# Can space and time separately be curved?

How can I imagine curved time, if it is not a part of four dimensional spacetime? Similarly for space.

What are the measurable, observable consequences of these two phenomena in a laboratory or in everyday life?

• Space & time are not separate, so your question doesn't really make sense. – Kyle Kanos Oct 10 '15 at 11:15
• You cannot consider a hyperplane in $R^4$? – user2925716 Oct 10 '15 at 12:03
• Possible duplicate of What's the difference between space and time? – user81619 Oct 10 '15 at 12:05
• I do not think that this question should be closed. See my answer below. – Valter Moretti Oct 10 '15 at 13:54
• The spatial metric in a rotating coordinate system has non-zero curvature although the space is flat! Furthermore the duplicate is not a duplicate but a only somewhat related question (and the answers there do not answer this question). – Sebastian Riese Oct 10 '15 at 17:30

This is not a trivial question, in my opinion. It applies to spacetimes where there is a natural distinction between space and time. Such spacetimes are called static spacetimes (even if this class could be enlarged to various directions, in particular including stationary spacetimes, describing the metric of rotating gravitational sources, and conformally static spacetimes as the cosmological models used in modern cosmology).

Sticking to static spacetimes, in particular the local spacetime around our Sun with a good approximation, the natural separation between space and time is due to the presence of a particular Killing symmetry which defines a natural time direction in spacetime. Here, $3$-spaces ortogonal to these temporal directions are the natural notion of physical space. Moreover, the fact that the metric is static implies that the metric properties of these natural spaces and of the curves describing the time evolution do not depend on the introduced notion of time. In this sense, it makes here sense to speak about two reciprocally independent geometries, regarding space and time respectively.

The metric takes the form, referring to adapted coordinates

$$ds^2 = g_{00}(\vec{x}) dt^2 + \sum_{i,j=1}^3g_{ij}(\vec{x}) dx^i dx^j\:.$$ Above $t=x^0$ is the said notion of time, $\vec{x}=(x^1,x^2,x^3)$ denote the spatial coordinates in the natural rest frame orthogonal to the natural notion of time. This orthogonality is mathematically represented by the absence of the terms $g_{0i}=g_{i0}$ of the metric. As I said, the temporal metric $g_{00}$ and the spatial one $g_{ij}$ ($i,j=1,2,3$) do not depend on the natural notion of time $t$, but depend on the place $\vec{x}$ in space.

It makes sense to study the geometrical properties of $g_{00}$ and $g_{ij}$ ($i,j=1,2,3$) separately. We can say that the physical space is curved if the metric $h= \sum_{i,j=1}^3g_{ij}(\vec{x}) dx^i dx^j$ cannot be transformed, by a change of coordinates, into the standard Euclidean one. In this case, some property of Euclidean geometry must fail to hold. For instance the sum of internal angles of a triangle constructed by spatial geodesics could be different from $\pi$, and the value may depend on the triangle itself. This property must not depend on time for it to be a spatial property.

The physical interpretation of the temporal geometry is more complicated. I cannot say that what I am about discussing technically means that time is curved''. However it should substantially provide an answer to your question.

Whenever $g_{00}(\vec{x}) \neq -1$ (the latter being the value in flat spacetime''), several physical phenomena arise. What we can do, for instance, is to compare the length of a natural time interval $\Delta t$ with the interval of time $\Delta \tau$ evaluated by an ideal clock at rest in a position $\vec{x}$ $$\Delta \tau = \sqrt{g_{00}(\vec{x})} \Delta t\:. \tag{2}$$ All physical phenomena are stationary with respect to $t$, not $\tau$. The fact that the geometry along time direction is not trivial and depends on the position $\vec{x}$, regarding propagation of light and taking (2) into account, gives rise to the known gravitational redshift.

A much more evident phenomenon due to the fact that $g_{00}(\vec{x}) \neq -1$, which takes place even if the spatial geometry is flat, is the acceleration of bodies in geodesical motion evaluated with respect to $t$. In other words, the fact that free falling bodies accelerate in a gravitational field, also including the motion of the planets around the Sun, in the relativistic framework, is due to the presence of a $g_{00}$ different from $-1$ and depending on the spatial position. In fact, up to multiplicative constants and with a physically sound approximation, $\nabla_{\vec{x}}g_{00}$ coincides with the classical gravitational acceleration $\vec{g}(\vec{x})$, at $\vec{x}$, which is responsible for the motion of planets in the Newtonian picture.

The question What makes a coordinate curved? asks whether it is possible for the time coordinate to be curved while the spatial coordinates are flat.

The discussion there is probably a little too involved for this question, but the summary is that no, you cannot have a coordinate system in which the curvature is only in the time coordinate.

Actually, this is a trivial question. Time cannot be curved independently, but space can be. Consider a piece of paper. You physically crushing the paper is an example of curving space but keeping time as it is.