Can the gravitational potential energy of an object ever be negative? ?

Please explain how.Can the gravitational potential energy of an object ever be negative? ?
Within Newtonian mechanics, gravity is described by a central conservative force field of the form: -





F = -dV/dr





The negative sign indicates that the force of gravity is an attractive one - towards the centre! Thus, the negative sign is a 'sign convention'!!!





Thus, Newton found the equation of universal gravitational attraction to be: -





F(r) = -G.m.M/r虏





Hence, by integration we can find the gravitational potential at a point V(r) as: -





V(r) = -G.m.M/r





It can also be shown that for a uniform sphere of mass 'M' and radius 'R', that at a distance 'r', from the centre, the potential V(r) is given by: -





V(r) = -G.m.M.(3.R虏 - r虏)/(2.R鲁)





Thus, the gravitational potential energy always has a negative sign within Newtonian theory and has a maximum value at the centre of a gravitating sphere of mass 'M' (see Reference 1).





However, Newton has not had the last 'word' upon gravitational theory!





After publishing his General Theory of Relativity in 1915, Einstein explored the consequences of his gravitational field equation.





G = 8蟺T





In a paper titled 'Kosmologische Betrachtungen zur allgemeinen Relativitatsctheorie' (Cosmological Considerations on the General Theory of Relativity) published in 1917, Einstein found that to model a steady state universe, which was the then favoured model, he had to bastardise his field equation into the parameterised form: -





G(渭谓) - 位g(渭谓) = -魏(T(渭谓) - 陆g(渭谓)T)





Where the constant(?) '位' is the cosmological constant or a 'fudge factor' that Einstein introduced to make his field equation work for a steady state universe. When Edwin Hubble discovered, in 1925, that the universe was expanding, Einstein commented that this constant was the greatest blunder of his life. It seems that Einstein ignored the implications, implicit, within his field equation that the universe is expanding (see Reference 2) !





However, a modern interpretation of this constant is that it represents a repulsive aspect to the gravitational force. The repulse force could be viewed as anti-gravity!





Wikipedia, the free encyclopaedia comments, 'In 1998, observations of type Ia supernovae (';one-A';) by the Supernova Cosmology Project at the Lawrence Berkeley National Laboratory and the High-z Supernova Search Team suggested that the expansion of the universe is accelerating'. this acceleration is attributed to the presence of 'dark energy'. However, Wikipedia, the free encyclopaedia further comments, 'The simplest explanation for dark energy is that it is simply the ';cost of having space';: that is, a volume of space has some intrinsic, fundamental energy. This is the cosmological constant, sometimes called Lambda (hence Lambda-CDM model) after the Greek letter 位, the symbol used to mathematically represent this quantity. Since energy and mass are related by E = mc虏, Einstein's theory of general relativity predicts that it will have a gravitational effect. It is sometimes called a vacuum energy because it is the energy density of empty vacuum.....The cosmological constant has negative pressure equal to its energy density and so causes the expansion of the universe to accelerate.'





By Newtonian analogy, a repulsive form of gravity must have a positive potential value!








I hope this is of some help!Can the gravitational potential energy of an object ever be negative? ?
Whether it is positive or negative depends upon what is taken as reference point. Consider a stone of mass m on earth of mass M. Assume that there is nothing else in the world. Where ever I take it, there will be some finite force acting on it due to earth towards its centre. I will have to do some work for taking it to infinite distance or very large distance from earth. If that hypotetical point is taken as zero for potential energy. Then it is obvious that the object will lose potential energy as it comes to any distance from earth, say r. The objectwill gain kineic energy equal to the loss of potential energy. Because that point is taken as zero the potential energy of the stone at distance r will be -GMm/r.