Is the time "direction" in General Relativity equivalent to a spatial volume

Question

Most G.R.textbooks introduce time as an extra dimension, i.e. -ict. (see EDIT below for clarification). So although I can not mentally imagine this, I think of it as an extra line, "orthogonal" to the 3 normal space dimensions.

Then the text books introduce the equivalence principle and use it, involving, if I understand it correctly, purely the time direction, to illustrate the deflection by the Sun of say, a photon from a distant object and they state that, without also allowing for the curvature of space near a massive object, we will measure only half of the actual deflection observed.

So can I take it that my naive idea of a "time" as simply a linear "direction" is actually equivalent to a volume of space, as the deflection of a photon through time is 1/2 of it's actual total deflection?

If my assumption is incorrrect, could somebody explain more regarding the subtleties involved in the physical intuition behind 4-D spacetime.

I hope this question is clearly put.

Also, I see a possible duplicate in What's the difference between the equivalence principle and curvature of spacetime? and although I'm not sure it answers my question directly, I shall delete my question if necessary.

EDIT: If you see ict in a textbook on G.R, it's time for a new textbook, as, following the points in the comments below, this terminology is both outdated and unnecessarily confusing, at least for me.

This post might help a bit. It's not exactly what you were asking, but it might give better insight and allow you to ask the question more directly — Jim, May 19 '15 at 13:39
When you say time is equivalent to a volume of space are you thinking of the difference between vectors and one forms? As in a vector can be visualized as an arrow through space time, and a one form like a series of parallel volumes of space that the arrow penetrates. — mmesser314, May 19 '15 at 13:58
@ACuriousMind thank you for your time, the book I am reading is Cheng, Ta Pei, published 2005 and true enough, he just uses -ct in the metric. Perhaps my question was not clear and I am thinking (wrongly) in physical rather than math terms. I do know the basic ideas behind a manifold and it's dimensions. As far as co-ords on it go, I know you can use whatever set is most useful. I will read up more on manifolds, I now have a glimmer of light about your point about regarding spacetime in this way rather than in the physical way my question was phrased and more reading will help. — , May 19 '15 at 13:59
@mmesser314 no thank you, I know the basics of differential forms versus vectors but ACuriousMind has 100% pointed me in the right direction in regarding this problem and I shall follow though on that. — , May 19 '15 at 14:02
From a physical point of view time is nothing like the spatial directions, as an object can only move along in a single direction and at a "fixed rate". It is an interesting artifact of the theory that it can be treated SIMILAR to a spatial dimension, but I wouldn't go overboard identifying them too closely, otherwise you are missing most of what nature is really trying to tell you. — CuriousOne, May 19 '15 at 18:37
@CuriousOne thank you, yes I do realise, after reading the above comments, including your own, that trying to impose a physical picture to non-intuitive notions can/will lead you badly astray. My current mental picture of spacetime is as badly skewed as those tv programs showing you the big bang as a rather large fireball, from the outside:). The manifold idea is the one I will follow, with possibly a tesseract based physical picture coming closest to reality and even then I,m probably wrong to even try. — , May 19 '15 at 20:39

score 2 · Accepted Answer · answered May 19 '15 at 20:56

Most G.R.textbooks introduce time as an extra dimension... So although I can not mentally imagine this, I think of it as an extra line, "orthogonal" to the 3 normal space dimensions.

Don't. It's a dimension in the sense of measure, not in the sense of freedom of motion: I can hop forward a metre but you can't hop forward a second. Think about what a clock really does. It doesn't literally measure the flow of time like some kind of cosmic gas meter. It features some kind of regular cyclical internal "local" motion, be it of cogs or a crystal, and it shows you a cumulative result that you call the time. Then think about what we're really doing with all this. We're modelling motion through space "over time". Einstein gave us the equations of motion. That extra dimension is there because ours is a world of space and motion. When you look up at the sky you don't see worldlines and light cones, you see planes and meteors and planets and stars and light, and everything is moving. Have a look at A World without Time: The Forgotten Legacy of Godel and Einstein. It's like time exists like heat exists. Temperature is an average measure of motion, time is a cumulative measure of motion, and just as you can't literally climb to a higher temperature, you can't literally travel to another time. Because you can't move through a measure of motion.

Then the text books introduce the equivalence principle

The equivalence principle was an "enabling" principle, wherein Einstein's happiest moment was when he realised the falling man was not subject to a force. However he said this in 1916:

"This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes".

That's because the equivalence principle applies to an infinitesimal region only. So it doesn't apply to the room you're in. Being in a real (="quite special") gravitational field is like accelerating through space, but it isn't exactly the same.

and use it, involving, if I understand it correctly, purely the time direction, to illustrate the deflection by the Sun of say, a photon from a distant object

Light curves because the speed of light varies with position. Search the Einstein digital papers for Einstein saying that. Also see Shapiro's 4th test of General Relativity: "the speed of a light wave depends on the strength of the gravitational potential along its path". You can liken this variable coordinate speed of light to a "tilt" in your time dimension, or "spacetime tilt", see this article for a mention of tilted light cones. But as far as I know the spatial element to the Shapiro delay contributes only a tenth of the total. As far as I know light is deflected twice as much as matter because of the wave nature of matter.

and they state that, without also allowing for the curvature of space near a massive object, we will measure only half of the actual deflection observed.

Can you give me a reference? It might be an idea if you asked a dedicated question on this.

So can I take it that my naive idea of a "time" as simply a linear "direction" is actually equivalent to a volume of space, as the deflection of a photon through time is 1/2 of it's actual total deflection?

I'm sorry, but what you're saying doesn't sound right.

If my assumption is incorrect, could somebody explain more regarding the subtleties involved in the physical intuition behind 4-D spacetime.

4D spacetime is a mathematical model for motion through space over time, starting with the motion of light. IMHO it's easier than you think.

score 1 · Answer 2 · answered May 19 '15 at 14:13

Your understanding of time is roughly correct - it is another direction, "orthogonal" to our usual 3 space directions. But keep in mind that you can think of it purely mathematically as well. The position of an object as a function of time can be written as $x(t)$, $y(t)$, and $z(t)$, but why not write that exact same thing in four dimensions? $$(t,x,y,z)$$ Having both an intuitive understanding as well as a formal understanding is important.

The connection to volume is not right, however. It's not "adding time as a forth dimension" which predicts the deflection of light rays from the Sun. Understanding spacetime as a single 4-dimension entity is Special Relativity, but the bending of light comes from General Relativity. Mass "bends" spacetime, and changes distances which objects travel. Essentially, in flat space we have the Pythagorean theorem (with the negative sign for time): $$ds^2=-dt^2+dx^2+dy^2+dz^2$$ But in curved space we have a metric which modifies the Pythagorean theorem, giving us a more general metric (pick any metric in your text for examples). It is the bending of spacetime ("Changing of the Pythagorean theorem") which causes the deflection, not any intuitive understanding of the time direction.

thank you, I would upvote but no reputation currently. – May 19 '15 at 14:18 — , May 19 '15 at 14:18

Is the time "direction" in General Relativity equivalent to a spatial volume

2 Answers2