# Curvature of Time?

#### sanman

We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic.

But what does curvature of time look like?

How do we experience it?

We typically experience the passage of time in what seems to be a forward linear manner. The forward part seems to be due to how our nervous system works, thus giving a chronological bias towards causality in our perception.

But if we can see how gravity curves space, then how do we percieve how it affects time?

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Gold Member
Time dilation.

Garth

#### sanman

But Time Dilation isn't supposed to happen unless we approach some significant fraction of the speed of light. Meanwhile, we can observe the curvature of space even when standing totally still, can't we?

Are we saying that acceleration towards the speed of light is similar to an object undergoing freefall, whereby it moves according to the natural geodesic curve of space? So then "time dilation" or curvature of time, becomes apparent or manifest to us due to lightspeed being a reference frame analogous to freefall?

Due to the constraints of our nervous system, we tend to perceive T as an ordinal axis, compared to X,Y,Z where we have full degrees of freedom along each axis.
So when we don't have full freedom on the T-axis, and can only experience it in a "forward time" direction, then we can only experience this time dilation as a deviation or discrepancy in the passage of time.

If Time could move backwards, we could experience gravity as a repulsive force, and then presumably would we similarly experience the curvature of time as "time contraction" instead of time dilation?

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#### Thrice

We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic.
No that involves the curvature of both space and time.

#### chroot

Staff Emeritus
Gold Member
But Time Dilation isn't supposed to happen unless we approach some significant fraction of the speed of light.
No. Time dilation happens anytime there's gravity, and anytime any two things are moving in relation to one another. You're experiencing a kind of time dilation right now by sitting in the Earth's gravitational well. When you fly in an airplane, you're experiencing another kind of time dilation.

The magnitude of these time dilations is very small by human standards, so you don't really notice it. It is observable with atomic clocks, however.

Meanwhile, we can observe the curvature of space even when standing totally still, can't we?
The force keeping your butt in the chair is, in fact, a result of the curvature of space. You might need to consider that you cannot have curvature in space without curvature in time.

Are we saying that acceleration towards the speed of light is similar to an object undergoing freefall, whereby it moves according to the natural geodesic curve of space?
No, objects that are accelerated do not follow geodesics.

So then "time dilation" or curvature of time, becomes apparent or manifest to us due to lightspeed being a reference frame analogous to freefall?
Light speed is not a frame of reference. A reference frame is nothing more than a coordinate system, generally chosen so that some observer is at its origin.

Due to the constraints of our nervous system, we tend to perceive T as an ordinal axis, compared to X,Y,Z where we have full degrees of freedom along each axis.
Every technical term you used here (ordinal, degree of freedom) is used incorrectly, so I have no idea what you're trying to say.

So when we don't have full freedom on the T-axis, and can only experience it in a "forward time" direction, then we can only experience this time dilation as a deviation or discrepancy in the passage of time.
If you change all positive charges to negative charges, flip all movements as in a mirror image, and then reverse time, you might be surprised -- nothing changes. Things keep on doing what they were doing before. The physical laws are invariant until these transformations, collectively called CPT (charge, parity, and time). Thus, the "direction" of time is an arbitrary choice, at least in the way that the physical laws operate.

If Time could move backwards, we could experience gravity as a repulsive force, and then presumably would we similarly experience the curvature of time as "time contraction" instead of time dilation?
Something like that. The physical laws would not be the same if you only reversed time, so the physics would actually be completely different. I'd have to think about it a bit to explicitly figure out all the consequences.

- Warren

#### pervect

Staff Emeritus
But Time Dilation isn't supposed to happen unless we approach some significant fraction of the speed of light. Meanwhile, we can observe the curvature of space even when standing totally still, can't we?
Gravitational time dilation can be experienced even when "sitting still". Time dilation happens anytime one is near a large mass, not only when one is moving.

The following quote may be of some help. It's excerpted from

http://www.eftaylor.com/pub/chapter2.pdf

You keep talking about “curvature” of spacetime. What is curvature?

