Why is the linearized vierbein a 2nd rank tensor in flat spacetime?

[DrDu]

Why are gravitons tensorial spin 2 particles while Newtonian gravity is a scalar?

 

[Dickfore]

Because the spacetime metric is a tensor. And spin-2 comes from the fact that
(1/2,1/2) x (1/2, 1/2) = (0,0) + (1,0) + (0,1) + (1,1)

so a symmetric traceless tensor is in the (1,1) representation, whose spatial part corresponds to S=2 rep. of the so(3) ~ su(2) Lie algebra of spatial rotations.

EDIT:
Scalar gravitational potential from Newton's theory is the g₀₀ perturbation to the Minkowski metric.

 

[DrDu]

Thank you Dick!

 

[haushofer]

The Newton potential is not a scalar under general coordinate transformations, only under the galilei group. More precisely, as soon as you make boosts quadratic in time or more, which are (time dependent) accelerations, the potential transforms inhomogeneously. You can derive this from the fact that the potential comes from the i00 component of the connection.

 

[atyy]

I don't think there is a "why". It's just like that.

Nordstrom's second theory is a scalar relativistic theory of gravity that preceded general relativity. It satisfies the equivalence principle, and has a geometric formulation. It happens to predict the wrong perihelion precession.

http://arxiv.org/abs/gr-qc/0611100
http://arxiv.org/abs/gr-qc/0405030

Also relevant is exercise 24.1c in http://www.pma.caltech.edu/Courses/ph136/yr2006/0424.1.K.pdf about a scalar theory of gravity: "Explain why this prediction implies that there will be no deflection of light around the limb of the sun, which conflicts severely with experiments that were done after Einstein formulated his general theory of relativity. (There was no way, experimentally, to rule out the above theory in the epoch, ca. 1914, when Einstein was doing battle with his colleagues over whether gravity should be treated within the framework of special relativity or should be treated as a geometric extension of special relativity.)"

 

[dextercioby]

Why are gravitons tensorial spin 2 particles while Newtonian gravity is a scalar?

The why is answered in a simple manner: Gravitons per definition are excitations of the quantized graviton field* (in exactly the same manner in which photons are excitations, i.e. 1-particle states of the quantized electromagnetic field in vacuum) which is described through a tensorial object carrying 2 flat space-time indices on the same position, covariant, and which is symmetric wrt interchange of the indices. An object with 2 tensorial indices <downstairs> automatically contains a spin 2 field (the spin 1 content is removed through symmetrization and the spin 0 content by redefining the field to be traceless) as it follows from the theory of non-unitary finite dimensional vectorial representations of the connected component of the Lorentz group.

*The graviton field is a first perturbation of the curved spacetime metric around the flat Minkowski background is conventionally called <the Pauli-Fierz field> and obeys the same quantum dynamics as a linearized vierbein (the antisymmetric components of the linearized vierbein do not propagate) which is used in supergravity theories.

 

[haushofer]

What exactly do you mean by "antisymmetric components of the vierbein"? The vierbein has two different components, namely a flat and a curved one, so it doesn't really make sense to take the antisymmetric part of it; then you're already talking about the metric, isn't it? :)

 

[andrien]

I don't think it is appropriate to say that because of two indices on some stress-energy tensor type thing,one can say that it is spin 2.One say that with electromagnetic field the origin is charge current which has one index so it represents spin-1.this is really not right.If I remember it, then in feynman lectures on gravitation it is pointed out that spin 2 is the lowest possible spin which can be chosen,and is satisfactory.

 

[haael]

The question was about the difference between Newton and Einstein theories of gravity. In Newton's the gravitational potential is scalar. Why is it a tensor in Einstein's?

One simple answer might be like this:
In Newton's gravity the source of gravitational interaction is mass. Mass is a scalar. To describe an interaction intensity between two scalars, we need one scalar.

In Einstein's gravity the source of interaction is the whole energy-momentum 4-vector. To describe interaction between components of two 4-vectors we need a 4-tensor. Some other arguments are needed to show that the tensor must be symmetric.

This is not a mathematical derivation, rather an intuitive explanation. The very same line may be answered to a question why interactions of spinors are described by gauge-covariant vector fields.

 

[dextercioby]

What exactly do you mean by "antisymmetric components of the vierbein"? The vierbein has two different components, namely a flat and a curved one, so it doesn't really make sense to take the antisymmetric part of it; then you're already talking about the metric, isn't it? :)

The linearized vierbein (the quotation left the essential word <linearized> out) is a genuine 2nd rank tensor in flat spacetime. It has a 0 + 1 + 2 spin content wrt both the connected component of O(1,3) and SL(2,C).