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lalbatros
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If I am asking this question, this is maybe a proof that I need a strong "back-to-the-basics".
Could you give the way?
Thanks,
Michel
Could you give the way?
Thanks,
Michel
... in other words the force density acting on a freely falling particle is zero, hence it suffers is no acceleration in the freely falling frame of reference ...
Yes it is obvious - if you think about it a bit!lalbatros said:Garth,
I don't understand why you can conclude this:
I think that the answer may be simple or even obvious. I thought first that the answer would be obtained by comparing motions in two frames (accelerated without gravitation and another with gravity) and concluding that physics is the same. I also considered that it could be simply related to the invariance of the equations.
Thanks to help me on that,
Michel
Demystifier said:The Einstein equation, which is the equation of motion of the gravitational field, does NOT imply the equivalence principle. Instead, it is the equation of motion of MATTER that implies the equivalence principle, provided that matter is MINIMALLY coupled to gravity (i.e. to the metric tensor).
The conservation of energy-momentum implies the equivalnce principle only if [tex]T_{\mu\nu}[/tex] is specified and equal to the quantity that corresponds to the minimal coupling of matter to gravity.
Did you ever heard about the notion of "minimal coupling"?Garth said:Further more, the statement "matter is MINIMALLY coupled to gravity (i.e. to the metric tensor)" would imply the metric does not influence the paths of matter at all.
Not really.lalbatros said:Would you have a reference on the web that shows that explicitely?
Agreed, that is why I said "such as"Demystifier said:There are many many other solutions besides the perfect-fluid solution.
Agreed, varying the action in 4D space-time necessarily results in the conservation of energy-momentum because the method is true for generalised coordinates.Einstein equation is obtained by varying action with respect to metric. To obtain the matter equation of motion, you need to vary action with respect to matter degrees of freedom.
In which case, in the case of no external forces acting, you disagree with Wald "General Relativity" see Eq.4.3.6 and the discussion on page 73.The energy-momentum conservation is NOT the equation of motion.
Obviously in GR it does locally.Think also about the following. One of the solutions of the energy-momentum conservation equation is the "cosmological constant" solution
[tex]T_{\mu\nu}(x)=g_{\mu\nu}(x)\lambda .[/tex]
Does it obey the equivalence principle?
I am familiar with the expression being used referring to a scalar field, in which case if the scalar field is minimally coupled then its presence affects only the curvature of space-time and does not perturb test particles from their geodesic world-lines through that space-time.Demystifier said:Did you ever heard about the notion of "minimal coupling"?
I have never used "minimal coupling" in that sense and I find its use a little over-redundant. Which authors do use it as such?In the case of gravity, minimal coupling means that the only way the nontrivial metric enters the equations is through replacing the ordinary derivatives by covariant derivatives.
1. Of course, but I was talking generally, not about the special case without external forces.Garth said:1. In which case, in the case of no external forces acting, you disagree with Wald "General Relativity" see Eq.4.3.6 and the discussion on page 73.
2. Could you clarify what you meant by "the equation of motion of the gravitational field"? Thank you.
This terminology perhaps is not common in GR, but is common in gauge theories. And you certanly know that GR can be viewed as a gauge theory.Garth said:I have never used "minimal coupling" in that sense and I find its use a little over-redundant. Which authors do use it as such?
Agreed, and [itex]G_{\mu\nu}[/itex] describes that curvature.robphy said:It's not so much that it's the field equations G=kT themselves that imply the Equivalence Principle. It is more about that those equations (or alternate equations like them) are formulated on a generally-curved spacetime.
http://relativity.livingreviews.org/open?pubNo=lrr-2001-4&page=node3.html
Garth said:Agreed, and [itex]G_{\mu\nu}[/itex] describes that curvature.
Garth
Agreed, the Riemannian fully describes the curvature.robphy said:[itex]G_{\mu\nu}[/itex] describes part of that curvature.
Yes indeed, the WEP is satisfied in Newtonian theory and the Brans-Dicke Field Equations equations satisfy the SEP, but noting G varies with position in that theory.The point I was making is that the Einstein Field Equations are not required for the so-called Equivalence Principle.
The equation G=8pT is known as the Einstein field equation, which describes the relationship between the curvature of spacetime (represented by G) and the distribution of matter and energy (represented by T). This equation is critical in understanding the principle of equivalence, as it shows that gravitational force is not a separate force, but rather a result of the curvature of spacetime caused by mass and energy.
The equation G=8pT serves as a mathematical representation of the equivalence principle, which states that the effects of gravity are indistinguishable from the effects of acceleration. This equation shows that the curvature of spacetime, which is responsible for the effects of gravity, is directly related to the distribution of matter and energy in that spacetime.
The Einstein field equation, G=8pT, is derived from general relativity, which is the theory that describes the relationship between gravity and the curvature of spacetime. This equation shows that the curvature of spacetime is due to the presence of matter and energy, supporting the principle of equivalence by demonstrating that gravity is not a separate force, but rather a result of the curvature of spacetime caused by mass and energy.
An example that illustrates how G=8pT implies the equivalence principle is the phenomenon of free-fall. According to the principle of equivalence, an object in free-fall experiences no gravitational force, but rather is following a geodesic (a straight line) in curved spacetime. The Einstein field equation, G=8pT, shows that the curvature of spacetime is directly related to the distribution of matter and energy, and in the case of free-fall, the object is following the curvature of spacetime caused by the mass of the planet or other large object.
The equation G=8pT has been extensively tested and verified through various experiments and observations. One of the most well-known examples is the observation of the bending of starlight by the Sun's gravitational field during a solar eclipse, which confirmed the prediction of general relativity and the Einstein field equation. Additionally, the precision of the equation has been tested through the precise measurements of the orbits of planets and other celestial bodies, which have all shown consistent results with the principles of general relativity and the equivalence principle.