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

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?

## marked as duplicate by ZeroTheHero, Jon Custer, Kyle Kanos, Prahar, David HammenMay 11 '17 at 22:44

• I've removed a comment discussion that seemed to be deteriorating. Please use comments to suggest clarifications and improvements to the question; use answers to provide (complete or partial) answers to questions; use Physics Chat to have discussions. In all cases, remember to be nice. – rob May 3 '17 at 1:11
• Let me remind people of the "Be nice." rule, and in particular that casting aspersions on another users statement on the basis of level of education is not only not nice, it's deeply unsound logic. – dmckee May 3 '17 at 1:42
• Individual space-time dimensions DO NOT CURVE. It is a physically, mathematically and logically meaningless thing to say. Spacetime, as a whole, can have curvature. Dimensions do not have curvature. – Bence Racskó May 4 '17 at 13:16

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?

• Thank you @BobBee ! So how would you answer the question? In general relativity, is it proper to say that mass curves the three spatial dimensions and also that mass curves the fourth dimension? – girlphysicsmajor May 3 '17 at 2:00
• No, at least if talking scientifically. More correctly, that all forms of mass-energy curves spacetime. How it curves it, calculate. It's been said, but maybe not an exact quote: matter tells spacetime how to curve, and spacetime tells matter how to move. In specific conditions you can define a good useful time coordinate, or spatial sections, and then how those behave. But curvature is defined by invariant quantities that do not depend on which coordinate system one uses – Bob Bee May 3 '17 at 2:10
• So would @BobBee say that, scientifically speaking, "it is FALSE to say that mass curves the three spatial dimensions and also that mass curves the fourth dimension in general relativity." --BobBee – girlphysicsmajor May 3 '17 at 2:12
• Mass can curve everything. Not a lawyer. – Bob Bee May 3 '17 at 2:17

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 ($x_{1..4}$).