The word curvature is an analogy, a visual way of extending ideas about three dimensional space to the four dimensions of spacetime. Travelers detect curvature—in both three and four dimensions—by the gradual increase or decrease of the “distance” between “straight lines” that are initially parallel. In three space dimensions, the actual paths in space converge or diverge. Think of two travelers who start near one another at the equator of Earth and march “straight north.” Neither traveler deviates to the right or to the left, yet as they continue northward they discover that the distance between them decreases, finally reaching zero as they arrive at the north
pole.
How does this relate to gravitational time dilation? Attempt to form a square in space-time, by moving 1 meter up, 1 second into the future, 1 meter south, 1 second into the past.

You don't wind up at your starting point in space-time, because (to oversimplify a bit) clocks at different altitudes don't tick at the same rate due to gravitational time dilation.

This is very similar to the way that one does not wind up at one's starting place by starting at the equator on the Earth, going 1 meter north, 1 meter east, 1 meter south, and 1 meter west. The reason is the same - curvature.

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#### Demystifier

2018 Award
In Riemann geometry, a 1-dimesional curve or a 1-dimensional manifold cannot be curved. Thus, time as a 1-dimensional entity cannot be curved either. What can be curved is space or spacetime.

#### MeJennifer

In Riemann geometry, a 1-dimesional curve or a 1-dimensional manifold cannot be curved. Thus, time as a 1-dimensional entity cannot be curved either. What can be curved is space or spacetime.
Note that relativity works with pseudo-Riemannian manifolds.
Most of the rules are the same but for instance in a pseudo-Riemannian manifold the arc length between two points can be longer than the length of any geodesic between them.

#### Jheriko

No, objects that are accelerated do not follow geodesics.

This is true, however an object following a geodesic path may appear to be accelerating in some respect due to the curvature of space-time. e.g. the bending of light around the sun can be interpreted as the light accelerating towards the sun. If we throw a ball it ends up on a geodesic, yet it appears to accelerate towards the Earth etc..

#### Jheriko

Note that relativity works with pseudo-Riemannian manifolds.
Most of the rules are the same but for instance in a pseudo-Riemannian manifold the arc length between two points can be longer than the length of any geodesic between them.
If this geodesic doesn't minimise the arc length, then how is it defined? A path that parallel transports its tangent vector into itself? I was under the impression that these definitions are identical... not sure where that came from though...

According to wikipedia the definition of a geodesic on a (psuedo-)Riemannian manifold is "Just as in a standard metric space, a geodesic on a (pseudo-)Riemannian manifold M is defined as a curve γ(t) minimizes the length of the curve."

Wikipedia can be wrong though (the grammar in the above quote is, at the least!)

#### robphy

Homework Helper
Gold Member
chroot said:
No, objects that are accelerated do not follow geodesics.
This is true, however an object following a geodesic path may appear to be accelerating in some respect due to the curvature of space-time. e.g. the bending of light around the sun can be interpreted as the light accelerating towards the sun. If we throw a ball it ends up on a geodesic, yet it appears to accelerate towards the Earth etc..
Chroot's comment refers to the observer-independent spacetime 4-acceleration rather an observer-dependent spatial 3-acceleration [familiar to Galilean/Newtonian kinematics].

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#### daniel_i_l

Gold Member
But if we can see how gravity curves space, then how do we percieve how it affects time?
because of gravity, clocks in sattalites go faster than clocks on earth - this is taken into account in GPS systems.

#### Demystifier

2018 Award
Note that relativity works with pseudo-Riemannian manifolds.
Most of the rules are the same but for instance in a pseudo-Riemannian manifold the arc length between two points can be longer than the length of any geodesic between them.
What I said above about Riemannian geometry is true also for the pseudo-Riemannian geometry. Your observation on the arc length above does not imply that a curve may be curved. The pseudo-Riemann curvature of a 1-dimensional curve is zero.

#### Schrodinger's Dog

I always like the use of imaginary numbers here to explain the time axis of the graph.

I imagine a graph with an (x,y,z,t) axis. You can use whatever you like to represent the fourth axis, but imaginary numbers are an already oft used axis so it makes sense to use it in discussion about time and space curvature. In this context I can equally equate a curve in space with a curve in time prvided I've set the graph up correctly. I was thinking about this with relation to time after looking into complex numbers, couldn't you call this time I thought? And a friend said yes someones already done it, seems obvious.

there's the much more simple representation with minowski diagrams. But if I want to get at least an intuitive grasp of 4D space-time I think about it something like that.

