The curvature of space and curvature of spacetime

[PainterGuy]

Hi,

The quote below has been taken from this article, https://math.ucr.edu/home/baez/einstein/node2.html, which I came across.

Similarly, in general relativity gravity is not really a `force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial. If you toss a ball, it follows a parabolic path. This is far from being a geodesic in space: space is curved by the Earth's gravitational field, but it is certainly not so curved as all that! The point is that while the ball moves a short distance in space, it moves an enormous distance in time, since one second equals about 300,000 kilometers in units where

. This allows a slight amount of spacetime curvature to have a noticeable effect.

The quote doesn't make any sense to me, especially the part in boldface. Could you please help me with it?

 

[Ibix]

It's a bit misleading to say that the ball moves at all if you are taking the four dimensional view. There is only the worldline, one "slice" through which is what we call "the ball, now". But the point he's making is that "the ball now" and "the ball one second later" are one second apart along the ball worldline. The constant $c$ is the natural scaling between units of distance and units of time, so one second and one light second (300,000km) are the same size.

The relevance of this is that the ball's spatial position has changed a tiny amount between those two events on its worldline, just a few meters, but hundreds of thousands of kilometres in the time direction. So curvature involving timelike directions is much more important.

In fact, completely neglecting spatial curvature is a big part of approximating Newton's theory of gravity out of GR. It's only in fairly extreme circumstances with very precise measurements that you need worry about spatial curvature.

 

[PainterGuy]

Thank you!

It still doesn't make much sense to me but it's my own shortcomings to be blamed.

The relevance of this is that the ball's spatial position has changed a tiny amount between those two events on its worldline, just a few meters, but hundreds of thousands of kilometres in the time direction. So curvature involving timelike directions is much more important.

How is the ball moving at hundreds of thousands of kilometers in the time direction?

Th quote below says that Milky Way is moving at the speed of 2.1 million miles per hour is 938.784 km/s. If this is assumed that everything on Earth is moving at almost 939 km/s even then it's hundreds of thousands of kilometers. Could you please guide me with this?

And how fast is the Milky Way Galaxy moving? The speed turns out to be an astounding 1.3 million miles per hour (2.1 million km/hr)! We are moving roughly in the direction on the sky that is defined by the constellations of Leo and Virgo.

Source: https://nightsky.jpl.nasa.gov/docs/HowFast.pdfHelpful links:
/watch?v=F5PfjsPdBzg (put www.youtube.com in front)

 

[PeterDonis]

How is the ball moving at hundreds of thousands of kilometers in the time direction?

One second of time is three hundred thousand kilometers (one light second) in the time direction.

 

[malawi_glenn]

How is the ball moving at hundreds of thousands of kilometers in the time direction?

In the time direction. Read again. In the time direction.

Here is a space-time diagram for an object that is at rest in a certain reference frame (vertical line)
One can interpert this as the object is moving in the time direction.

Here is a space-time diagram for a beam of light, traveling in the x-direction in that same reference frame (45 degree angle straight line)

Here is a space-time diagram for an object moving with less than speed of light

If the velocity of an object in that frame is say 1 m/s in the x-direction, then delta x = 1 m when delta t = 1s, but in the time direction this is 300 000 000 m.

 

[PeroK]

"The point is that while the ball moves a short distance in space, it moves an enormous distance in time, since one second equals about 300,000 kilometers in units where

. This allows a slight amount of spacetime curvature to have a noticeable effect."

This just shows that you can't do physics by soundbite!

 

[PainterGuy]

One second of time is three hundred thousand kilometers (one light second) in the time direction.

I think one can define 'c' as anything but does the 'physical' ball really move hundreds of thousands of kilometers in any 'real' time direction over its one second journey in the air?

 

[256bits]

but it is certainly not so curved as all that!The quote doesn't make any sense to me, especially the part in boldface. Could you please help me with it?

This might help.
To you, when the ball is tossed in a parabolic path, it looks as if there is a lot of curvature from the gravitational pull of the earth.
Then, toss a bit of light, and the path looks quite straight, But, the light will also be attracted to the Earth and fall towards the Earth at the same rate as the ball. Since the light has traveled 300,000 km /sec, the parabolic path of the light is barely noticeable, at least not to the naked eye.

 

[malawi_glenn]

I think one can define 'c' as anything but does the 'physical' ball really move hundreds of thousands of kilometers in any 'real' time direction over its one second journey in the air?

In a Minkowski spacetime diagram yes.

 

[PainterGuy]

Then, toss a bit of light, and the path looks quite straight, But, the light will also be attracted to the Earth and fall towards the Earth at the same rate as the ball. Since the light has traveled 300,000 km /sec, the parabolic path of the light is barely noticeable, at least not to the naked eye.

Thanks! Are you suggest a ball made up of pure light?! I think the light won't fall back to Earth but yes, it'd get red-shifted on its way moving away from earth.

In a Minkowski spacetime diagram yes.

Thank you! So, according to Minkowski spacetime diagram, the 'physical' ball really moves hundreds of thousands of kilometers in the time direction over its one second journey in the air. The time direction being "ct" with unit of meter or light-second.

