Relative mass and Gravity

Question

According to relative mass effect predicted by Einstein's relativity theory, the mass of an object increases with increase in it velocity given by the equation

$$ M = \frac{M_0}{\sqrt{(1 - v^2 / c^2)}}. $$ This is also one of the reason why an object can never attain the speed of light. This phenomena has also been experimentally observed in particle accelerators. My question is, with increase in mass of the objet say at 98% speed of light, will their be an increase in its gravitational strength or Field around the object? If so, has this been experimentally observed and confirmed?

Answer 1

No. The kinetic energy of an object does not gravitate, only the rest mass, $M_0^2 = M^2- p^2/c^2$.

(The formula you wrote above for $M$ is a little obsolete. It is the rest mass plus the kinetic energy of the object, from the famous formula $E=Mc^2$. Nowadays people would just call it $E/c^2$, and use the symbol $M$ to mean $M_0$.)

As an example of this answer, consider the stress-tensor for a pressureless fluid: $$T^{\mu\nu} = \rho u^\mu u^\nu.$$ Now the Einstein equation says $$R= -4\pi G T,$$ and here $$T={\rm tr}T^{\mu\nu} = g_{\mu\nu}T^{\mu\nu} = \rho u^2 = \rho$$ where $u^2=1$ by normalization of 4-velocity. Thus the curvature scalar is always given by $-4\pi G\rho$, independent of the object's velocity.