Maxwell's Equations (ME)

[Alan Grayson]

Maxwell's Equations are written in vector form, and vectors are tensors, and tensors are invariant under change of coordinates. It is known that ME are invariant under the Lorentz transformation, and predict that EM waves travel at the velocity of light regardless of the coordinate system. So, applying instead the Galilean transformation to ME, shouldn't they also predict the same velocity of light as the Lorentz transformation? TY AG

[John Clark]

On Thu, Sep 26, 2024 at 6:22 AM Alan Grayson <agrays...@gmail.com> wrote:

> Maxwell's Equations are written in vector form, and vectors are tensors, and tensors are invariant under change of coordinates. It is known that ME are invariant under the Lorentz transformation, and predict that EM waves travel at the velocity of light regardless of the coordinate system. So, applying instead the Galilean transformation to ME, shouldn't they also predict the same velocity of light as the Lorentz transformation? TY AG

Yes and that's why Maxwell's Equations needed no modification to be consistent with Special Relativity or General Relativity, although they are inconsistent with Quantum Mechanics. Maxwell's theory predicted what the speed of light would be but it didn't say what specific thing that speed was relative to; before Einstein came along some thought that was a flaw in the theory, but it turned out to be its greatest triumph. 

John K Clark    See what's on my new list at  Extropolis

cgr

 

[Alan Grayson]

Does your "Yes" mean that ME will remain invariant under a Galilean transformation (which I doubt), and make the same prediction for the Sol (which I also doubt)? Or did you forget or ignore my question? AG  

[Alan Grayson]

So what does your "Yes" mean? Please clarify. TY, AG 

[John Clark]

On Sun, Sep 29, 2024 at 6:57 AM Alan Grayson <agrays...@gmail.com> wrote:

>> Yes and that's why Maxwell's Equations needed no modification to be consistent with Special Relativity or General Relativity, although they are inconsistent with Quantum Mechanics. Maxwell's theory predicted what the speed of light would be but it didn't say what specific thing that speed was relative to; before Einstein came along some thought that was a flaw in the theory, but it turned out to be its greatest triumph. 

> Does your "Yes" mean that ME will remain invariant under a Galilean transformation

No. Galilean Relativity assumes there is no speed limit in the universe, and velocities always combine linearly even if they're going close to the speed of light, and simultaneity is an objective fact because time is the same for all observers.  Einsteinian Relativity makes none of these assumptions. So Maxwell's Equations are inconsistent with Galilean Relativity but not with Einsteinian relativity. According to Galileo you could travel as fast as a beam of light and if you did you would see a frozen electromagnetic wave that did not change with time, but Maxwell had no equation that described such a thing. That inconsistency was the primary motivation Einstein had for developing Special Relativity.

John K Clark    See what's on my new list at  Extropolis

der

 

 

[Alan Grayson]

But ME are written in tensor form. Doesn't that mean the equations are invariant under coordinate transformations? If so, shouldn't ME be invariant under the Galilean transformation, which is a coordinate transformation? Where has my logic gone awry? TY, AG. 

[John Clark]

On Sun, Sep 29, 2024 at 8:41 AM Alan Grayson <agrays...@gmail.com> wrote:

> But ME are written in tensor form.

Yes. 

> Doesn't that mean the equations are invariant under coordinate transformations?

Yes.

>If so, shouldn't ME be invariant under the Galilean transformation, which is a coordinate transformation?

NO. Galilean relativity is a very good approximation of reality as long as the speeds don't become too high, and it would also be completely invariant under coordinate transformation IF Galileo's assumptions were correct; namely that there is no speed limit in the universe, and velocities always combine linearly even if they're going close to the speed of light, and simultaneity is an objective fact because time is the same for all observers.  To put it another way, invariant Galilean transformations are mathematically consistent BUT experiment shows they are NOT physically consistent because Galileo's physical assumptions were NOT correct. It is necessary that a physical theory be mathematically consistent but it is not sufficient.

