- #1
JustinLevy
- 895
- 1
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?
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?