But what does happen in reality? That's the point I was trying to make in post #3. A ball is thrown upward from earth, it goes up and down following a parabolic path during the period of one second. Where is the journey of hundreds of thousands of kilometers of 'real' distance? Could you please guide me?

 

[JandeWandelaar]

The question is though why we move from past to future and not from future to past (which would make us feel like a puppet with a clockwork, unwinding from future to a big crunch). But this is a different question all together.

 

[PeterDonis]

All particles move at the speed of light through spacetime. A photon moves with c through space, a particle standing still moves with c through time.

This is a fairly common pop science claim but it is wrong.

The correct statements that this wrong pop science claim is a garbled version of are:

(1) An object with nonzero rest mass moves on a timelike worldline, and its 4-velocity vector has a magnitude of $c$ (in conventional units, or $1$ in "natural" units in which $c = 1$). This could be interpreted, generously, as the object's "speed through spacetime" being $c$, but this only works for objects with nonzero rest mass.

(2) An object with zero rest mass (like a photon, or more correctly a light pulse) moves on a null worldline, and the tangent vector to its worldline has magnitude zero (not $c$). The object's 3-velocity in any inertial frame has magnitude $c$ (or $1$ in "natural" units), but this is not a "speed through spacetime" in any sense. It is a "speed through space" for the chosen inertial frame's definition of "space", but "space" is frame-dependent and there is no way to translate this into a "speed through spacetime".

 

[PeterDonis]

This is a fairly common pop science claim but it is wrong.

Moderator's note: A number of posts arguing about this wrong pop science claim have been deleted.

 

[JandeWandelaar]

This could be interpreted, generously, as the object's "speed through spacetime" being c, but this only works for objects with nonzero rest mass.

Exactly. An object has no speed or motion through spacetime at all.

 

[PeterDonis]

The correct statements that this wrong pop science claim is a garbled version of are

I should also add, to forestall any discussion of another common wrong pop science claim:

(3) There is no continuity between the tangent vectors to worldlines for objects with nonzero rest mass (timelike worldlines) and objects with zero rest mass (null worldlines). You cannot gradually change frames and say that a timelike object moves "more through space" and "less through time" as its 3-velocity in your chosen frame increases. Nor can you say that a light pulse is a "limiting case" of this where the object moves entirely through "space" and not at all through "time". This model is based on an assumed continuity between timelike and null worldlines that does not exist. The action of Lorentz transformations on timelike and null vectors is fundamentally different (it hyperbolically rotates the former but dilates the latter).

 

[PeterDonis]

Exactly. An object has no speed or motion through spacetime at all.

That's not what you said in the post that started the series of posts that I have now deleted. I would strongly suggest that you stop posting on this topic altogether until you understand it better. You are very close to a warning at this point.

 

[PeterDonis]

toss a bit of light, and the path looks quite straight, But, the light will also be attracted to the Earth and fall towards the Earth at the same rate as the ball.

This is only true for light moving tangentially, i.e., perpendicular to the "vertical" direction. If you "toss" a beam of light straight up from the Earth, it won't "fall back" to the Earth. It won't even "slow down". It will just redshift very slightly as it escapes vertically.

 

[PeterDonis]

A ball is thrown upward from earth, it goes up and down following a parabolic path during the period of one second. Where is the journey of hundreds of thousands of kilometers of 'real' distance?

In spacetime. The ball's path in spacetime is 300,000 km long. Actually, if the "one second" is according to the clock of the Earth observer who throws the ball, then the length of the ball's path in spacetime is a tiny bit longer than 300,000 km, since the ball will have slightly more elapsed proper time (and the "length" in spacetime of a timelike path is the elapsed proper time along that path--one second of elapsed proper time is 300,000 km of "length" along a timelike path).

 

[PeterDonis]

The question is though why we move from past to future and not from future to past (which would make us feel like a puppet with a clockwork, unwinding from future to a big crunch). But this is a different question all together.

Yes, it is, and it belongs in a separate thread if you want to discuss it (but I would recommend searching the forums for previous threads first as this topic has been discussed here before).

 

[JandeWandelaar]

That's not what you said in the post that started the series of posts that I have now deleted. I would strongly suggest that you stop posting on this topic altogether until you understand it better. You are very close to a warning at this point.

I know, but now I realize it was nonsense I wrote. Objects just don't move in spacetime.

 

[PeterDonis]

Objects just don't move in spacetime.

In the geometric sense, this is true, yes. An "object" is a worldline--i.e., a curve in spacetime (or, if the object is large enough that its extension must be included in the model, it is a "world tube", a family of curves occupying a more or less "cylindrical" region of spacetime). This curve does not "move", it is just there, part of the overall geometry.

However, for many purposes, it is fine to think of an object as "moving" along its worldline, in the sense that it is "at" different points on its worldline at different instants of time by its own clock. One just has to be aware of the limitations of this concept of "movement".

 

[PeroK]

I know, but now I realize it was nonsense I wrote. Objects just don't move in spacetime.

"Nonsense" is a bit harsh. I'd say you simply have to distinguish between.

a) the three-velocity and coordinate time of the IRF ( or any RF), which is essentially a classical approach.

b) the four dimensional worldline, where essentially you take a geometric approach.