Dear Pmb,

Gravitational time dilation equation is:

[tex]t(m,r) = t \cdot \sqrt{ 1- \frac{2 \cdot G \cdot m}{c^2 \cdot r}}[/tex]

We have a body with a mass [tex]m_1[/tex]. We observe some point which is at a distance [tex]r_1[/tex] from that body. The time equation at that point is

[tex]t_1 = t \cdot f(m_1, r_1)[/tex]

If another body with a mass [tex]m_2[/tex] comes (from very far away) at some distance [tex]r_2[/tex] from the observed point, the time equation at that point becomes

[tex]t_2 = t_1 \cdot f(m_2, r_2) = t \cdot f(m_1, r_1) \cdot f(m_2, r_2)[/tex]

So, [tex]N[/tex] bodies will produce the following time behavior at the observed point:

[tex]\displaystyle \tau = t \cdot \prod_{i=1}^N f(m_i, r_i) \Rightarrow \frac{\tau}{t} = \prod_{i=1}^N f(m_i, r_i) = \prod_{i=1}^N \sqrt{1 - \frac{2 \cdot G \cdot m_i}{c^2 \cdot r_i}}[/tex]

[tex]r_i[/tex] is the distance of the i-th body from the observed point

Hence, the superimposed time-behavior at some point which is equally distant from two or more objects **will not be the same** as the time-behavior produced at the same distance from one object with the mass equal to the mass-sum of those two or more objects:

[tex]\frac{\tau}{t} = \sqrt{1 - \frac{2 \cdot G \cdot \sum_{i=1}^N {m_i}}{c^2 \cdot r }} [/tex]

How does this fit with the continuity principle, that is, with the mass (energy) conservation principle?