# Can spacetime as a rubber sheet be taken literally or merely as pedagogic?

#### EnumaElish

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Can "spacetime as a rubber sheet" be taken literally or merely as pedagogic?

Doesn't a literal interpretation imply that gravity is not a "force" in the usual sense, but merely a consequence of how the space is shaped? Unless the potential energy content of the rubber is counted, perhaps.

(I think a rather famous Russian physicist came up with that one. Gamov?)

Related Special and General Relativity News on Phys.org

#### pervect

Staff Emeritus
If you take the idea literally, you have to take it very literally, and say that not space is warped, but space-time is warped.

When you take this very literal view, you can describe Newtonian gravity as being due to the geometry of space-time.

The time in space-time is very important. Models of space-time that mimic only the Newtonian aspects of gravity have all of the "curvature" in the "time" aspect, while space itself remains perfectly flat.

Full GR predicts that both space and space-time are curved, but the curvature of space itself can usually (though not always) be considered to be small.

#### JesseM

One problem with the rubber sheet model is that people tend to think of objects being pulled "down" into depressions on the sheet, when really it's just the curvature itself that's important, you could just as easily imagine masses creating humps on the rubber sheet. Another problem is that a geodesic on an ordinary rubber sheet would be path between two points that has the shortest length, whereas in GR a geodesic is a path through spacetime with the greatest proper time.

#### EnumaElish

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Can this be a realistic model of gravity?

I came across the following model on a Web page built by some Russian pysicist, I think. (Posted in English, though -- I can't read Cryllic.) This model begins with the usual rubber sheet model. But then goes a step further when it poses the question "what if the equivalance principle of the theory of general relativity was not just an equivalance but an identity?" That is, what if gravity is acceleration?

The model proposes the following mental picture (as a pedagogic device, it better be): there is a Big Iron Horse (these are my words; Big Train in the original) accelerating constantly (I guess this translates as "jerk > 0 at all times" in physicspeak) on the Long Iron Road (railroad). Our whole universe is pictured as an elastic Big Banner (2-D spacetime, as it were) attached to the top of the train and facing the front of the Iron Horse (all vector spaces spanning the elastic sheet are perpendicular to the Train's trajectory). Galaxies, etc. are pictured as objects on (in?) the Big Banner.

As the Train accelerates, inertia acts on all objects on the banner. More massive the object, bigger the inertia. As a result, bigger objects create more curvature on the Big Banner.

My questions are:
1. excluding the Train and the Railroad aspects, can anyone think of a theoretical objection to this (metaphysical) model, along the lines of (but beyond) JesseM's objection to the rubber model as a literal model of reality (problem with geodesics)?
2. in your view, would a circular Railroad (spacetime having angular momentum) make the picture a little more realistic compared to a non-circular (e.g., linear) trajectory?

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#### pervect

Staff Emeritus
This does not appear to be a pedagogic device that attempts to model general relativity. It seems to be some other theory of gravity entirely.

The idea of moving space-time with a railroad really doesn't make any sense at all.

Movement is the rate of change of something with respect to time. But time is a part of space-time. "Moving time" does not make any sense - what would it move through? "Moving time" does not create any sensible mental picture.

If one removes space-time from the picture, and replaces it with space, one gets a false picture of gravity based on warped space. We know that this isn't right - space may not be absolutely flat, but spatial curvature alone (wihtout the time component) is not nearly large enough to explain the motion of the Earth around the sun.

The railroad does absolutely nothing for the picture. The standard approach, as I mentioned earlier, is to imagine space-time on a curved surface.

This is really only possible (in the sense of being able to visualize it) with a 2-d space time, one space dimension, and one time. To be able to envision "curved time", one draws the space-time diagram on a sheet of paper, so that the time dimension is represented by a spatial dimension. Then one draws the resulting graph on a sphere or other curved spatial surface.

[clarify]
Mathematically it is possible to rigorously define curvature and to be able to apply it to spaces of higher than 2 dimension. It's genearlly not possible to intuitively visualize higher-deimensional curved spaces, though.

Going back to the simple picture of a 1-space+1-time, which can be visualized as a plane,
there are various posts which talk about this approach here on the forum, and one does indeed find that bodies travelling along geodesics in such a curved space-time appear to attract each other.

For a textbook reference which takes this approach, see for instance MTW's "Gravitation", which supports it with the necessary mathematics (tensor calculus). Non-mathematical readers would have to ignore the math without letting it scare them away to get anything out of the book.

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#### EnumaElish

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Thanks. (Apparently, a message with < 10 characters will not post to these forums, so I am adding this sentence.)

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#### amt

The Rubber sheet analogy is an excellent source of example. It should be noted that, even though Mass warps space, the reason we are falling towards Earth is because Space is Pushing us towards Earth and not not because we are sucked into Earth.

Here is a question I have on space-time:
Mass warps space-time. I can understand space being warped, but what about time? How is space-time effecting us in our daily lives in terms of 'time'?

Thanks.

i never thought of it like that

amt said:
Here is a question I have on space-time:
Mass warps space-time. I can understand space being warped, but what about time? How is space-time effecting us in our daily lives in terms of 'time'?
gravity "slows down" time.

