Re: Black holes radiate - the end of GR is there

Koobee Wublee <kublai@xxxxxxx> wrote:

> <carlip-nospam@xxxxxxxxxxxxxxxxxxx> wrote in message
> news:di6e66$dnm$1@xxxxxxxxxxxxxxxxxxxxxx
>> Koobee Wublee <kublai@xxxxxxx> wrote:

>>> And I have to disagree that proper spacetime hold the key to describe the
>>> minimal action of an event. If I choose to minimize either the proper
>>> time or the observed time, I get the same answer as to minimize the
>>> proper spacetime.

>> You will find a discussion of this in Perlick's Living Reviews article,
>> http://relativity.livingreviews.org/Articles/lrr-2004-9/index.html
>> and in Class. Quant. Grav. 7 (1990) 1319. For a certain class of
>> metrics -- those that are conformal to static metrics -- your method
>> will work, that is, it will give null geodesics. For a generic metric,
>> it won't.

> Your generic metric of minimizing proper spacetime would not work for the
> study of photon deflection unless you abide to Ciufolini/Wheeler's method
> of creative minimization.

You can either define a null geodesic as a limit of non-null ones, or
as a null auto-parallel curve, or by using Fermat's principle *correctly*
(that is, in a way that doesn't depend on a coordinate choice), as described
in the articles I cited by Perlick.

>> Let me repeat one more time, since you seem to have missed this very
>> basic point: the geodesic equation does *not* have to be assumed
>> separately in general relativity. It follows as a consequence of the
>> field equations. If you want to reject the geodesic equation (for those
>> metrics for which extremizing time and extremizing proper time give
>> different results), then you necessarily must reject the Einstein field
>> equations as well.

> OK, let me try one more time to understand what you are saying. I
> construde you mean the following.

> "Since the field equations were derived with the proper spacetime as the
> representation of minimal action, one must minimize the proper spacetime to
> arrive at the solutions to the field equations."

I don't know what this means -- I don't understand the phrases "the proper
spacetime as the representation of minimal action" and "minimiz[ing] the
proper spacetime." (How in the world do you "minimize a spacetime"?) I
suspect, though, that you've mixed up the field equations and the geodesic
equations.

What I mean is this:

General relativity contains two basic ingredients: the Einstein field
equations, which determine the gravitational field (or spacetime geometry)
due to a given configuration of mass and energy, and the geodesic equations,
which determine how objects move in that gravitational field (or spacetime
geometry). The field equations can be written as G_{ab}=8pi T_{ab}, while
the geodesic equations are the equations of motion we have been discussing.
If this is confusing, think of the electromagnetic analog: Maxwell's
equations determine the electric and magnetic fields produced by a given
charge and current distribution (field equations), while the Lorentz force
law, together with Newton's second law, determine the motion of a charged
particle in those fields (equations of motion).

The statement is that in general relativity, these two ingredients are not
independent -- the field equations have a solution only if the objects that
act as sources of the gravitational field also move along geodesics. In
particular, if you want to change the geodesic equation (say, by minimizing
some time coordinate instead of proper time), you must also change the
Einstein field equations.

But if you change the Einstein field equations, there's no reason to talk
about the Schwarzschild metric to begin with -- it's only important because
it's the solution of the field equations for a spherically symmetric mass.

> If so, in my own study, the real field equations are still out there to be
> written down based on the minimal action of elapsed time and not proper
> spacetime.

If you are saying that the correct field equations are not the Einstein field
equations, why are you talking about the Schwarzschild metric at all?

>> [...]

>>>> Note that Krantiotis and Whitehouse give the *exact* solution, to *all*
>>>> orders. Furthermore, E and L are determined *exactly* by the aphelion
>>>> and perihelion -- see eqns. (25) and (26). (The quantities e_2 and e_3
>>>> are the roots of a cubic polynomial that depends on L and E.)

>>> It is funny how you would think that they have the exact solution. On
>>> the bottom of page 7, they wrote "We will show... there is a small
>>> parameter space for L and E that reproduces [the observed orbital
>>> anomaly of Mercury]...".

>> How does that contradict anything I said? They obtain the exact solution,
>> which depends on two integration constants, E and L. For a certain small
>> range of values of these constants, the exact solution agrees with the
>> observed perihelion advance.

>>> This means they can come up with values of L and E that will explain the
>>> 43" per century of orbital anomaly. And therefore GR's prediction of
>>> Mercury's orbital anomaly is achieved by allowing L and E to be certain
>>> values that are possibly under the constraint of all these equations.
>>> This, to me, is not a solid proof. A solid proof is to derive thorughly
>>> and exactly what L and E are in terms of value.

>> As I said, E and L are determined *exactly* by the aphelion and
>> perihelion -- see eqns. (25) and (26). The equations aren't going to
>> tell you these values, because they're different for each planet. So
>> you observe the value of the aphelion and perihelion (*not* the advance
>> of the perihelion) to determine E and L, plug them into the equation,
>> and get the right expression for the advance.

> For some certain range of E and L? On page 9 of the same paper, notice E
> and L are carried out to 8+ digits after decimal. Thus, I have to correct
> what you said.

