the thing though. C1 can be any value, and it will not affect the vacuum solutions, since the vacuum solutions solve:
[tex]R^{\mu\nu}=0[/tex]
Regardless of C1.
So we wouldn't have to reject GR. In fact, as far as I know (and hence the question), there doesn't seem to be an easy way to measure C1. Altabeh's ideas along the lines of conservation of energy or momentum experiments seem like a decent path, but it sounds like the deviations due to C1<>1/2 would be too small to measure to put much constraint on C1

No! Actually either putting [tex]C_1\neq1/2[/tex] or thinking that a free-to-be-chosen constant [tex]C_3[/tex] in the EFE equations will make the theory lose its self-consistency and this is highly rejected by today's physics because they all try to find a unified equation to explain all of the forces we have in physics together. You're just forcing the theory to take a big detour from its main road by allowing such things to be possible. Furthermore, this also kills the conservation laws of GR quickly and then I'm so afraid to say that then the fate of the new theory by these assumptions will be doomed to getting thrown in the trash can.

How? How do we get C_3 if we ignore the Newtonian limit?

You must get the idea of setting C_3 as a "coupling constant": If [tex]Lambda\neq 0[/tex] then C_3 can be readily modified according to our observational data. In this case no reduction to Newton's theory will be possible because the weak field cannot be retrieved due to the precense of the Lambda term. (Do not think of a very small Lambda to justify C_3 can be the relativistic gravitational constant this makes the term ineffective.) But if [tex]Lambda= 0[/tex] then the theory can reduce to Newton's theory whenever we follow three important steps 1) a universe with a non-relativistic material distribution 2) particles moving at non-relativistic velocities 3) the Minkowski spacetime being perturbed by a weak field around a gravitating matter. Note, however, that in this case you can also have the theory based on C_3 as a coupling constant which is more efficient than the modified EFE with a non-vanishing Lambda term.

Furthermore, it still seems strange to me that in the "normal approach" we need to use the Newtonian limit twice. First to set C3, and then again after solving the differential equations for the metric in order to set the integration constants. It's almost as if C_3 alone can't set the "coupling" constant and we still another equation.

You just set C_3 once that all field equations are written extensively because otherwise it cannot be meaningful and it of course will look like you're having a guess about something serious in a fundamental equation and then turn to construct the model based on this probably void guess.

Do we actually need more than the EFE, and the equation of geodesic motion, to solve for orbital motions around the sun? It seems like the answer is yes, using the above arguments. What am I missing? Is there another constraint that would help us determine the integration constants without having to refer to Newtonian limits again?

As you said, the answer to the first part of your question is yes but this is always the case. We must refer to geodesic equations when it comes to "motions" around sun or other gravitating bodies. But unfortunately we don't have another alternative to find out what value C_3 would have to get the EFE to involve Newton's theory within it. This is just done through the use of geodesic equations along with following the above steps. So you're not missing anything here.