Jordan And Einstein Frames
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The
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
in scalar-tensor theory can be expressed in the Jordan frame or in the Einstein frame, which are field variables that stress different aspects of the gravitational field equations and the evolution equations of the matter fields. In the Jordan frame the scalar field or some function of it multiplies the
Ricci scalar In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...
in the Lagrangian and the matter is typically coupled minimally to the metric, whereas in the Einstein frame the Ricci scalar is not multiplied by the scalar field and the matter is coupled non-minimally. As a result, in the Einstein frame the field equations for the space-time metric resemble the
Einstein equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the ...
but test particles do not move on geodesics of the metric. On the other hand, in the Jordan frame test particles move on geodesics, but the field equations are very different from Einstein equations. The
causal structure In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified according to the types of causal structures they admit (''c ...
in both frames is always equivalent and the frames can be transformed into each other as convenient for the given application. Christopher Hill and Graham Ross have shown that there exist "gravitational contact terms" in the Jordan frame, whereby the action is modified by
graviton In theories of quantum gravity, the graviton is the hypothetical elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with re ...
exchange. This modification leads back to the Einstein frame as the effective theory. Contact interactions arise in
Feynman diagrams In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
when a vertex contains a power of the exchanged momentum, q^2, which then cancels against the
Feynman propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In ...
, 1/q^2, leading to a point-like interaction. This must be included as part of the effective action of the theory. When the contact term is included results for amplitudes in the Jordan frame will be equivalent to those in the Einstein frame, and results of physical calculations in the Jordan frame that omit the contact terms will generally be incorrect. This implies that the Jordan frame action is misleading, and the Einstein frame is uniquely correct for fully representing the physics.


Equations and physical interpretation

If we perform the
Weyl rescaling In theoretical physics, the Weyl transformation, named after German mathematician Hermann Weyl, is a local rescaling of the metric tensor: g_ \rightarrow e^ g_ which produces another metric in the same conformal class. A theory or an expressi ...
\tilde_=\Phi^ g_, then the Riemann and Ricci tensors are modified as follows. :\sqrt=\Phi^\sqrt :\tilde=\Phi^\left R + \frac\frac -\frac\left(\frac\right)^2 \right/math> As an example consider the transformation of a simple Scalar-tensor action with an arbitrary set of matter fields \psi_\mathrm coupled minimally to the curved background :S = \int d^dx \sqrt \Phi \tilde + S_\mathrm tilde_,\psi_\mathrm=\int d^dx \sqrt \left R + \frac\frac - \frac\left( \nabla\left(\ln \Phi \right) \right)^2\right+ S_\mathrm Phi^ g_,\psi_\mathrm/math> The tilde fields then correspond to quantities in the Jordan frame and the fields without the tilde correspond to fields in the Einstein frame. See that the matter action S_\mathrm changes only in the rescaling of the metric. The Jordan and Einstein frames are constructed to render certain parts of physical equations simpler which also gives the frames and the fields appearing in them particular physical interpretations. For instance, in the Einstein frame, the equations for the gravitational field will be of the form :R_ - \frac R g_= \mathrm\,. I.e., they can be interpreted as the usual
Einstein equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the ...
with particular sources on the right-hand side. Similarly, in the
Newtonian limit In physics, the Newtonian limit is a mathematical approximation applicable to physical systems exhibiting (1) weak gravitation, (2) objects moving slowly compared to the speed of light, and (3) slowly changing (or completely static) gravitational f ...
one would recover the Poisson equation for the Newtonian potential with separate source terms. However, by transforming to the Einstein frame the matter fields are now coupled not only to the background but also to the field \Phi which now acts as an effective potential. Specifically, an isolated test particle will experience a universal four-acceleration :a^\mu= \frac \frac(g^ + u^\mu u^\nu), where u^\mu is the particle four-velocity. I.e., no particle will be in free-fall in the Einstein frame. On the other hand, in the Jordan frame, all the matter fields \psi_\mathrm are coupled minimally to \tilde_ and isolated test particles will move on geodesics with respect to the metric \tilde_. This means that if we were to reconstruct the Riemann curvature tensor by measurements of geodesic deviation, we would in fact obtain the curvature tensor in the Jordan frame. When, on the other hand, we deduce on the presence of matter sources from gravitational lensing from the usual relativistic theory, we obtain the distribution of the matter sources in the sense of the Einstein frame.


Models

Jordan frame gravity can be used to calculate type IV singular bouncing cosmological evolution, to derive the type IV singularity.


See also

*
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
*
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...


References

* Valerio Faraoni, Edgard Gunzig, Pasquale Nardone, Conformal transformations in classical gravitational theories and in cosmology, ''Fundam. Cosm. Phys.'' 20(1999):121, . * Eanna E. Flanagan, The conformal frame freedom in theories of gravitation, ''Class. Q. Grav.'' 21(2004):3817, . General relativity Tensors {{relativity-stub