Logic of GR as mathematically derived

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In summary, the conversation discusses the attempt to reverse engineer General Relativity in order to determine its basic concepts and mathematical logic. The values for time dilation and length contraction are being sought, and a list of assumptions about the field and motion of particles through the field is provided. The concept of invariants is also introduced and discussed. The conversation ends with a formula for locally measured acceleration in the presence of special relativity.
  • #36
PeterDonis said:
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.

Never mind; I see now how to rewrite the formula. We take the formula for z:

[tex]z = \sqrt{1 - \frac{2M}{r}}[/tex]

and rearrange to:

[tex]r = \frac{2M}{1 - z^2}[/tex]

and then substitute into the formula for a to obtain:

[tex]a = \frac{M}{r^2 z} = \frac{M \left( 1 - z^2 \right)^2}{4 M^2 z} = \frac{\left( 1 - z^2 \right)^2}{4 M z}[/tex]

which is what you wrote a few posts ago. :redface:
 
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  • #37
Well sure, you could do it like that too I suppose, but only if you like doing things the easy way. :smile:
 
  • #38
Ah, okay. For the past week I have been trying to incorporate assumption C for the conservation of angular momentum m_p v'_t r', where m_p here is the mass of the particle, which is considered invariant so divided out for convenience, leaving the constant of motion P = v'_t r', which is measured the same locally at every shell depending upon how a particle is originally set in motion. Okay, so by reverse engineering the solution to the SC metric in another thread, I had gained P = v'_t r' = v'_t r / sqrt(1 - 2 m / r), which I took to mean r' = r / L, using the inferred radius by extending a local ruler at r all the way along r radially that is contracted radially by a factor of L = sqrt(1 - 2 m / r). One problem I have had with that, though, upon seeing that the quantities z, L / dr, and L_t / r are invariant for a particular shell (or coordinate independent), is that P is not invariant as it is currently expressed, although it should be.

I realize now that with the tangent motion of a particle, the local shell is not using some locally inferred radius r', but rather C' / (2 pi), where C' is the locally measured circumference of the shell at r. C' / (2 pi) ≠ r / L here of course, so with the distant observer inferring a circumference of the shell of C = 2 pi r, the local static observer with a tangent contraction of rulers of L_t, will physically measure C' = (2 pi r) / L_t by placing infinitesimal rulers end to end around the circumference, which is invariant. So to conserve the locally measured angular momentum of a particle, we would have

p'_angular = p'_t r' = [m_p v'_t / sqrt(1 - (v'/c)^2)] (C' / 2 pi) = constant

where from assumption B we gained sqrt(1 - (v'/c)^2) = z / K, so

= m_p v'_t K ((2 pi r / L_t) / 2 pi) / z

= m_p v'_t K (r / L_t) / z

and upon dividing out the invariant m_p and constant of motion K, we gain another constant of motion

P = v'_t r / (L_t z)

which still works out to P = v'_t r / sqrt(1 - 2 m / r) as found by reverse engineering SC, but is now invariant. This is the new corrected equation for assumption C. I will have to go back to a couple of other threads where I got this far and then got stuck at this point also.
 
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  • #39
grav-universe said:
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?

[tex]\rm{4 \dot{A} B^2 - 2 r \ddot{B} AB + r \dot{A} \dot{B}B + r \dot{B} ^2 A=0}[/tex]

[tex]\rm{r \dot{A}B + 2 A^2 B - 2AB - r \dot{B} A=0}[/tex]

[tex] \rm{- 2 r \ddot{B} AB + r \dot{A} \dot{B}B + r \dot{B} ^2 A - 4\dot{B} AB=0}[/tex]
Well, since L_t / r is an invariant, and those equations should represent invariants, I tried simply replacing r with r / L_t in the equations. They all work out to zero with Schwarzschild, as they should anyway since L_t is just 1, but they also all work out to zero with GUC, where L_t is non-unity. GUC is the same as SC but all of the shells are moved uniformly closer to the center while keeping the same coordinate distance between the shells and cutting out the volume inside the event horizon, so it has the same value for L at any r as SC, so the same z and L for any shell, and so the same A and B, only represented with the conversion for r1 instead of r. Apparently then, L_t / r is the correction factor for A and B when changing coordinates systems of the form r1 = r - n m from SC, simply sliding the shells uniformly inward or outward, where n has any numerical value. They didn't work out for EIC for some reason though, not sure why. I need those same equations but including C for tangent length.
 
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  • #40
Ahah! Yay. :) I can now see where one of the solutions to the Ricci tensors comes from. From the relationship we found before

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

which for convenience I will write dz / dr as just z' with second derivatives double primed.

We can re-arrange to gain

L_t^4 = z'^2 L^2 r^4 / m^2

The variables in the tensors are

A = 1 / L^2

B = - z^2

and B' = d(-z^2) = - 2 z z'

z' = - B' / (2 z)

z'^2 = B'^2 / (4 z^2) = - B'^2 / (4 B)

so we can rewrite the relationship once more to

L_t^4 = - B'^2 r^4 / (4 m^2 A B)

Finding the derivative for that using Wolfram, we get

d(L_t^4) = d[- B'(r)^2 r^4 / (4 m^2 A(r) B(r))] = r^3 B' (r B A' B' + A(r B'^2 - 2 B (r B" + 2 B')) / (4 m^2 A^2 B^2)

Now, if as a coordinate choice, we make L_t = 1, then its derivative is zero. So now we have

(0) (4 m^2 A^2 B^2) / (r^3 B') = 0 = r B A' B' + A(r B'^2 - 2 B (r B" + 2 B'))

r B A' B' + r A B'^2 - 2 r A B B" - 4 A B B' = 0

This is the same as the third Ricci tensor solution in the last post. :) Now I just need something similar to find the other two.
 
