The Lagrangian of a tachyonic scalar field T can be expressed as
LT = - V(T)( 1 - du T du T)1/2
When T only varies with the cosmological time t, the energy of density is
rhoT = V(T)( 1 - dtT dtT)-1/2
where dtT is the tachyonic velocity.
Velocity depends on the path in curved spacetime. Cf. https://arxiv.org/abs/gr-qc/9712019
And the statement complementing the question isn't relevant to the question, since it doesn't show why the field T would describe an instability-which is what a tachyon does.
particle having its velocity less than that of light is called mass particle and mass is>0
........................................equal to,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.photon.............................= 0
........................................more.........................................tachyon...........................< 0
In its evolution the speed of the particle depends on the curvature of space time, but not so its maximum velocity, which given the shape of the energy-moment tensor can not be greater than that of light. Where was the tachyonic way of life?
The velocity (as well as its norm, the speed) doesn't depend just on the curvature, but on the path used, in curved spacetime. This-and more- is explained in any course in general relativity, cf. the one linked to above, for instance. There isn't any such notion as the shape of the energy-momentum tensor. It would be better to study a textbook or lecture notes on general relativity-there's no point trying to guess the answer, especially since the question doesn't make sense. So it's useful to understand what's the meaningful question to ask.
Stam Nicolis
Can you explain why the Lagrangian of the tachyonic scalar field in the FRW metric has the form expressed in the question above? Can you deduce why the square of the Lagrangian has that form and not the opposite in sign through the study of a textbook or lecture notes on general relativity? That is, why is
L2 = V2(T)( 1 - du T du T) and not
L2 = - V2(T)( 1 - du T du T) ?
Such a problem might be circomvented upon dispensing with secondary concepts as metrics, length, time intervals, velociity etc,. beginning with actual problem statemet. See
http://www.ptep-online.com/complete/PiP-2016-02.pdf
Well the short answer is that the potential term of the first expression is negative definite and of the second is positive definite.
A particle's a tachyon if it has imaginary mass-its mass squared is negative-and it's a ghost if it has a negative definite kinetic term. So one has to be careful how the metric's signature is defined.
The longer answer involves writing down the Lagrangian for a scalar field in curved spacetime and computing the mass of its excitations through the asymptotic behavior of the energy-momentum tensor.
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