 John K Clark    See what's on my new list at  Extropolis

nsf

[Alan Grayson]

Something's still awry. You agree that tensor equations are invariant under coordinate transformations, and that the Galilean transformation is a coordinate transformation, so that should be enough for ME to be invariant under the GT. Yet you assert it isn't. It's as if the deficiencies of the GT are sufficient to undo what seems immediately true. This is the inverse of another situation I've come across in the definition of the tangent plane in GR. Here, one uses all velocities on the spacetime manifold through the tangent point P, to define the vector space on the tangent plane on which the metric tensor is defined. But notice that for this to work, all velocities are used through P, including those with v > c. That is, in this case, the condition of reality is relaxed to get the desired result -- the definition of the tangent plane -- but in the case of ME, the condition of reality is strongly enforced. Hopefully, you can say something about these situations, that will make me feel I understand what's going on. At present I don't. TY, AG  

[John Clark]

On Sun, Sep 29, 2024 at 2:07 PM Alan Grayson <agrays...@gmail.com> wrote:

 >> Galilean relativity is a very good approximation of reality as long as the speeds don't become too high, and it would also be completely invariant under coordinate transformation IF Galileo's assumptions were correct; namely that there is no speed limit in the universe, and velocities always combine linearly even if they're going close to the speed of light, and simultaneity is an objective fact because time is the same for all observers.  To put it another way, invariant Galilean transformations are mathematically consistent BUT experiment shows they are NOT physically consistent because Galileo's physical assumptions were NOT correct. It is necessary that a physical theory be mathematically consistent but it is not sufficient.

> Something's still awry. You agree that tensor equations are invariant under coordinate transformations, and that the Galilean transformation is a coordinate transformation, so that should be enough for ME to be invariant under the GT. Yet you assert it isn't.

All Galilean tensor equations are 100% mathematically consistent, but physically they are not even approximately correct unless the speeds are very low, and if you're talking about Maxwell's Equations the speeds are very high. You can have Galilean transformations that are mathematically invariant with any change of coordinates, but  they are not physically consistent with experimental results because they contain assumptions which turned out not to be physically true. 

 John K Clark    See what's on my new list at  Extropolis 

npy

[Alan Grayson]

Sorry, but this doesn't work for me. The physical deficiencies of the Galilean transformation should not have any effect of the non-invariance of ME when it is applied. So the solution must relate to some error in my concept of the invariance of tensors under coordinate transformations. AG

[Brent Meeker]

A Galilean boost could be an accurate transformation in some other world, but it's not in this world.  It's only an approximation at small boosts.  There can be more than one mathematically consistent transformation but only one physically realized one.

Brent

[Alan Grayson]

On Sunday, September 29, 2024 at 8:04:32 PM UTC-6 Brent Meeker wrote:

A Galilean boost could be an accurate transformation in some other world, but it's not in this world.  It's only an approximation at small boosts.  There can be more than one mathematically consistent transformation but only one physically realized one.

Brent

I think you and JC are assuming ME are "true" and the GT is "false", and from this arises an inconsistency which manifests in a non-invariance of the former when the latter are applied. But the fact is that ME are not true, insofar as they assume EM waves are continuous. Consequently, I think I misinterpreted how equations written in tensor form transform under coordinate transformations. That is, there's a subtle but important difference between coordinate transformations, and frame of reference transformations. Specifically, the GT is NOT a coordinate transformation. For example, in coordinate transformations, such as between spherical and rectangular coordinates, all the coordinates are well-defined and fixed, but due to the motion assumed in a GT, this is not the case. AG

[John Clark]

On Mon, Sep 30, 2024 at 3:14 AM Alan Grayson <agrays...@gmail.com> wrote:

> there's a subtle but important difference between coordinate transformations, and frame of reference transformations

That's very true.  A rank 1 Tensor (a.k.a.  a vector) is not necessarily invariant under changes in the coordinate system, instead it transforms in a specific, consistent way. For example angular momentum is not invariant under coordinate translations, but that’s OK because it's a feature of the physical situation and is not a sign of inconsistency. Changing the origin alters the angular momentum calculation but this is physically consistent because angular momentum is inherently tied to the reference point. There is no such thing as absolute angular momentum; it depends on where you measure it from.

So strictly speaking angular momentum is not a tensor it's a pseudo-tensor, or if you prefer a pseudo-vector because it has most of the properties of a tensor but not all of them. A true tensor remains unchanged under parity inversion (the letter O looks the same in a mirror) but a pseudo tensor does not (the letter L does not look the same in a mirror).