#### MeJennifer

I always like the use of imaginary numbers here to explain the time axis of the graph.

I imagine a graph with an (x,y,z,t) axis. You can use whatever you like to represent the fourth axis, but imaginary numbers are an already oft used axis so it makes sense to use it in discussion about time and space curvature. In this context I can equally equate a curve in space with a curve in time prvided I've set the graph up correctly. I was thinking about this with relation to time after looking into complex numbers, couldn't you call this time I thought? And a friend said yes someones already done it, seems obvious.

there's the much more simple representation with minowski diagrams. But if I want to get at least an intuitive grasp of 4D space-time I think about it something like that.
This is also called a Wick rotation.
But while a Wick rotation works in flat space-time it does not so in curved space-time. Which is one of the current problems in the development of a quantum theory of gravity.

#### Schrodinger's Dog

This is also called a Wick rotation.
But while a Wick rotation works in flat space-time it does not so in curved space-time. Which is one of the current problems in the development of a quantum theory of gravity.
Yeah it's only a way of dealing with it in your head(intuitively) I assume you mean that the bend caused by the magnitude of the vector and the relation to time doesn't match up with Gravity and it's relation because of the warping of space itself. Interesting.

Just think of gravities bend in space and time in the same way as you would the relation to c, and time. Even though one cannot be mapped in a relation to the other, one is a direct consequence of the other. I'm sure there are better analogies.

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#### grant9076

If Time could move backwards, we could experience gravity as a repulsive force, and then presumably would we similarly experience the curvature of time as "time contraction" instead of time dilation?
Something like that. The physical laws would not be the same if you only reversed time, so the physics would actually be completely different. I'd have to think about it a bit to explicitly figure out all the consequences.
I never really took physics but I think that the direction of gravity (like other fields) would stay the same regardless of whether you go backwards or forward in time. For example:

If someone threw a ball upward, it would decelerate until it reached the apex and then accelerate downward until it hit the ground. If you filmed the event and played it backwards, you will see the ball jump from the ground at a given velocity and decelerate until it reached the apex and then accelerate downward until it landed in the person's hand. In either case, the acceleration is downward. When I use the terms 'accelerate' and 'decelerate', I am referring to the apparent behavior of the ball as it travels along the geodesic.
However, I believe that what I said above still holds true from the standpoint of Einstein's field equations. The time component of spacetime curvature (if I'm not mistaken) is expressed as a function of -dt^2 (or dt^2 depending on convention). Which means the sign (+ or -) should not affect the equations.

Also, the only law that I can think of that reverses with time reversal is the second law of thermodynamics.

It's been 23 years since I tested out of physics at college so please be patient with me if I am dead wrong.

#### Truth Finder

To get the effect of moving backward in time, on curvature; substitute with "-t" in equations instead of "t" in the Curvatur Tensor. To get the effect of moving backward in time, on gravity; substitute with "-t" in equations instead of "t" in the final free fall equations.

Wonna add, curvature in GR, is a 4-dimensional concept. It simply means, that transformations from certain frame to another are flat (i.e. roughly speaking; not so simple as SR's). So, I agree with your point of view grant9076 in checking the equations like this. But ........ you have a very good mathematical engineer! .. Are you a mathematician or an engineer?

Schwartz VANDSLIRE.

#### grant9076

Are you a mathematician or an engineer?
Thanks but actually, I'm a pilot by trade. Although I did well when I got my bachelors in engineering, it has been a couple of decades and I really haven't thought about that stuff until I first visited this site a few months ago. So, I cannot consider myself to be either.

However, although testing out of physics never gave me the opportunity to study relativity in detail, it always seemed to make perfect sense. Personally, I think that it is only a matter of time before some genius out there unifies Einstein's theory with quantum mechanics.

