If someone draw a M-R curve for a model of compact star and find maximum mass from the plot and corresponding radius from this plot where maximum mass attained, is it necessary to satisfy M/R ratio the Buchdahl's limit i.e., M/R is less than 4/9 ?
The answer to your question is: Yes, if the following conditions are satisfied:
1) The fluid distribution is spherically symmetric and static.
2) The physical and metric variables are regular everywhere within the fluid distribution.
3)The energy density is positive, and larger than (or equal to) the pressure.
4) The pressure gradient is negative.
5) The pressure is isotropic.
6) The fluid sphere is finite, outside which the spacetime is described by the Schwarzschild metric.
If you relax any of the above conditions, then you may have fluid distributions violating the Buchdahl’s limit.
So when I am finding maximum mass from M/R curve and from the curve allowable radius then also M/R should be less than 4/9...Is it correct?
The answer to your question is: Yes, if the following conditions are satisfied:
1) The fluid distribution is spherically symmetric and static.
2) The physical and metric variables are regular everywhere within the fluid distribution.
3)The energy density is positive, and larger than (or equal to) the pressure.
4) The pressure gradient is negative.
5) The pressure is isotropic.
6) The fluid sphere is finite, outside which the spacetime is described by the Schwarzschild metric.
If you relax any of the above conditions, then you may have fluid distributions violating the Buchdahl’s limit.
@Prof. Herrera: can we say that the Buchdahl limit is relaxed in the presence of charge seeing that we can treat charged compact fluids as anisotropic matter distributions up to some finite radius?
Yes you are right @Piyali Bhar , however in the regime of General Relativity. If the gravity theory is different than Einstein gravity, then the Buchdahl limit can be relaxed. For example in presence of extra dimension, the maximum value of 2M/R goes beyond 8/9 and reaches up to unity. This in fact may act as a possible testbed of extra dimension - if the ratio 2M/R is found to lie between 8/9 and 1 , then it may reflect the existence of extra dimension. For the demonstration of Buchdahl limit with extra dimension, see arXiv: 1801.04123
just stick to the corresponding Buchdahl limit for different cases like, isotropic, charged-isotropic, ansicotropic, charged-anisotropic, higher dimensions etc. For the corresponding theory, there is a corresponding modified limits.
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