#### pervect

Staff Emeritus
gravity "slows down" time.
Yes. To borrow an example from Feynman, if you throw a clock up in the air, ask what path it would take to maximize the amount of time that would pass.

As the clock moves higher, it ticks faster, so the more quickly the clock can gain altitude, the better.

But to get the clock to a high altitude quickly, it has to move quickly, and time dilation will slow down it's reading, which is bad.

The path that a clock should follow to maximize its reading is just the natural motion that it follows in a gravitational field.

Thus the fact that time moves more slowly deep in a gravity well crucially influences how clocks move, given that they follow geodesics which must extremize "proper time". We call this influence "gravity".

#### EnumaElish

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pervect said:
Thus the fact that time moves more slowly deep in a gravity well crucially influences how clocks move, given that they follow geodesics which must extremize "proper time". We call this influence "gravity".
Is it not a wonder, then, within special relativity what slows down time is a first-order change (velocity) but within general relativity time is (also?) slowed down by a second-order change (gravity)?

Assuming, that is, gravity can fairly be called a second-order change.

Can the orders of magnitude of the two slow-down effects (the one arising from v under SR, and the other arising from G under GR) be meaningfully compared?

#### pervect

Staff Emeritus
EnumaElish said:
Is it not a wonder, then, within special relativity what slows down time is a first-order change (velocity) but within general relativity time is (also?) slowed down by a second-order change (gravity)?

Assuming, that is, gravity can fairly be called a second-order change.

Can the orders of magnitude of the two slow-down effects (the one arising from v under SR, and the other arising from G under GR) be meaningfully compared?
In GR, time is slowed down by gravitational potential energy, not by gravity per se.

For example, take a look at
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

we let the negative of the potential erergy function per unit mass be U

U = 2GM/R

and time dilation in terms of U becomes

T_{infinity} = T_0 / sqrt(1-2U/c^2)

In special relativity, we see that time dilation is

T = 1/sqrt(1-v^2/c^2)

U has units of energy/mass = velocity^2, so U/c^2 is dimensionless
v^2/c^2 is also dimensionless.

So we see that the dimensional dependence of time dilation due to gravity and speed is very similar, both being dimensionless quantites. Even the structure of the equations is very similar.

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#### amt

I like the answers, especially the ones with formulas. Let me ask another question please. How different would our lives be if we lived in a gravity free place? Let's assume we live on a planet, but this particular planet does not warp space/time. Would our lives go by faster?

Thanks.

Staff Emeritus
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Let's make that a little more definite; suppose we live on a space station which is not massive enough to warp space significantly, and the station is located somewhere high in the gravity potential of the galaxy, away from the stars. Would our time go by faster? Well faster than what? The people down in their gravity wells would see us aging faster, and we sould see them aging slower, provided that we could see each other over those distances.

#### amt

So, people down in their gravity wells would see us aging faster, and we would see them aging slower, provided that we could see each other over those distances.

Now, does this mean that we are REALLY aging faster and that the people in the gravity wells are REALLY aging slower? Or is it just the apparent visual effect?

Sorry to digress from the original question posted here.

it is time not an optical illusion

#### pervect

Staff Emeritus
amt said:
So, people down in their gravity wells would see us aging faster, and we would see them aging slower, provided that we could see each other over those distances.

Now, does this mean that we are REALLY aging faster and that the people in the gravity wells are REALLY aging slower? Or is it just the apparent visual effect?

Sorry to digress from the original question posted here.
What is "REAL"? It seems to me that this thread is getting bogged down in philosphy.

Are you "REAL"? How do you know that you are "REAL"? Am I "REAL"? How do you know if I am "REAL"?

This can be debated endlessly, without any resolution. But preferrably it can be debated in the philosophy forum :-).

To attempt to answer the question on what we observe, we've already noted two important points:

1) A, at the high gravitational potential, sees B, at the lower gravitational potential, age more slowly.

2) B, at the low gravitational potential, sees A, at the higher gravitational potential, age more rapidly.

To these two important points, I will add a third point that has not been mentioned but hopefully will clarify the situation.

3) The time it takes light to travel from "A" to "B" and from "B" to "A" can be made constant, so that the time it takes for a round trip from A to B and back to A is always constant, as is the time it takes for a round trip from B to A and back to B. Furtermore, we can arrange matters so that the path the light takes is also always the same path (except for occuring at different times). When we arrange things in this manner, both of the previous statments (1 & 2) are still true.

This third statment hopefully makes it clear that the difference in aging is not due to the travel time of the light - because the travel time of the light is constant.

#### EnumaElish

Homework Helper
pervect said:
So we see that the dimensional dependence of time dilation due to gravity and speed is very similar, both being dimensionless quantites. Even the structure of the equations is very similar.
Thank you pervect. I have a follow up:

How does physics deal with constant velocity in a gravitational field? Are SR effects compounded with GR effects? For example, would one say "for starters I will assume away gravity and impose the SR effects (e.g. $$t_1 = t_0 / \sqrt{1-v^2/c^2}$$, replace inertial mass with relativistic mass, etc.); once I am done with SR I can impose gravitational effects (e.g. $$t_2 = t_1 / \sqrt{1-2U/c^2}$$, etc.)"?

