In general relativity, is it proper to say that mass curves the three spatial dimensions and also that mass curves the fourth dimension? [duplicate]

Question

In general relativity, is it proper to say that mass curves the three spatial dimensions (x1, x2, x3) and also that mass curves the fourth dimension x4?

Answer 1

As @Jaywalker said you are essentially right. But careful that space and time are mixed in in General Relativity(GR), you can have coordinate transformations that mix the two.

Also, just as an aside, it is not just mass the curves spacetime, it is any form of energy, momentum and stress also. The source term is called the stress energy tensor. The diagonal components are energy and momentum.

But space and time have limits on how much they can mix. There are 4 dimensional (in spacetime) distances, call them separate actions between two spacetime events (event is a spatial point at a specific time, so x,y,z,t), and they can be spacelike, timelike or lightlike. If light like it means only light could get from one to the other. If timelike it means you and I could. If spacelike it means we can never travel fast enough to get from one to the other, so they are causally unrelated. So you can only transform coordinates and keep saying time and space to a certain extent.

So, it is not always correct to say it curves space or time, though the terms are loosely used in more popular writings.

But indeed you can measure cases where time is 'bent' and where space is 'bent'. For time it means that time runs faster or slower in that gravitational field. GR predicts it. In the earth's gravitational field, and for instance similarly for Black Hole (a much bigger effect) time runs slower for someone close to the surface (of the earth, or at the Black Hole horizon) that further away. It's called time dilation. Near the earth, the GPS satellite clocks run faster than a clock on the surface of the earth. Outside a Black Hole's horizon, someone close to the horizon with a clock has that clock run much slower than someone pretty far away. And interestingly, GR fixes this problem in that the observer closer to the earth, or closer to the Black Hole horizon, still Experiences time running normally, never notices any anomaly. Each observer has his/her clock, and body times, and everything he/she experiences, just happening at a normal rate. GR explains this, calling it their proper time. Proper time is invariant. Still, that person will notice other things happening faster, they will notice an observer further away, aging faster. see Wikipedia at https://en.m.wikipedia.org/wiki/Gravitational_time_dilation

Cosmology is a different example. The standard model for the universe is called a Robertson-Walker-Friedman-Lamaitre model (different people contributed different aspects). Spacetime expands, and the physical distances between far-off galaxies increases, so space is expanding. Not only that, space can have positive, negative or zero curvature - though it seems observationally that it very close to zero. There are other cases, for instance outside a rotating Black Hole, space is dragged around with the rotation.

But in general to have any curvature in GR one has to have curvature along two different spacetime directions. Just time curvature is not possible, and curvature in only one spatial direction also not. You really have to go by the math, and calculate the effects. See for instance a similar question asked a few years ago in Physics Stack Exchange, and some of the examples and math. See it at How do spatial curvature and temporal curvature differ?

Answer 2

Yes. The essence of the gravitational attraction is that movement on the time coordinate transforms to movement on a spacelike one.

It happens on all the four coordinates ($x_{1..4}$).