Gravitomagnetic effect in magnetized neutron stars

Rotating bodies in General Relativity produce frame dragging, also known as the gravitomagnetic effect in analogy with classical electromagnetism. In this work, we study the effect of magnetic field on the gravitomagnetic effect in neutron stars with poloidal geometry, which is produced as a result of its rotation. We show that the magnetic field has a non-negligible impact on frame dragging. The maximum effect of the magnetic field appears along the polar direction, where the frame-dragging frequency decreases with increase in magnetic field, and along the equatorial direction, where its magnitude increases. For intermediate angles, the effect of the magnetic field decreases, and goes through a minimum for a particular angular value at which magnetic field has no effect on gravitomagnetism. Beyond that particular angle gravitomagnetic effect increases with increasing magnetic field. We try to identify this `null region' for the case of magnetized neutron stars, both inside and outside, as a function of the magnetic field, and suggest a thought experiment to find the null region of a particular pulsar using the frame dragging effect.