# Does acceleration curve spacetime?

#### DaniV

When Einstein wrote the special realvity he said that inertional acceleration and gravitational acceleration are the same,
my question:

Does it mean that any source of somthing that making body accelerate (force) is also curving the space-time?
for example- spaceship that accelerating in empty space without influence from another masses, by their fuel supply.

Does the space-time curving more then it curves because of the spaceship mass,
due to the fact that the spaceship accelerating?

Related Special and General Relativity News on Phys.org

#### Nugatory

Mentor
Acceleration does not curve spacetime, only gravity does.

The equivalence principle does not say that gravity and acceleration are the same, it says that if you choose a region of spacetime small enough that tidal effects can be ignored, then gravity and acceleration will behave the same way.

#### SlowThinker

Acceleration does not curve spacetime, only gravity does.
I hope this question does not derail the theme, but... I often read that what looks like gravity, is in fact objects moving straight in curved spacetime.
Is it really the curvature of spacetime, that is pulling me down as I write this? It seems I'm not being pulled by tidal effects, am I? If I close my eyes, I can't really tell if I am on an accelerating spaceship.

Wouldn't deformed spacetime be more accurate description?

#### phinds

Gold Member
I hope this question does not derail the theme, but... I often read that what looks like gravity, is in fact objects moving straight in curved spacetime.
Is it really the curvature of spacetime, that is pulling me down as I write this? It seems I'm not being pulled by tidal effects, am I? If I close my eyes, I can't really tell if I am on an accelerating spaceship.

Wouldn't deformed spacetime be more accurate description?
The phrase "curved spacetime" really comes from applying the human standard of Euclidean Geometry to space-time when it is more properly described by Riemann Geometry. A straight line in Riemann Geometry is called a geodesic when applied to space-time and it you look at it from the point of view of Euclidean Geometry it is curved. The Earth's gravitational field as applied to you is actually straight in Euclidean geometry because it goes from the center of the Earth radially outward.

The light from a distant star can be "curved" around another celestial object such as our sun because the geodesic does not follow a Euclidean straight line from the star to us here on Earth. This is in fact exactly how the Theory of General Relativity was finally confirmed to the satisfaction of everyone but Einstein who ALREADY knew it was right. (Actually that's tongue in cheek. My understanding is that he waited with baited breath for the results to come in and it took years for the experiment to finally be pulled off).

And by the way, you ARE being affected by tidal forces but they are SO trivially small that I'm not even sure they could be measured with existing technology.

#### A.T.

Is it really the curvature of spacetime, that is pulling me down as I write this? It seems I'm not being pulled by tidal effects, am I?
Yes, exactly. The intrinsic curvature is related to tidal effects. Gravitational acceleration doesn't require intrinsic curvature. The cone surface in the clip below doesn't have intrinsic curvature:

#### Nugatory

Mentor
Tidal effects are the way that gravity behaves differently from being in an accelerated spaceship. If you hold a coin in each hand, extend your arms, and drop the coins you might say that both fall straight down and follow parallel paths to the floor. That's not quite right - they follow slightly converging paths that would intersect at the center of the earth if the floor didn't get in the way, and the distance between where they land on the floor is slightly less than the distance beween your outstretched hands. That's a tidal effect, and it wouldn't happen if you were standing in an accelerated spaceship instead.

To see how curved spacetime produces the effect that we perceive as a gravitational force pulling us towards the surface of the earth... Look for a video made by member @A.T. - it's linked from many older threads.
[Edit: looks like he posted it while I was writing this post]

#### SlowThinker

The phrase "curved spacetime" really comes from applying the human standard of Euclidean Geometry to space-time when it is more properly described by Riemann Geometry. A straight line in Riemann Geometry is called a geodesic when applied to space-time and it you look at it from the point of view of Euclidean Geometry it is curved.
Yes, exactly. The intrinsic curvature is related to tidal effects. Gravitational acceleration doesn't require intrinsic curvature.
I guess it is too much to ask for a clear answer to an unclear question
So, do I understand your answers correctly in that there are 2 kinds of curvature being mixed together?
1. A thrown/falling object follows a curved path in space, which is a straight path in spacetime. This is the strong effect we all feel.
2. Then there is the intrinsic (Riemann) curvature of spacetime. Physicists speak about this one when saying that "the spacetime is curved".

The two are related like (1) a function and (2) its derivative, in the sense that (2) can be computed from (1), and (1) can be partially computed from (2), except that any constant acceleration can be added to the result.

#### phinds

Gold Member
I guess it is too much to ask for a clear answer to an unclear question
So, do I understand your answers correctly in that there are 2 kinds of curvature being mixed together?
1. A thrown/falling object follows a curved path in space, which is a straight path in spacetime. This is the strong effect we all feel.
2. Then there is the intrinsic (Riemann) curvature of spacetime. Physicists speak about this one when saying that "the spacetime is curved".

