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If negative mass really existed and somehow a very fast traveling negative mass object reached near the black hole's event horizon.

What would happen when it crosses the event horizon? According to Newton's law the negative mass should be repelled and only an object that can travel faster than light can escape a black hole. So, will the object be ejected out of the event horizon with a speed greater than light and that would lead to the object's mass becoming -∞ (minus infinity) according to theory of relativity and disobey laws of physics as we know it.

Any Explanations?

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"According to Newton's law the negative mass should be repelled" -- Nope, in both Newtonian physics and in general relativity, negative mass would be attracted gravitationally to positive mass, although negative mass would exert a repulsive gravitational effect on positive mass (but if the negative mass is small compared to the mass of the black hole this latter effect is negligible). In Newtonian physics this is not too difficult to derive, the Newtonian gravitational force law indicates the gravitational force vectors between a positive and negative mass would point away from each other, so the positive mass is obviously repelled, but for the negative mass the acceleration is in the opposite direction of the force due to the negative mass in F=ma, so the negative mass is attracted. In general relativity the analysis is obviously more complicated, but Hermann Bondi showed negative mass would have the same basic properties in GR, see this article. Note that if negative mass didn't fall downwards in gravity just like positive mass this would be a violation of the equivalence principle, since being in a chamber at rest in a gravitational field is supposed to be equivalent to being in a chamber accelerating in deep space, and if you let go of both a positive and negative mass in such a chamber they should naturally just move inertially while the floor of the chamber accelerates up to meet them.

The situation of negative mass falling into a black hole does have one important consequence though, in GR it's the only way for the event horizon of a black hole to shrink rather than expand, and for this reason a dynamical black hole metric (the Vaidya metric) with negative mass falling into it is sometimes used when trying to model the long-term behavior of a black hole that is "evaporating" due to continually emitting Hawking radiation (since this is a quantum effect, and general relativity is not fully compatible with quantum mechanics, this evaporation should ultimately require a full theory of quantum gravity to model it completely accurately, but it seems reasonable to expect that the earlier stages of evaporation, before the size of the black hole and the energy density approach the Planck scale where quantum gravity effects are expected to become significant, should have some close analogue in classical general relativity). See for example the paper here, whose abstract says "the black hole evaporation due to the Hawking radiation that is modeled by the Vaidya metric with a negative mass", or section IV of this paper which uses the Vaidya spacetime to model a black hole and says on p. 4 " This matter energy is negative near the event horizon. In the dynamical horizon equation, if black hole absorbs negative energy, black hole radius decreases. This is one of the motivations to use the negative energy tensor."

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    $\begingroup$ If negative mass is attracted to positive mass, while positive mass is repelled from negative mass, can't you tie a string between them and set up a perpetual motion machine? The combined object will accelerate because both objects will be accelerated in the direction of the positive mass. Of course, this doesn't actually increase the energy of the system because the combined object has zero mass. $\endgroup$ – Peter Shor Dec 15 '14 at 18:13
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    $\begingroup$ @Peter Shor - Yes, this would be a weird consequence of negative mass--as you point out it wouldn't violate conservation of energy, likewise with conservation of momentum. It's mentioned by John Cramer in the article I linked to: "There is a curious corollary of this result, which Bondi pointed out in his paper. Consider a pair of equal and opposite positive and a negative mass placed close to each other. The negative mass is attracted to the positive mass, while the positive mass is repelled by the negative mass." $\endgroup$ – Hypnosifl Dec 15 '14 at 18:32
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    $\begingroup$ (continued) "Thus the two masses will experience equal forces and accelerations in the same direction (in violation of Newton's third law) and the system of two particles will accelerate, seemingly without limit. The negative mass will chase the positive mass with constant acceleration." I would quibble with Cramer's statement that this violates Newton's third law though, since it's still true that $m_1 a_1 = - m_2 a_2$--the force vectors do point in opposite directions, it's just that the negative mass accelerates in the opposite direction of its force vector. $\endgroup$ – Hypnosifl Dec 15 '14 at 18:32

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