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#### grant9076

If someone threw a ball upward, it would decelerate until it reached the apex and then accelerate downward until it hit the ground. If you filmed the event and played it backwards, you will see the ball jump from the ground at a given velocity and decelerate until it reached the apex and then accelerate downward until it landed in the person's hand. In either case, the acceleration is downward.
In addition to the above thought experiment, another unsophisticated thought experiment that I did as a teenager that cemented my belief in general relativity was the following:

If a particle with negative mass was in a gravitational field, it would sense it as repulsion. However, because it has negative mass, it will react opposite to the force and 'accelerate' downward just like a particle with positive mass. I reasoned at the time that if the reaction to gravity is the same for a negative mass as it is for a positive mass, then it must be true for every mass in between including zero mass. Based on these 2 thought experiments, I concluded that whether something has positive mass, negative mass or no mass, whether it travels forward or backward in time, if it exists, it has to react to gravity in exactly the same way.

I saw no other option but to conclude that gravity is a distortion of spacetime and that Einstein was fundamentally right. These thought experiments made the concept more intuitively obvious to me than any article or book that I have read on the subject prior or since.

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#### Chris Hillman

Hi all, this thread has been rather confused, I think, so I've been avoiding comment, but maybe I can help a bit after all:

The time component of spacetime curvature (if I'm not mistaken) is expressed as a function of -dt^2 (or dt^2 depending on convention).
Don't get that. But for what it is worth, in geometric units in which (G=c=1), path curvature (acceleration of a particle) has units of $L^{-1}$, while sectional curvature (components of Riemann curvature tensor) have units of $L^{-2}$, where $L$ stands for some length unit, such as cm. In Einstein's field equation, we form the Einstein tensor by a kind of generalized "trace" of the Riemann tensor, so its components are expressed in the same units, so these must agree with the units of mass-energy density, pressure, and stress (as in the stress tensor from elastodynamics). And they do!

Which means the sign (+ or -) should not affect the equations.
I don't understand what the first sentence means, but the signature used is a convention and of course making a different choice slightly changes our description of the physics/geometry, but doesn't change the physics/geometry we are describing!

Also, the only law that I can think of that reverses with time reversal is the second law of thermodynamics.
You mean, the only law you can think of which is not symmetric under time reversal? (There are some more subtle examples.)

If a particle with negative mass was in a gravitational field, it would sense it as repulsion. However, because it has negative mass, it will react opposite to the force and 'accelerate' downward just like a particle with positive mass. I reasoned at the time that if the reaction to gravity is the same for a negative mass as it is for a positive mass, then it must be true for every mass in between including zero mass. Based on these 2 thought experiments, I concluded that whether something has positive mass, negative mass or no mass, whether it travels forward or backward in time, if it exists, it has to react to gravity in exactly the same way.
I am not sure why this is supposed to establish that gtr is a reasonable theory, but for others, I think that grant is saying that in Newtonian gravity, if you consider a pair of pointlike objects, one with positive mass $m_1 > 0$ and one with negative mass $m_2 < 0$, then the "gravitational force" on the second particle reverses the expected direction (but has the expected magnitude) in Newton's inverse square force law $F_2 = \frac{G \, m_1 \, m_2}{r^2} = -\frac{G \, m_1 \, |m_2|}{r^2}$, but the response to this force, given by $F_2= m_2 \, a_2 = -| m_2 | \, a_2$, is to accelerate in the opposite of the expected direction (but with the expected magnitude); these two reversals cancel out, so that the second particle falls toward the first, like $a_2 = G \, m_1/r^2$. But, by the same reasoning, the first particle falls away from the second (that is, the inverse square force on the first particle reverses the expected direction, but not its response to this force), like $a_1 = -G \, |m_2|/r^2$. If $m_2 = -m_1$, the two particles will in fact maintain constant distance, so that we have a zero mass system which spontaneously accelerates indefinitely with constant acceleration. No doubt this disconcerting runaway acceleration is why Newton forbade negative mass.

Since gtr has a Newtonian limit, we should expect to encounter the same problem in gtr unless we forbid negative mass-energy. This is more or less what we do, modulo the awkward fact that in classical terms, one would have to say that the energy density in between the two plates in the Casimir effect is negative.