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  • #41
In terms of solving for C(r), for cases where C(r) is non-unity, with the metric of the form

ds^2 = A(r) dr^2 + B(r) c^2 dt^2 + C(r) dO^2 r^2 {as Wiki has it but with c^2 drawn out of B}

we would have

C(r) = 1 / L_t^2, giving

d[L_t^4] = d[1 / C^2] = - 2 C' / C^3

and

d[- B'^2 r^4 / (4 m^2 A B)] = r^3 B' (r B A' B' + A(r B'^2 - 2 B (r B" + 2 B'))) / (4 m^2 A^2 B^2) from before, so

- 2 C' / C^3 = r^3 B' (r B A' B' + A(r B'^2 - 2 B (r B" + 2 B'))) / (4 m^2 A^2 B^2)

- 8 m^2 A^2 B^2 C' / (r^3 C^3 B') = r B A' B' + A(r B'^2 - 2 B (r B" + 2 B'))
 
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  • #42
Actually, here's a simpler one involving C. From the last post, we had

1 / C^2 = -B'^2 r^4 / (4 m^2 A B)

1 = -B'^2 C^2 r^4 / (4 m^2 A B)

d(1) = 0 = d[-B'^2 C^2 r^4 / (4 m^2 A B)]

= r^3 C B'(r B C A' B' + A(r C B'^2 - 2 B(r C B" + r B' C' + 2 C B'))) / (4 m^2 A^2 B^2)

r B C A' B' + r A C B'^2 - 2 r A B C B" - 2 r A B B' C' - 4 A B C B' = 0
 
<h2>1. What is the "Logic of GR as mathematically derived"?</h2><p>The "Logic of GR as mathematically derived" refers to the logical and mathematical principles that underlie the theory of General Relativity (GR). These principles were developed by Albert Einstein in the early 20th century and provide the framework for understanding gravity as a curvature of spacetime.</p><h2>2. How is the "Logic of GR as mathematically derived" different from other theories of gravity?</h2><p>The "Logic of GR as mathematically derived" differs from other theories of gravity, such as Newton's theory of gravity, in that it is based on the principle of general covariance, which states that the laws of physics should be the same for all observers regardless of their relative motion. This leads to a more comprehensive and unified understanding of gravity.</p><h2>3. What are the key mathematical concepts in the "Logic of GR as mathematically derived"?</h2><p>The key mathematical concepts in the "Logic of GR as mathematically derived" include the use of tensor calculus, which is a mathematical tool for describing the curvature of spacetime, and the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy.</p><h2>4. How has the "Logic of GR as mathematically derived" been tested and verified?</h2><p>The "Logic of GR as mathematically derived" has been tested and verified through numerous experiments and observations, such as the bending of light by massive objects, the precession of the orbit of Mercury, and the gravitational waves detected by the LIGO experiment. These confirm the predictions made by the theory and provide strong evidence for its validity.</p><h2>5. What are some current challenges and open questions in understanding the "Logic of GR as mathematically derived"?</h2><p>Some current challenges and open questions in understanding the "Logic of GR as mathematically derived" include the search for a theory of quantum gravity that can reconcile GR with the principles of quantum mechanics, and the study of extreme conditions, such as black holes, where the effects of gravity are most pronounced. Additionally, the precise nature of dark matter and dark energy, which are believed to make up a large portion of the universe, remains a mystery within the framework of GR.</p>

1. What is the "Logic of GR as mathematically derived"?

The "Logic of GR as mathematically derived" refers to the logical and mathematical principles that underlie the theory of General Relativity (GR). These principles were developed by Albert Einstein in the early 20th century and provide the framework for understanding gravity as a curvature of spacetime.

2. How is the "Logic of GR as mathematically derived" different from other theories of gravity?

The "Logic of GR as mathematically derived" differs from other theories of gravity, such as Newton's theory of gravity, in that it is based on the principle of general covariance, which states that the laws of physics should be the same for all observers regardless of their relative motion. This leads to a more comprehensive and unified understanding of gravity.

3. What are the key mathematical concepts in the "Logic of GR as mathematically derived"?

The key mathematical concepts in the "Logic of GR as mathematically derived" include the use of tensor calculus, which is a mathematical tool for describing the curvature of spacetime, and the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy.

4. How has the "Logic of GR as mathematically derived" been tested and verified?

The "Logic of GR as mathematically derived" has been tested and verified through numerous experiments and observations, such as the bending of light by massive objects, the precession of the orbit of Mercury, and the gravitational waves detected by the LIGO experiment. These confirm the predictions made by the theory and provide strong evidence for its validity.

5. What are some current challenges and open questions in understanding the "Logic of GR as mathematically derived"?

Some current challenges and open questions in understanding the "Logic of GR as mathematically derived" include the search for a theory of quantum gravity that can reconcile GR with the principles of quantum mechanics, and the study of extreme conditions, such as black holes, where the effects of gravity are most pronounced. Additionally, the precise nature of dark matter and dark energy, which are believed to make up a large portion of the universe, remains a mystery within the framework of GR.

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