No. See tables 1-3.

> "For a certain [precise] values of these constants, the exact solution
> agrees with the observed perihelion advance."

> So, Krantiotis/Whitehouse worked it backwards, they knew the answer is 43"
> per century. In doing so, they calculated what E and L are to 9+
> significant digit of precision.

This is completely wrong. That's not where their values of E and L came
from. There are two numbers here, and *three* observable quantities, the
perihelion r_P, the apehelion r_A, and the perihelion advance Delta. By
using the observed values of r_P and r_A -- that is, the observed mimimum
and maximum distance of Mercury from the Sun -- you can *compute* L and E,
and therefore Delta.

Look at the explanation in section 3.2. Kraniotis and Whitehouse say very
clearly that E is determined by the observed semimajor axis of Mercury's
orbit, as reported in _Allen's Astrophysical Quantities_, and that L in
table 2 is determined from the observed eccentricity of Mercury's orbit.

> So, if E and L are what they wrote down,
> then Mercury's orbital anomaly is observed. Granted that they are able to
> show other parameters such as the perihelion and aphelion, the proof of
> Mercury's orbital anomaly has to address what E and L are head on.
> Krantiotis/Whitehouse, Ciufolini/Wheeler, and others' solutions do not
> calculate precisely what E and L are directly.

You *can't* "calculate precisely what E and L are." They're *different*
for different planets. What you can do is observe a planet, measure its
apehelion and perihelion, and use that to predict the advance of its
perihelion.

> One good approach is to eliminate the dependence on E and L totally by
> taking derivatives. After arriving at (dt/ds) or (ds/dt) through the
> Fermat-Lagrangian method or the variational method, one plugs the result
> back into the spacetime equation with Schwarzschild metric. Collect the
> constant E to one side of the equation, and take the derivative.
> Incidentally, the result all contains the derivative of the equation
> yielding L.

>> Fine, So you converted first-order differential equations to second-order
>> ones. That means that when you solve the equations, you will have new
>> constants of integration. How do you fix their values?

> I went through this with Bilge already. You can see where I am coming from
> from the following two posts.

> http://groups.google.com/group/alt.sci.physics.new-theories/msg/4d729310873fb1e7?dmode=source

> http://groups.google.com/group/alt.sci.physics.new-theories/msg/dfee758f6a8a1639?dmode=source

I'm sorry, but I could not follow this at all.

Let me try one more time. (After this, I give up...)

1. The Schwarzschild metric is a solution of the Einstein field equations. There's
no particular point in thinking about it if you reject the field equations.
2. The field equations require that objects move along geodesics.
3. Kraniotis and Whitehouse, among others, have found the *exact* solution to the
geodesic equations for the Schwarzschild metric.
4. This solution depends on two integration constants. (It must, since different
planets move in different orbits -- there can't possibly be a unique solution.)
5. The two integration constants can be determined exactly by specifying a planet's
apehelion and perihelion (or semimajor axis and eccentricity).
6. Given such a determination, the *exact* solution of the field equations predicts
a calculable perihelion advance.
7. The predicted amount agrees with observation for Mercury, Mars, the asteroid
Icarus, and several binary pulsar systems.

>>>>> What I found is not what is commonly believed.

>>>> Do you claim you found something different from the results of
>>>> Krantiotis and Whitehouse? If so, you've made a mistake in the math.

>>> The first two tries I show two different values. However, from the third
>>> try and on to about a dozen, I consistantly get zero anomaly with a
>>> solution independent of L and E. Yes, I have discovered that by write
>>> down a solution to Mercury's orbital anomaly independent of E and L I
>>> get no anomaly at all.

>> What values did you choose for your integration constants? The solution
>> of the geodesic equation *must* have such constants -- otherwise, you could
>> solve the equation and determine, for example, Mercury's eccentricity,
>> which depends on these constants. (In case it's not obvious, you *can't*
>> do that in the real world. Mercury's eccentricity is determined by
>> initial conditions, and equations of motion can't determine these.)

> The answer should not depend on the integration constants.

The integration constants determine, among other things, the size of the
orbit (the semimajor axis) and the eccentricity. Are you seriously claiming
that the perihelion advance should be the same regardless of the orbit?

> The eccentricity
> should not play a major role in orbital anomaly. Just take a look at the
> table containing all major planets in the folling link.

> http://www.mathpages.com/rr/s6-02/6-02.htm

You mean, where it says, "The effect is most noticeable for objects near the
Sun with highly elliptical orbits, but it can be seen even in the nearly
circular orbits of Venus and Earth, although the discrepancy isn't nearly so
large as for Mercury"?

Steve Carlip



.

Relevant Pages

  • Re: Black holes radiate - the end of GR is there
    ... The null geodesics is defined to ... clarification as observed by an observer), and proper spacetime. ... >> "Since the field equations were derived with the proper spacetime as the ... >>> observed perihelion advance. ...
    (sci.physics.relativity)
  • Re: Black holes radiate - the end of GR is there
    ... you are really confusing me of what you are talking ... You seem to be under the impression that "proper spacetime" is a standard ... The Einstein field equations are *not* derived by minimizing proper time, ...
    (sci.physics.relativity)