What is 'Curvature' of Spacetime

[cbd1]

We have described the distortion in spacetime which Einstein derived in GR as a "curvature" of spacetime. This is barely more descriptive than "warping" spacetime. I understand that what this means is that spacetime varies from being Euclidean, having distortion caused around objects of mass. But what is this curvature? What does it really mean logically?

Any experts want to share their rationalization of how they picture what the curvature of spacetime actually is in their own mind?

 

[pervect]

While one could go into the mathematics that define curvature (it may seem like a loosely defined concept, but it's not - but the mathematics of describing the Riemann tensor is not something one can realistically do in a short "feel-good" post), there may be a less technical alternative.

It is both necessary, and sufficient, to draw space-time diagrams on curved sheets of paper (rather than flat ones) to accurately describe the measurements of clocks and rulers in the presence of the gravitational field of a single massive body.

This is more of a teaching tool than the way that people actually compute things, but it's an interesting and not terribly technical way of talking about GR.

The particular approach I'm talking about is due to Donald Marolf, "Spacetime Embedding Diagrams for Black Holes", http://arxiv.org/abs/gr-qc/9806123.

In general, embedding diagrams are just a visualization aid for the mathematics. This approach is limited in what it can do - embedding diagrams aren't a replacement for learning the math, they are more or less just a visual aid. Conceptually, though, if you know how to draw a space-time diagram, and know what a Lorentz transform is, you just need to do the same on the curved surface presented by Marlof. If you _don't_ already know how to do a Loretnz transform :-(, some reading up on the basics of SR is going to be needed before this approach is going to be very helpful. You'll need a bit of understanding of SR first to understand GR.

To be just a bit more technical, on any sufficiently small area of Marolf's diagram, the curvature can be ignored, and you can consider yourself as working in a plane, where you do your space-time diagrams as usual, including the Lorentz transform. This would be the "tangent plane" to the surface.

On any larger diagram, you need to consider the effects of curvature, and replace the concept of "straight line" with "geodesic" to get anywhere. Unfortunately, Marolf's diagram won't tell you how to actually calculate geodesics.

 

[Dale]

Curvature in spacetime is simply the fact that two objects with 0 proper acceleration can still accelerate relative to each other.

 

[Daverz]

I think the problem for most physics students is that they don't have experience with the curvature of surfaces that one gets studying classical differential geometry (we all know about curvature of curves from calculus.) This intuition carries over to abstract manifolds. Most GR books will just try to make it seem plausible that the Riemann tensor is related to some kind of concept of curvature. The nice books will relate it back to Gaussian curvature, but that won't resonate if you don't have prior experience with what it means for surfaces. So my advice is to pick up a book like Struik's Lectures on Classical Differential Geometry that covers local surface theory (it won't matter for this purpose that the book is a little old fashioned).

 

[cbd1]

Thanks guys. I understand that the mathematics *describes* the curvature of space, but what about spacetime gives it the ability to "curve". Or I guess more at what I'm getting at is what does spacetime curve into? The questions does not need mathematics to answer; rather, it is conceptual. What is physically going on with this curvature? Do geodesics become relatively closer together towards the objects of mass? Could it be described as a compacting of space around objects of mass which results in this curvature?

 

[pervect]

Thanks guys. I understand that the mathematics *describes* the curvature of space, but what about spacetime gives it the ability to "curve". Or I guess more at what I'm getting at is what does spacetime curve into?

While it is possible to embed space-time into something with more dimensions, such an embedding is not necessary, nor is it unique, and it may in general require a fairly large number of dimensions. The example I gave embeds the r-t plane of the Schwazschild black hole into a 3-d space, needing only one more dimension, but in general you may need more.

So, even at the conceptual level, you can't get away totally from the math. The concept of curvature that we use in GR are not those of extrinsic geometry, but the intrinsic geometry as "ants living on the curved surface" do. We don't need to speculate about what lies "outside of space" to be able to see how it is curved.

One of the classic motivators to curved-space time is to consider the behavior of two clocks at different heights. Consider the following time-space diagram. Time runs from left to right, so AB is the time-like worldline of our first clock, and CD is the time-like worldline of a second clock. Because of gravitational time dilation, the two clocks don't run at the same rate.

(A)x------------------x (B)
    \                         
      \                          
       \                           
    (C)x-------------------x (D)

The first clock sends out a signal at event A, and again at event B. It records some time interval AB

The second clock receives them at times C and D. The jagged line from A to C is the space-time plot of the light signal that goes from the first clock at A to the second clock. There's another similar line from B to D.

We have a parallelogram on our space-time diagram. AB is parallel to CD, and AC is parallel to BD.

But because of gravitational time dilation, the length of AB, the elapsed time on the first clock, is not equal to the length of CD, the elapsed time on the second clock.

In a Euclidean geometry, we know that the opposite sides of parallelograms must be equal.

We therefore conclude that we can't draw our space-time diagram on a flat, Euclidean sheet of paper in the presence of gravity. (Unless our notion that the lines are parallel is somehow off, at least).

