Curvature of Spacetime and its Impact on the Speed of Light

[Arjun Kumar]

As per Maxwell's equations the speed of light is

c = 1/√(permeability X permittivity)

I find that both permeability and permittivity have pi in their calculations. Since spacetime is not flat, the value of pi would change due to the curvature of space near large masses. Does this mean that the speed of light would also be altered by the curvature of spacetime?

 

[mathman]

π is constant! It is a mathematical constant, unaffected by the physics of spacetime or anything else.

 

[Bill_K]

As per Maxwell's equations the speed of light is

c = 1/√(permeability X permittivity)

The so-called permeability of free space is also a mathematical constant, an artifact of the SI system of units, having nothing at all to do with physics. It is exactly 4π x 10^-7 by definition. (Next, someone will say the same thing about c.)

 

[Arjun Kumar]

Pi is constant only in Euclidian geometry / space. Pi is defined as the number of times the diameter of a circle divides into the circumference. If flat Euclidean space that is 3.14 and remains constant. However, space time is not a flat Euclidean space. It can be curved (by the presence of mass/energy). In such cases pi will not equal 3.14. For example if you draw a circle on the surface of a sphere, the diameter of the circle (while remaining on the surface of the sphere) would be larger than in flat space and so pi would be less. (for the equator, on the surface of the earth, pi would be 2!). So Pi is not always constant. Thus I am still looking for an answer to my question.

 

[Doc Al]

Pi is constant only in Euclidian geometry / space. Pi is defined as the number of times the diameter of a circle divides into the circumference. If flat Euclidean space that is 3.14 and remains constant. However, space time is not a flat Euclidean space. It can be curved (by the presence of mass/energy). In such cases pi will not equal 3.14. For example if you draw a circle on the surface of a sphere, the diameter of the circle (while remaining on the surface of the sphere) would be larger than in flat space and so pi would be less. (for the equator, on the surface of the earth, pi would be 2!). So Pi is not always constant. Thus I am still looking for an answer to my question.

The value of pi we use is the one from Euclidean geometry. As pointed out, it's a constant. While the ratio of circumference to diameter of a circle may well change in different geometries, we wouldn't call such ratios pi.

 

[Arjun Kumar]

I was reading up on this at the link below.

http://www.maxwells-equations.com/materials/permeability.php

at this site I saw that permeability in a vacuum is defined as

μ=4(pi).10(exp-7)

which is what got me confused?

 

[pervect]

As per Maxwell's equations the speed of light is

c = 1/√(permeability X permittivity)

I find that both permeability and permittivity have pi in their calculations. Since spacetime is not flat, the value of pi would change due to the curvature of space near large masses.

Even on a curved manifold, the value of the circumference of a circle to its radius will approach pi locally for a small enough circle, because the tangent space to the manifold is flat.

As others point out , "Pi" really is defined as a constant, but even if you look at the geometry, the ratio of circumference to diameter approaches a constant even for non-euclidean geometries.

(There are a few pathological exceptions, such as if the circle encloses a singularity, but they don't change the argument).

Does this mean that the speed of light would also be altered by the curvature of spacetime?

Nope

 

[Arjun Kumar]

I agree that the value of the circumference of a circle to is radius will approach pi locally for a small enough circle. While this is true and that every 'point' in space is flat, beyond a 'point' space is not flat and can take different forms. (Every point on the surface on Earth is flat, but we know that the surface forms a sphere - both views are correct!)
My confusion was that if pi, (being in my belief a ratio and not a constant) can change, then by also being a part of the equations for the speed of light, the speed of light would also be subject to change.

 

[vanhees71]

The choice of the values for $\epsilon_0$ and $\mu_0$ is some men made convention to get convenient quantities for currents, voltages, and charges in electrical engineering, something Nature couldn't care less about.

The only fundamental constant that enters the theory of electromagnetism compared to Newtonian mechanics is the universal speed of light in the vacuum. As it has come out through Einstein's SRT paper of 1905, this is a much more universal speed than just the speed of light in vacuum.

More recent analyses of the special relativity principle has shown that there are, up to equivalence, only two space-time manifolds that obey the special relativity principle (i.e., the indistinguishablitiy of all inertial frames, i.e., the impossibility to measure an absolute constant velocity) are either Galilei-Newton space-time (a fibre bundle, i.e., you simply pin 3D Euclidean spaces at each point along the time axis) or Einstein-Minkowski space-time (a pseudo-Euclidean affine manifold with a fundamental form of signature (1,3) or equivivalently (3,1), depending on which sign convention you prefer). The latter implies a universal speed $c$, which to the best of our knowledge is the phase velocity of electromagnetic waves in the vacuum. This is because empirically there is no hint for a non-zero photon mass (to put it in modern QFT languague).