The two are related like (1) a function and (2) its derivative, in the sense that (2) can be computed from (1), and (1) can be partially computed from (2), except that any constant acceleration can be added to the result.
No, both of the things you are talking about follow geodesics. Ballistic objects, such as a thrown ball, follow geodesics in space-time. This will be a straight path in Riemann geometry and a curved path in Euclidean Geometry.Your body, standing on the surface of the Earth, is doing its best to follow a geodesic to the center of the Earth but it is being inhibited by the surface of the Earth, as will be the thrown ball when it lands.

Things in freefall, such as a thrown ball, are MOVING along a geodesic. A body being held in place by an external force is on a geodesic but not moving along it.

#### PeterDonis

Mentor
Is it really the curvature of spacetime, that is pulling me down as I write this? It seems I'm not being pulled by tidal effects, am I? If I close my eyes, I can't really tell if I am on an accelerating spaceship.
Locally, you're right, you can't tell; for all you know, you could be in an accelerating spaceship.

But if you and someone on the other side of the Earth compare notes, you will realize that you are both accelerating in opposite directions, yet you remain the same distance apart. Then you know you're not in an accelerating spaceship in flat spacetime, but on a planet in curved spacetime.

So it isn't the acceleration you yourself feel that tells you spacetime is curved; it's the comparison of your acceleration (strength and direction) with that of others in different spatial locations that does so. In other words, it's seeing that you and everyone else on Earth are all accelerating in different directions, yet you all stay in the same places.

Wouldn't deformed spacetime be more accurate description?
Not unless you just interpret "deformed" as a synonym for "curved".

do I understand your answers correctly in that there are 2 kinds of curvature being mixed together?
1. A thrown/falling object follows a curved path in space, which is a straight path in spacetime. This is the strong effect we all feel.
2. Then there is the intrinsic (Riemann) curvature of spacetime. Physicists speak about this one when saying that "the spacetime is curved".
There are indeed two kinds of curvature involved, and you have the second one right, but not the first. The other kind of curvature is path curvature, i.e., non-geodesic motion, i.e., nonzero proper acceleration. If you drop a rock, the one whose path is curved in spacetime is you, not the rock--because you feel acceleration and the rock doesn't.

The way the two kinds of curvature work together can be seen in the example I gave above: you and everyone else on Earth all have curved paths, and the path curvatures are all in different directions. But you all stay in the same places, because the spacetime you are all in is also curved, so paths that are curved in different directions can still remain the same distance apart.

#### SlowThinker

Wouldn't deformed spacetime be more accurate description?
Not unless you just interpret "deformed" as a synonym for "curved".
So it would be incorrect to say that "something is wrong with spacetime in an accelerating rocket and that's why things fall down"? Is it just "an accelerating rocket in perfectly ordinary spacetime", end of story?

1. A thrown/falling object follows a curved path in space, which is a straight path in spacetime.
There are indeed two kinds of curvature involved, and you have the second one right, but not the first. The other kind of curvature is path curvature, i.e., non-geodesic motion, i.e., nonzero proper acceleration. If you drop a rock, the one whose path is curved in spacetime is you, not the rock--because you feel acceleration and the rock doesn't.
Uh-huh. So I am accelerating to stay in place.
So the correct answer to "Why do things fall down?" is "They don't!"?

#### PeterDonis

Mentor
So it would be incorrect to say that "something is wrong with spacetime in an accelerating rocket and that's why things fall down"? Is it just "an accelerating rocket in perfectly ordinary spacetime", end of story?
If by "something is wrong with spacetime" you mean "spacetime is not flat", and by "perfectly ordinary spacetime", you mean "flat spacetime", then yes, the above would be incorrect. But most GR experts would not agree that something is "wrong" with curved spacetime and only flat spacetime is "perfectly ordinary".

I am accelerating to stay in place.
Yes.

So the correct answer to "Why do things fall down?" is "They don't!"?
Yes, that's one way to look at it, and a way that is often very helpful.

Another way is to define "down" as "the direction opposite to the acceleration I feel" instead of "the direction of gravity". Then, even if you were in an accelerating rocket instead of on a planet, you could still say that a dropped object would fall down. That can often be a helpful way to look at it too.

Both ways, as you can see, focus attention on the acceleration you feel, and to remove all talk of a "force of gravity" (and in the standard view of GR, there is indeed no such thing as a "force of gravity"). That lets you distinguish, without any unwanted baggage, between the things that the two scenarios (accelerating rocket and planet) have in common (the acceleration you feel, and its local effects like what happens to objects you drop) and the things that differ between them (your relationship to other people feeling acceleration at other locations).

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