Yes, you are correct. Clocks can indeed measure both time and space curvature due to the phenomenon of time dilation predicted by the theory of general relativity.
According to general relativity, the presence of a gravitational field can cause time to pass at different rates depending on the strength of the gravitational field. This effect, known as gravitational time dilation, means that clocks closer to a massive object will tick more slowly compared to clocks farther away from it.
Additionally, general relativity also predicts that the curvature of spacetime due to gravity affects the spatial dimensions. In the presence of a strong gravitational field, spacetime becomes curved, and distances are altered. This curvature is what causes objects to follow curved paths in the presence of gravity.
Clocks that are in different gravitational fields or experiencing different accelerations will exhibit time dilation relative to each other. This effect has been observed and measured in various experiments, including high-precision atomic clocks.
For example, clocks on satellites in orbit around the Earth experience a slightly weaker gravitational field compared to clocks on the Earth's surface. As a result, the clocks in orbit tick slightly faster due to the lower gravitational time dilation. This effect needs to be accounted for in satellite navigation systems like GPS to ensure accurate positioning.
So, by measuring the time dilation between clocks in different gravitational fields, we indirectly measure the curvature of spacetime caused by gravity. This demonstrates the intimate connection between the measurement of time and the understanding of spacetime curvature in the framework of general relativity.