Rather than actually trying to draw our space-time diagrams on an actual curved surface, we instead introduce a metric, so we can draw them on a flat surface, with arbitrary coordinates, but still get the right answers for distance - much as we might make a flat map of the curved surface of the Earth. The distances on such a map won't be "to scale", but we can compensate for that with a metric.

So, the metric tells tie difference between the coordinate time, on our sheet of paper, and the actual time that arise on real clocks. The difference is ascribed to "gravitational time dilation".

The questions does not need mathematics to answer; rather, it is conceptual. What is physically going on with this curvature? Do geodesics become relatively closer together towards the objects of mass? Could it be described as a compacting of space around objects of mass which results in this curvature?

Two geodesics of the space-time from a large central mass (such as a black hole, the ubiquitous Schwarzschild solution) accelerate away from each other.

It turns out not to be quite so simple as a "compacting of space". Though such an idea would be an example of an embedding of a space-time in a space of higher dimension, and therefore a curved space-time.

There is one approach that manages to do a lot of what GR does without geometry (it provides equivalent results, though it assumes an underlying flat topology which is both it's strength and weakness at the same time).

Unfortunately it's rather technical on its own right. See for instancehttp://arxiv.org/abs/grqc/9512024" by John F. Donoghue

Einstein thought about these matters, and he decided that he didn't really need to know whether there was some sort of underlying geometry with different rulers and clocks that might make space-time "flat". You'll see this when he talks about whether or not we might think of our rulers "expanding with heat". He decided that it was just as valid to study the geometry of space-time using physical rulers and clocks rather than imaginary ones, and the result of this study is GR.

 

[andrewkirk]

Another example I find useful is that if you have three observers stationary relative to one another and equidistant from each other and each shines a laser at the other two then the angles between the two directions in which any of the users laser beams emits her laser beams will not be 60 degrees. The laser beams follow geodesics and the beams form an equilateral triangle but the angles are not 60 degrees because space is not flat.

Similarly, if you take two lasers L metres apart and shine them into space, in directions perpendicular to the line segment connecting the two lasers, and parallel to one another, then send an observer to stand in the path of the first laser a long way off and measure the distance to the beam of the second laser (ie the distance to the closest point in the path of the second laser beam to where she is standing), that distance will not equal L, as it would in flat space.

 

[LostConjugate]

When working with gravity you are generally working with large curved objects. It is not a surprise that the mathematical description of the field is curved.

 

[K^2]

Thanks guys. I understand that the mathematics *describes* the curvature of space, but what about spacetime gives it the ability to "curve". Or I guess more at what I'm getting at is what does spacetime curve into?

That's the whole point. It doesn't actually curve. That's just a visualization tool.

What we really see is changes in the metric of space-time. Metric determines distance between two events separated by infinitesimal distance in space and time. The reason why we call that curvature is because when you take a sheet of some material, to curve it, you must stretch or shrink the material at certain points, and that's very similar to what we are observing.

The problem is that the general solution to the Einstein Field Equation cannot be represented as curvature in any finite number of dimensions. So in all likelihood, the changes in metric are not due to actual deformations. Something else is going on.

Unfortunately, that's pretty much all we know about it. We have an equation that relates metric to energy stress tensor, and very little idea on why that is actually so.

 

[Q-reeus]

Thanks guys. I understand that the mathematics *describes* the curvature of space, but what about spacetime gives it the ability to "curve". Or I guess more at what I'm getting at is what does spacetime curve into? The questions does not need mathematics to answer; rather, it is conceptual. What is physically going on with this curvature? Do geodesics become relatively closer together towards the objects of mass? Could it be described as a compacting of space around objects of mass which results in this curvature?

Not sure if what you're really asking is "how can vacuum/empty space curve". If so I agree it is nonsensical, and as K^2 said "something else is going on". Having banished the notion of an aether in formulating SR, there is a quote somewhere to the effect that in GR Einstein slipped it back in another guise. I believe he stated that in a very low key way in 1920. Just how can matter here curve spacetime over there without introducing some notion of spacetime as an actual physical medium? Wasn't action-at-a-distance dispensed with for good by Faraday/Maxwell? Some theorists take seriously the belief spacetime has physical 'elastic' properties and claim doing so can 'improve' on GR; example:http://arxiv.org/abs/0911.3362" I toss that in as example of what's out there without endorsing.

 

[ZirkMan]

The problem is that the general solution to the Einstein Field Equation cannot be represented as curvature in any finite number of dimensions. So in all likelihood, the changes in metric are not due to actual deformations. Something else is going on.

Is this really true? I always though that the result of the GR equations are vectors (tensors) values which determine the size and direction of curvature for the given point of spacetime in presence of energy. The sum of those tensors will give a curved surface which we cannot directly imagine because it is curved in 4D instead of our familiar 3D and thus the problem with conceptual imaginative understanding. So this is wrong? Science popularization where this explanation is often used is (yet again!) misleading instead of explaining the reality?

I also want to thank you and pervect (and others too) for the effort to try to explain this very important subject with insights that those of us with lesser mathematical training would not be able to get otherwise. This thread has already opened my eyes to new possibilities of what I have been considering a dead end.