[Alan Grayson]

Interesting. TY. My conclusion, in part, is that the LT and the GT are not coordinate transformations, and therefore we can't assume that ME will be invariant under those transformations, simply because ME are written in tensor form.  But the original question more or less remains; namely, why are ME invariant under the former transformation, but not the latter? AG

[Alan Grayson]

On Monday, September 30, 2024 at 1:41:16 PM UTC-6 John Clark wrote:

On Mon, Sep 30, 2024 at 3:14 AM Alan Grayson <agrays...@gmail.com> wrote:

> there's a subtle but important difference between coordinate transformations, and frame of reference transformations

That's very true.  A rank 1 Tensor (a.k.a.  a vector) is not necessarily invariant under changes in the coordinate system, instead it transforms in a specific, consistent way. For example angular momentum is not invariant under coordinate translations, but that’s OK because it's a feature of the physical situation and is not a sign of inconsistency. Changing the origin alters the angular momentum calculation but this is physically consistent because angular momentum is inherently tied to the reference point. There is no such thing as absolute angular momentum; it depends on where you measure it from.

With further consideration, I think angular momentum would be invariant under a coordinate transformation because where you measure AM from, the reference point, won't change in a CT, since points don't move; they just get new labels. AG 

[John Clark]

On Mon, Sep 30, 2024 at 11:14 PM Alan Grayson <agrays...@gmail.com> wrote:

>> A rank 1 Tensor (a.k.a.  a vector) is not necessarily invariant under changes in the coordinate system, instead it transforms in a specific, consistent way. For example angular momentum is not invariant under coordinate translations, but that’s OK because it's a feature of the physical situation and is not a sign of inconsistency. Changing the origin alters the angular momentum calculation but this is physically consistent because angular momentum is inherently tied to the reference point. There is no such thing as absolute angular momentum; it depends on where you measure it from.

So strictly speaking angular momentum is not a tensor it's a pseudo-tensor, or if you prefer a pseudo-vector because it has most of the properties of a tensor but not all of them. A true tensor remains unchanged under parity inversion (the letter O looks the same in a mirror) but a pseudo tensor does not (the letter L does not look the same in a mirror).

> Interesting. TY. My conclusion, in part, is that the LT and the GT are not coordinate transformations, and therefore we can't assume that ME will be invariant under those transformations, simply because ME are written in tensor form.  But the original question more or less remains; namely, why are ME invariant under the former transformation, but not the latter? AG

The proximate cause is that velocities in Lorentz Transformations and in our physical world do not combine linearly like they do with Galilean Transformations. The ultimate cause nobody knows and there may not even be an ultimate cause. Nobody knows that either.

   John K Clark    See what's on my new list at  Extropolis

nny

[smitra]

Yes see:

https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime

So, the speed of light in a global coordinate system scales with the
square root of the determinant of the metric tensor. If you define
distances and time intervals locally using the metric tensor at each
point, the locally defined speed of light will be the same constant
everywhere.

Saibal

[Alan Grayson]

Note that the LT and the GT are frame transformations, not coordinate 

transformations (they're not the same) so even if ME are written in tensor

form (vectors are rank 1 tensors), there's no reason to expect ME to be

invariant under either transformation. But they are, only under the LT, but 

not under the GT. Do you know why? I don't see how your answer relates

to my question. Please clarify. TY, AG 

[smitra]

I see! You indeed need to transform both the 4-vector and the coordinate
system for the GR formalism to apply here.

Saibal

On 02-10-2024 00:45, Alan Grayson wrote:
> On Tuesday, October 1, 2024 at 6:44:04 AM UTC-6 smitra wrote:
>
>> On 26-09-2024 12:22, Alan Grayson wrote:
>>> Maxwell's Equations are written in vector form, and vectors are
>>> tensors, and tensors are invariant under change of coordinates. It
>> is
>>> known that ME are invariant under the Lorentz transformation, and
>>> predict that EM waves travel at the velocity of light regardless
>> of
>>> the coordinate system. So, applying instead the Galilean
>>> transformation to ME, shouldn't they also predict the same
>> velocity of
>>> light as the Lorentz transformation? TY AG
>>>
>>
>> Yes see:
>>
>>
> https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime
>>
>>
>> So, the speed of light in a global coordinate system scales with the
>>
>> square root of the determinant of the metric tensor. If you define
>> distances and time intervals locally using the metric tensor at each
>>
>> point, the locally defined speed of light will be the same constant
>> everywhere.
>>
>> Saibal
>

> Note that the LT and the GT are FRAME transformations, NOT coordinate

> transformations (they're not the same) so even if ME are written in
> tensor
> form (vectors are rank 1 tensors), there's no reason to expect ME to
> be
> invariant under either transformation. But they are, only under the
> LT, but
> not under the GT. Do you know why? I don't see how your answer relates
> to my question. Please clarify. TY, AG
>

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