This confusion can be avoided by using the more natural Gaussian or the rationalized Gaussian (Heaviside-Lorentz) system of units, where no artificial constants due to the choice of an arbitrary fourth independent unit for electric current (in the SI the unit Ampere) or charge (in the SI the unit Coulomb=Ampere times second) is introduced. That's why not long ago usually textbooks on theoretical classical electrodynamics were written using the Gaussian units and why in theoretical high-energy physics one usually uses the Heaviside-Lorentz system of units. Nowadays, unfortunately the textbooks on classical electromagnetism are written using the SI units, spoiling somewhat the beauty of Maxwell theory as a relativistic classical field theory.

In my opinion, the best compromise is found in the third edition of Jackson's textbook: He starts using the SI units in order to be compatible with the use of this units in experimental physics and engineering and then switching to Gaussian units in the chapters where the electrodynamics is treated in its true form as a relativistically covariant field theory, writing "SI" or "G" in the header line of the chapters :-).

 

[TrickyDicky]

I agree that the value of the circumference of a circle to is radius will approach pi locally for a small enough circle. While this is true and that every 'point' in space is flat, beyond a 'point' space is not flat and can take different forms. (Every point on the surface on Earth is flat, but we know that the surface forms a sphere - both views are correct!)
My confusion was that if pi, (being in my belief a ratio and not a constant) can change, then by also being a part of the equations for the speed of light, the speed of light would also be subject to change.

There is no confusion on your part, but there seems to have been some misunderstanding in the replies as they got stuck in a "constants discussion" that is unrelated to the OP.

The very first paper on GR by Einstein in 1911 "On the Influence of Gravitation on the Propagation of Light" addresses this issue. This is the one where he got the deflection of light wrong by a half since he didn't have the EFE yet, but the equivalence principle was there already.

So to make it short, just saying the speed of light is subject to change can be misunderstood. As you can see since c is referred to the local measure of c (where you have admitted as explained by others that pi is a constant) this speed doens't vary. However it is alright to say(with qualifiers) that in the presence of gravitational fields it does, and that's what we see as bending or deflection of light, gravitational lensing etc. In the words of Einstein "From the proposition which has just been proved, that the velocity of light in the gravitational field is a function of the location...".
So the conclusion you draw in the OP is right, but you have to be careful about what you mean. The speed of light c is a local constant just by the same reason pi is a local constant.

 

[Dale]

As per Maxwell's equations the speed of light is

c = 1/√(permeability X permittivity)

I find that both permeability and permittivity have pi in their calculations. Since spacetime is not flat, the value of pi would change due to the curvature of space near large masses. Does this mean that the speed of light would also be altered by the curvature of spacetime?

As Bill_K mentioned, c is a defined constant. It is like π in that its value is set by definition, but it is different from π in that its value depends on the system of units used. In natural units c=1 by definition. In SI units c=299792458 m/s by definition.

Now, whether or not the speed of light is equal to c is a slightly more interesting question. Again, the answer depends on your system of units. In SI units the speed of light is c also by definition. In other units the question is not a tautology, but nevertheless the answer tells you about the units rather than about the physics. A closely related physical question is whether or not the photon has mass, if it has mass then light of different wavelengths will travel at different speeds under any consistent system of units (and units that assume otherwise will be inconsistent).

Now, you asked about c in curved spacetime, where things get a little more complicated. You can talk about the coordinate speed of light, either locally or non-locally. Locally this can tell you some information about your coordinate system. Non-locally, I am not sure what you can learn from the coordinate speed of light. I suspect that it is still mostly information about your coordinates, however by combining local and non-local measurements maybe you could learn something about the curvature of spacetime along your non-local measurement also.

Instead of using coordinates you can use a reference frame (aka frame field or vierbein). However, a vierbein is defined as being orthonormal, and the normalization makes it so that c is defined once again. So even in curved spacetime the physical question remains whether or not the photon has mass.

 

[mathman]

Pi is constant only in Euclidian geometry / space. Pi is defined as the number of times the diameter of a circle divides into the circumference. If flat Euclidean space that is 3.14 and remains constant. However, space time is not a flat Euclidean space. It can be curved (by the presence of mass/energy). In such cases pi will not equal 3.14. For example if you draw a circle on the surface of a sphere, the diameter of the circle (while remaining on the surface of the sphere) would be larger than in flat space and so pi would be less. (for the equator, on the surface of the earth, pi would be 2!). So Pi is not always constant. Thus I am still looking for an answer to my question.

You are right in that the ratio depends on the geometry. However, π itself is a defined constant. The ratio in non-Euclidean geometry doesn't have a particular name.

As for the speed of light - I believe it remains unchanged.

 

[pervect]

I agree that the value of the circumference of a circle to is radius will approach pi locally for a small enough circle. While this is true and that every 'point' in space is flat, beyond a 'point' space is not flat and can take different forms. (Every point on the surface on Earth is flat, but we know that the surface forms a sphere - both views are correct!)
My confusion was that if pi, (being in my belief a ratio and not a constant) can change, then by also being a part of the equations for the speed of light, the speed of light would also be subject to change.

To use your example, if you reject the idea of taking the limit, what would you say the ratio of the circumference of a circle to its radius on the Earth's surface (or, if you prefer, a perfect sphere the same size as the Earth) is?