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#### grant9076

First, please pardon my lack of using the text tools (someday, I'll try to figure it out). What I was trying to say was that I always thought that spacetime was a 4-dimensional pseudo-riemannian manifold and that the metric tensor expresses the square of the shortest distance between 2 points (ds^2) as some function G (a diagonal matrix) of (-dt^2*c^2, x^2, y^2, z^2). What I was saying was that whether dt is negative or positive, -dt^2 will be the same and the overall equation will not change.

With regard to the particle, what I was trying to say is that:

If a particle with a positive mass started off at a given position and a given velocity with respect to a planet (or other celestial body), it will follow a particular path because of the gravitational field. Now if you replace the particle with one of negative mass and the same initial position and velocity, the path will be exactly the same. Because I considered at the time that the same would be true for every particle mass in between, then that path must be the real unaccelerated (geodesic) path for that given position and velocity. Because this path is curved and not straight, I reason that the curvature could only be caused by the gravity which is distorting the spacetime.

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You mean, the only law you can think of which is not symmetric under time reversal?

Yes, that is exactly what I was saying. For my own education, if you know of any other examples then please let me know.

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#### Chris Hillman

First, please pardon my lack of using the text tools (someday, I'll try to figure it out).
Nothing to it: surround your latex pseudocode by "tex" (inside brackets "[" and "]") and "/tex" likewise inside brackets.

What I was trying to say was that I always thought that spacetime was a 4-dimensional pseudo-riemannian manifold and that the metric tensor expresses the square of the shortest distance between 2 points ($ds^2$) as some function G (a diagonal matrix) times $-dt^2 \, c^2 +x^2 +y^2 +z^2$. What I was saying was that whether dt is negative or positive, -dt^2 will be the same and the equation will not change.
Did you forget some "d"s in that expression?

1. A spacetime model is "a 4-dimensional pseudo-riemannian manifold", correct.

2. A metric which is a scalar mutiple of the Minkowski metric,
$$ds^2 = G(t,x,y,z)^2 \, \left( -c^2 \, dt^2 + dx^2 + dy^2 + dz^2 \right)$$
where $G(t,x,y,z)$ is a scalar function on the spacetime, is a "conformally flat spacetime model", which is special property not enjoyed by all spacetimes. For example, the FRW dusts are conformally flat, but the Schwarzschild vacuum solution is not.

3. "whether dt is negative or positive, -dt^2 will be the same and the equation will not change", correct.

If a particle with a positive mass started off at a given position and a given velocity with respect to a planet (or other celestial body), it will follow a particular path because of the gravitational field. Now if you replace the particle with one of negative mass and the same initial position and velocity, the path will be exactly the same.
So these are positive or negative mass test particles, i.e. they assumed to have masses (in absolute value) sufficiently small that they do not disturb the ambient gravitational field. OK, I agree, then positive and negative mass test particles (regardless of mass, as long as its small enough to not disturb the ambient gravitational field, within the limits of measurement) will have "equivalent world lines" in a suitable sense (e.g. in a static solution like the Schwarzschild solution, we can imagine experimenting with first a positive and then a negative mass test particle, tossing them in the same direction with the same velocity from a static spaceship).

Because I considered at the time that the same would be true for every particle mass in between, then that path must be the real unaccelerated (geodesic) path for that given position and velocity. Because this path is curved and not straight, I reason that the curvature could only be caused by the gravity which is distorting the spacetime.
Hmm... I am still not sure I quite follow, but this sounds closely related to Einstein's own reasoning via the equivalence principle c. 1913, as he was searching for his geometrical theory of gravitation.

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You mean, the only law you can think of which is not symmetric under time reversal?

Yes, that is exactly what I was saying. For my own education, if you know of any other examples then please let me know.
The syntax is "QUOTE" and "/QUOTE" surrounded by brackets.

OK, another example: parity conservation turned out to be violated by the weak interaction http://en.wikipedia.org/wiki/Parity_conservation

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#### grant9076

Thanks for the example Chris. Please forgive my ignorance as I have been out of the science mindset for a very long time. I think that there is a thing or two that I need to teach myself about Hilbert spaces. Also, yes I did forget to put my d's in.:grumpy:

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