I went to the web link in your post and toured a little. Under "General Relativity" the website mentions gravitational time dilation. Is there a corresponding gravitational space compaction? E.g., $$L_2 = L_1 \sqrt{1-2U/c^2}$$ similar to SR's $$L_1 = L_0 \sqrt{1-v^2/c^2}$$?

#### pervect

Staff Emeritus
I'm going to be a bit busy for the next day or so, but I'll try and give a short answer

In special relativity, the metric coefficients are fixed. What is the same for all obsevers is the value of the Lorentz interval, given in geometric units (where c=1) by

ds^2 = dx^2 + dy^2 + dz^2 - dt^2

(Sign conventions vary - sometimes people use ds^2 = dt^2 - dx^2-dy^2-dz^2)

Time dilation can be derived most conveniently from the second form of the interval above. I was thinking of doing it to show how time dilation follows from the invariance of the Lorentz interval, but the post was getting too long.

General relativity calculates the Lorentz inteval with metric coefficients which can vary with one's position in space.

ds^2 = g_00 dt^2 + g_11 dx^2 + g_22 dy^2 + g_33 dz^2

(this is not the most general from of the metric possible, but it's a diagonal form that's very common and useful).

Here the metric coefficients are a function of one's position in space, and possibly of time as well.

Gravitational time dilation shows up in the term g_00. g_00 controls the rate at which coordinate time maps to the time a clock actually measures. If g_00 is 1, clocks keep the same time as coordinate time.

Gravitational length contraction shows up in the spatial terms of the metric, like g_11, g_22, g_33. If they are one, there is no length contraction.

The time dilation equations for moving clocks have to be re-derived from genreal considerations of the invariance of the Lorentz interval. In many circumstances you can multiply the gravitational time dilation by the SR time dilation factor, but you can't do this in all cases.

Doing special relativity is NOT a matter of replacing mass with relativistic mass.

Take a look at

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

for instance, especially the section about the relativistic version of F=ma at the end.

#### Unknowing

amt said:
Now, does this mean that we are REALLY aging faster and that the people in the gravity wells are REALLY aging slower? Or is it just the apparent visual effect?
Isn't that the relativity part? There is no absolute right answer, it all depends on your frame of reference.

#### pervect

Staff Emeritus
Unknowing said:
Isn't that the relativity part? There is no absolute right answer, it all depends on your frame of reference.
It's quite possible to work a problem in relativity and get an absolutely wrong answer. I should know, I've done it a time or two :-).

A theory that couldn't get wrong answers (that couldn't be falsified by comparing the predicted answers of the theory to experiment) would not qualify as a scientific theory. Relativity is a scientific theory, so it is capable of being falsified.

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#### EnumaElish

Homework Helper
pervect said:
... Doing special relativity is NOT a matter of replacing mass with relativistic mass.

Take a look at

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

for instance, especially the section about the relativistic version of F=ma at the end.
I looked at the link; thanks. I can see how directional mass can be confusing. OTOH, I remember the first time I read on the Internet a non-technical explanation of c being the universal speed limit, and it went something like, "Einstein predicted that as an object gains speed, its (relativistic) mass increases. As the mass gradually increases, an additional acceleration requires an ever increasing amount of energy. In the limit (v=c), its mass will reach infinity and threfore an infinite amount of energy will be required to get it there. That's why." It had made perfect sense!

#### EnumaElish

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JesseM said:
... Another problem is that a geodesic on an ordinary rubber sheet would be path between two points that has the shortest length, whereas in GR a geodesic is a path through spacetime with the greatest proper time.
Why is this a problem? Suppose the sheet is shaped like a saddle; and you have a "minimax" point. This way, one can minimize travel through space while at the same time maximizing travel through time.

#### JesseM

EnumaElish said:
Why is this a problem? Suppose the sheet is shaped like a saddle; and you have a "minimax" point. This way, one can minimize travel through space while at the same time maximizing travel through time.
What do you mean by "maximizing travel through time"? Are you talking the "proper time" that's measured by a clock which travels along a given path through curved spacetime, taking into account gravitation time dilation as well as velocity-based time dilation, or are you just talking about how much time an external observer would see an object take as it travelled along a path on a physical rubber sheet? I'm talking about the former.

Anyway, the problem is just that it is not true of all spacetimes that the shortest path in curved space is the same as the path that maximizes the proper time, although it can be true in some (like flat spacetime). For example, I'm pretty sure if you want to travel between opposite points on earth's orbit, you'd travel a smaller spatial distance if you went directly through the sun rather than following the earth's path.

#### EnumaElish

Homework Helper
JesseM said:
... Anyway, the problem is just that it is not true of all spacetimes that the shortest path in curved space is the same as the path that maximizes the proper time, although it can be true in some (like flat spacetime). For example, I'm pretty sure if you want to travel between opposite points on earth's orbit, you'd travel a smaller spatial distance if you went directly through the sun rather than following the earth's path.
So I am right in thinking that one can come up with a spacetime structure with that property; therefore it is not a problem per se.

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