# Two places having same gravitational acceleration - Do they also have same curvature of space time?

If there are two places (possibly on/above two different planets) having same gravitational acceleration (g), would it imply that the two places have same extent of space time curvature and will have same time dilation? Please answer with minimum required math, no need to put all the math, if lengthy math is involved.

• The density of the planet, therefore its radiud would also have an impact. If you squeeze the Earth to the size of a baseball it would become a black hole. – Peter R Feb 26 '16 at 21:14
• @PeterR: Baseballs are 9 mm? – Kyle Kanos Feb 26 '16 at 21:18
• small baseballs – Peter R Feb 26 '16 at 21:47
• I should have been more specific. The space-time interval under the Schartzchild solution has the Schwarzchild radius as one of the components, therefore, you could have the same gravitational acceleration for a more massive but less dense planet as compared to a denser but less massive planet. Each will have a different Schwarzchild radius. The important thing to remember is that mass of a planet increases with the cube of the radius while the gravitational acceleration decreases with the square of the redius. – Peter R Feb 26 '16 at 21:53
• @Peter-R: Looks like answer is - The two places can have different curvatures even with same gravitational acceleration. Right? – kpv Feb 26 '16 at 22:03

Since you mention planets in the question let's take the Schwarzschild metric that describes the geometry around a spherically symmetric object like (approximately) a planet. The acceleration measured by a stationary observer at a distance $r$ from the planet is:

$$a = \frac{GM}{r^2} \frac{1}{\sqrt{1-r_s/r}}$$

The time dilation measured by the same observer, relative to an observer at infinity is:

$$\frac{t_\infty}{\tau} = \frac{1}{\sqrt{1-\frac{r_s}{r}}}$$

So the ratio of the two is:

$$\frac{a}{t_\infty/\tau} = \frac{GM}{r^2}$$

And this varies with the mass of the object, so it is different for planets with different masses.

Although your question does specifically compare acceleration to time dilation you do also ask a more general question about whether the acceleration can be the same for different curvatures. However this can't be answered without defining what you mean by spacetime curvature. The curvature is described by a matrix (the metric tensor) not a single number so there isn't a simple comparison to make.