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Is it correct that the only way to have a theory of gravitation that fulfills the equivalence principle is to make use of a tensor as the source of gravity (and not a scalar or a vector, for example)? How can this be proven?
A scalar and a vector are both tensors. From what I recall there is a theory by Dicke (Brans too?) which is a relativistic theory of gravity which is consistent with the equivalence principle.hellfire said:Is it correct that the only way to have a theory of gravitation that fulfills the equivalence principle is to make use of a tensor as the source of gravity (and not a scalar or a vector, for example)? How can this be proven?
That's the kind of answer I am looking for… but with a proof.Chronos said:There is no way to avoid a tensor description of gravity geometrically.
You're incorrectly expecting proof where there is only postulate. The principle of general covariance requires the laws of nature to be the same in all coordinate systems. We give the name "tensor" to those objects which satisfy this property of covariance.hellfire said:That's the kind of answer I am looking for… but with a proof.
But a theory in which gravity couples to the trace of the energy-momentum tensor would be also covariant, as the trace is scalar (a rank 0 tensor). But it does not fulfill the equivalence principle.pmb_phy said:You're incorrectly expecting proof where there is only postulate. The principle of general covariance requires the laws of nature to be the same in all coordinate systems. We give the name "tensor" to those objects which satisfy this property of covariance.
Pete
Since when?hellfire said:But a theory in which gravity couples to the trace of the energy-momentum tensor would be also covariant, as the trace is scalar (a rank 0 tensor). But it does not fulfill the equivalence principle.
I am not aware of any error in what I wrote, but If I wrote something wrong, please correct me. That's the best way for me to learn.pmb_phy said:Since when?
Pete
That is not true. The Branse Dicke theory treats spacetime as a 4-d manifold and the description of the geometry is not frame dependant. However the field equations need not be a second rank tensor (e.g. Brans Dicke).Chronos said:What Pete said - GR treats spacetime as a four dimensional manifold. To describe the geometry of such a manifold without introducing frame dependence, you must use a rank 2 tensor.
What are these two separate field equations you speak of??Garth said:Hmmm... The Brans Dicke theory has two separate field equations and an equation of state, The gravitational field equation certainly is an equation of second rank tensors, the scalar field equation is an equation in which each term is a scalar, as is the equation of state, however, that is no different to GR, which also requires a 'scalar' equation of state. [Though no scalar field equation]
Garth
Garth said:Yes, the Brans Dicke theory takes the Einstein field equation and adds a scalar field coupled to the (rest) mass density of the universe that endows particles with inertial mass. Garth
hellfire said:a)...is a covariant theory, isn’t it?
b) but, however, it does not satisfy the equivalence principle, since light would not couple to a gravitational field. A photon would not be redshifted or blueshifted in a gravitational field, but it would be redshifted or blueshifted when emitted from an accelerated frame.
If all above is correct, what are the reasons, if any, for the need of having a rank 2 tensor in order to satisfy the equivalence principle?
Also Black Holes would not be coupled to a gravitational field. As the matter of a collapsed star became degenerate and relativistic its gravitational field would disappear!hellfire said:I am sorry but I still don’t get it, so please be patient with me.
A theory in which gravity would couple only to the trace of the energy-momentum tensor (such a theory was considered by Einstein before general relativity):
[tex]\square \phi = - 4 \pi G T_{\mu}^{\mu}[/tex]
With [tex]\inline g_{\mu \nu}[/tex] being diagonal with [tex]\inline \phi[/tex] or [tex]\inline - \phi[/tex] as diagonal elements...
a)...is a covariant theory, isn’t it?
b) but, however, it does not satisfy the equivalence principle, since light would not couple to a gravitational field. A photon would not be redshifted or blueshifted in a gravitational field, but it would be redshifted or blueshifted when emitted from an accelerated frame.
If all above is correct, what are the reasons, if any, for the need of having a rank 2 tensor in order to satisfy the equivalence principle?
The energy-momentum tensor is a mathematical quantity in physics that describes the distribution of energy and momentum in a given system. It is represented as a matrix with 4 rows and 4 columns, where each element represents a different component of energy or momentum.
The equivalence principle states that the effects of gravity are indistinguishable from the effects of acceleration. The energy-momentum tensor is used to describe the effects of gravity in Einstein's theory of general relativity, making it a fundamental concept in understanding the equivalence principle.
The energy-momentum tensor is significant because it allows us to accurately calculate and predict the behavior of systems in the presence of gravity. It is a crucial component in Einstein's theory of general relativity, which is the most accurate theory of gravity we have to date.
The energy-momentum tensor is calculated using mathematical equations that take into account the distribution of energy and momentum in a given system. These equations involve concepts such as spacetime curvature and the energy-momentum density of matter and energy sources.
The energy-momentum tensor can be applied to any system that involves energy and momentum, including both classical and quantum systems. However, it is most commonly used in the context of general relativity and gravitational systems.