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Boulware-Deser ghosts not a problem for General Lorentz Ether
#1
In the General Lorentz Ether, there is a parameter, \(\Upsilon\), which would give, for \(\Upsilon>0\), quite interesting effects, namely that it replaces the Big Bang by a Big Bounce, and stops the gravitational collapse. With \(\Upsilon<0\),  these effects would disappear, and the only effects different from GR would be nothing but some massless dark matter, without anything qualitatively much different.  So, there is some reason to prefer  \(\Upsilon>0\). 

Unfortunately, there was a serious counterargument against it, namely that in the local approximation around the vacuum solution the additional term with  \(\Upsilon>0\) looked like a particle with negative energy, a so-called ghost field.  Such ghost fields are considered the death penalty for a theory.  

I have nonetheless favored the  \(\Upsilon>0\) variant, and there was one argument which told me that the ghost is not as dangerous as it looks:  The ghost field would be ideal massless dark matter, which does not interact at all with any form of matter.  Once such dark matter does not interact with anything else, why should it be that deadly if it has to wrong energy sign?  Moreover, there was another argument, namely that not all field configurations for the additional "ghost field" would be allowed: The "ghost field" is the preferred time coordinate.  But not every function can be used as a time coordinate. 

Now I have tried to make this argument more precise and reached a lot more so that I can say now that the ghost field is really completely harmless.  

First, there is the additional argument that the classical energy is positive in GLE, because the energy-momentum tensor is the \(g^{\mu\nu}\sqrt{-g}\), which has the positive energy density \(g^{00}\sqrt{-g}=\rho>0\), which is simply the ether density, with zero in form of the GLE equations (which have a form similar to the Einstein equations, some \(T^{\mu\nu}-G^{\mu\nu}+F_{GLE}=0\)) added.  

Then, I have explicitly shown that the ghost mode has a quite small finite range where it defines a valid time coordinate.  And that one can compute the contribution of this term, and obtain a finite value for a finite volume.  Given that this holds independent of the momentum, there has to be even a maximal momentum - beyond this maximal momentum, even the first excited mode would no longer define a valid time coordinate.  

The paper can be found in arxiv:1711.09009.  For political reasons, I have added the observation that the same ideas can be used to save also other theories with Boulware-Deser ghost fields, in particular theories with massive gravity, and shown this explicitly for Logunov's "Relativistic Theory of Gravity".  This may be helpful for publication - it is one thing to reject a paper about an ether theory, and another one to reject one which shows a way to save a lot of theories of massive gravity from one important objection.  We will see.
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