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I'm a former physics major -- long long ago, no degree -- but do my best to read, understand and write about science. This one has been bugging me for a while.According to Einsteins General Relativity the presence of mass curves space which we have observed during eclipses and through gravitational lensing. So my question is (and maybe the assumption is wrong but...) If space is empty then what is being curved? Seems that it must be something, not nothing if it is being curved.

Let's step a bit back and see how the curvature of space can be introduced. The key point is the equivalence principle between gravitational field and the acceleration.

Let's consider an elevator that doesn't have any windows. The elevator moves along (say) 'z'-direction at an acceleration a=9.8m/s^2. This 'z' direction can be defined relative to an observer on Earth such that the elevator is moving perpendicular to Earth surface and away from Earth.…..In other words, the trajectory of the photon relative to the observer in elevator is not a straight line along a direction perpendicular to elevator but a curved one.

Next important ingredient is the equivalence principle: specifically, there is no way for the observer in the elevator to distinguish between moving at the uniform acceleration a=9.8m/s^2 or sitting on the surface of a planet under a gravitational field g=9.8m/s^2 (ignoring the very small tidal effect or the very small changes of g over the height of elevator).

However, if there is no experiment to distinguish between being uniformly accelerated at a=9.8m/s^2 [in empty space] and sitting at the surface of a planet under a field

Then is this 'curvature of space due to gravity' just a matter of pure geometry due to changing a system of coordinates [made of e.g. straight lines] with another system of coordinates [made of e.g. curved lines like those we can draw on a sphere]? Apparently yes, or so it seems to me.

That is incorrect. The key idea is tidal acceleration, not the equivalence principle

Curvature refers to entirely different phenomena. Its unfortunate that the term “curvature” has two meanings in this context since it confuses a lot of people.

This is also incorrect. An observer in such a field can tell that he’s not merely in an accelerating frame in flat spacetime by the presence of tidal acclerations

Sorry. But that’s all wrong. Fortunately there is a new text online which can explain all of this to you. See ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

I am not sure how the tidal acceleration has anything to do with the equivalence principle.

Why would we need the tidal acceleration to understand the space curvature?

Just out of curiosity: How would you explain the concept of space curvature in a more pedagogical/intuitive fashion?

What tidal acceleration? In the first case the observer is assumed to be in empty space sufficiently far from any source of gravitational field and moving with an acceleration a=9.8m/s^2.

I think it's important to make the distinction between the curvature of a particles trajectory and the intrinsic curvature of a manifold,to in this case space and spacetime.

Photons and particles in free fall follow the space curvature, right?

First off let’s be clear on one point. Einstein’s field equation of General Relativity relates mass to spacetime curvature, not merely spatial curvature.

If one wants see the curvature of space, then just follow the trajectory of a free fall particle.

I'm sorry but I don't know what "follow the space curvature" means.

QuoteI'm sorry but I don't know what "follow the space curvature" means.I meant so say that photons and free fall particles follows a geodesic.

The curvature is what we see when we map one kind of space-time grid onto a different kind of space-time grid. For analogy: Look what happens when you map the grid of log-log graph paper onto lin-lin paper, and vice versa.

Can space the thought in some sense as 'inhomogeneous' due to its intrinsically curvature [near planets]?

According to Einsteins General Relativity the presence of mass curves space which we have observed during eclipses and through gravitational lensing. So my question is (and maybe the assumption is wrong but...) If space is empty then what is being curved? Seems that it must be something, not nothing if it is being curved.

Think of building two concentric shells, a lower shell of reduced circumference rL and a higher shell of reduced circumference rH, such that the difference in reduced circumference rH-- rL equals 100 meters. Stand on the higher shell and lower a plumb bob, and for the first time measure directly the radial distance perpendicularly from the higher shell to the lower one. Will we measure a 100-meter radial distance between our two shells? We would if space were flat. But outside a massive body space is not flat. The relation between global differential dr and measured radial dierential distance d comes from the spacelike version of the Schwarzschild metric (3.6) with dt = d = 0.

Objection Are you refusing to answer my question? What CAUSES the discrepancy, the fact that the directly-measured distance between spherical shells is greater than the difference in r coordinates between these shells? WHY this discrepancy?ReplyA deep question! Fundamentally, this discrepancy shatters the notion of Euclidean space. We are faced with a weird measured result, which we can summarize with the statement, “Mass stretches space.” Your question “Why?” is not a scientific question, and science cannot answer it. We know only observed results and their derivation from general relativity. Does the following satisfy you? Space stretching causes the discrepancy! Section 391 3.8 exhibits one way to visualize this stretching.

Thanks everyone for your posts and discussion. Interesting stuff.

Would you like me to let you know when its finished?

QuoteWould you like me to let you know when its finished?Yes please!

The term spacetime curvature refers to the intrinsic curvature of a spacetime manifold as determined by the curvature tensor; also know as the Riemann tensor. It’s a term which is often misunderstood since in some cases the term curvature, when used unqualified, refers to the curvature of the worldline of a particle. Although Einstein used this term in his book Relativity: The Special and General Theory this referred to the curve and not spacetime. Spacetime curvature is a property of the spacetime manifold and not determined by trajectories of particles moving in said spacetime. When a free particle is moving in an inertial frame of reference the spatial trajectory will be that of a straight line. The particles worldline (i.e. spacetime curve) will also be straight. If I now change to a frame of reference moving uniformly relative to the first then the trajectory will still be a straight line. If I now change my frame of reference to that of one, which is accelerating relative to the original inertial then the particle, will be deflected and the spatial trajectory will be curved. This kind of curvature represents the curvature of the trajectory/curve and is unrelated to any property of the spacetime manifold.[tbd]

If space is empty then what is being curved?

It seems to me that noone in this forum has answered this question.

Everything will be crystal clear as soon as we can visualize four-dimensional curved spacetime. But we do not know anyone who can do this; we certainly cannot! So we compromise, we do our best to live with our limitations and develop intuition from the analogy to curved surfaces in space, such as the partial visualization of Schwarzschild geometry in the following sections.

This is pretty consistent with what I have found in my travels in physics.

It is possible that noone knows what spacetime curvature is all about but noone is gutsy enough to admit it!

In his book "Relativity Demistified", Wolfson states "We live, says Einstein, in a four-dimensional spacetime. The geometry of spacetime exhibits curvature and that spacetime curvature IS gravity".

Easy peesy lemon squeezy n'est pas.

I think not! Maybe all is needed is a great "explainer" to come along and elucidate the masses, or maybe the general understanding of the concepts involved has been lost in the mists of time.

In the meantime most of us will just continue to scratch our heads and wonder.

It is best to read as many different explanations to get different ways of viewing it.