# Coordinate transforms VS space-time transforms

4 views

### z@z

Jan 13, 2000, 3:00:00 AM1/13/00
to
The whole LET-SR-GR-GET debate (and probably also the corresponding
theories themselves) is overshadowed by the confusion between
coordinate transformations (Lorentz Ether Theory) and space-time
transformations (Special Relativity).

The cordinate transformation x'=2x means that in the coordinate
system S' a value of 10 m is attributed to the same objective
length to which 5 m are attributed in the system S. Rulers of S'
are contracted wrt the rulers in S.

The same space-time transformation x'=2x means exactly the contrary:
A length of 5 m in the system S is transformed into a length of 10 m
in system S'. Rulers of S' are expanded by factor 2 wrt the rulers
in S.

Whereas (original) LET uses a Lorentz coordinate transformation,
SR is fundamentally based on a Lorentz space-time transformation.

If we use a space-time transformation in LET, the correct
equations are not the SR equations

Dx' = [Dx - v Dt ] * gamma [1a]
Dt' = [Dt - v Dx /c^2] * gamma [1b]
Dx = [Dx'+ v Dt' ] * gamma [2a]
Dt = [Dt'+ v Dx'/c^2] * gamma [2b]

but

Dx' = [Dx - v Dt ] / gamma [1A]
Dt' = [Dx - v Dx/c^2] / gamma [1B]

and length contraction must be derived from [1A]

Dt = 0 --> Dx'= Dx / gamma

whereas in SR it is derived from [2a]

Dt' = 0 --> Dx = Dx' * gamma --> Dx' = Dx / gamma

Regards, Wolfgang

### Tom Roberts

Jan 13, 2000, 3:00:00 AM1/13/00
to
"z@z" wrote:
> The whole LET-SR-GR-GET debate (and probably also the corresponding
> theories themselves) is overshadowed by the confusion between
> coordinate transformations (Lorentz Ether Theory) and space-time
> transformations (Special Relativity).

Not really. Apparently you have never heard of the difference between
"active" transforms and "passive" transforms, and attempted to make
up a whole new vocabulary for them. In an active transform one uses
a single set of coordinates and performs some modification of the
physical system; in a passive transform one performs the inverse
modification to the coordinates.

So, for instance, an active transform could consist of a 90-degree
rotation of an object clockwise around its vertical axis. The
corresponding passive transform is a 90-degree rotation of the
coordinates counterclockwise around that same axis. The results
of each of these are the same (vis-a-vis the relationship between
globe and coordinates).

Both concepts are valid, and indeed are related by an isomorphism.
It does not matter which concept you use, as long as you remain
consistent throughout.

> If we use a space-time transformation in LET, the correct

> equations are not the SR equations [...] but

> Dx' = [Dx - v Dt ] / gamma [1A]
> Dt' = [Dx - v Dx/c^2] / gamma [1B]

^
+-- typo, should be t

You are wrong. Demonstrably. Look in Lorentz's 1904 paper. His
transforms multiply by gamma (his beta) just like SR's.

Why do you think they are called _LORENTZ_ transforms?
Hint: Lorentz used them in his 1904 paper, and 1904 came
before 1905.

Oh yes, your usage of "Dx" and "Dt" (etc.) are simply devices you
think highlight that you are discussing differences rather than
coordinates. In fact, the usual notation "x" and "t" also refer to
differences -- between the event in question and the event selected
as x=0,t=0.

This only holds in a linear theory, like SR and LET.

Tom Roberts tjro...@lucent.com

### Daryl McCullough

Jan 13, 2000, 3:00:00 AM1/13/00
to
Tom Roberts says...

>Both concepts [passive and active transforms]

>are valid, and indeed are related by an isomorphism.
>It does not matter which concept you use, as long as you remain
>consistent throughout.

I would like to point out that there *is* a difference
between the two concepts when it comes to invariance
properties. To say that the laws of physics should be
invariant under passive transformations is really to
say nothing at all. You can always write any laws
in such a way that any coordinate system can be
used. However, there is real physical content to
the claim that the laws of physics are invariant under
*active* transformations.

Daryl McCullough
CoGenTex, Inc.
Ithaca, NY

### Tom Roberts

Jan 14, 2000, 3:00:00 AM1/14/00
to
Daryl McCullough wrote:
> Tom Roberts says...
> >Both concepts [passive and active transforms]
> >are valid, and indeed are related by an isomorphism.
> >It does not matter which concept you use, as long as you remain
> >consistent throughout.
> I would like to point out that there *is* a difference
> between the two concepts when it comes to invariance
> properties.

I disagree. See below.

> To say that the laws of physics should be
> invariant under passive transformations is really to
> say nothing at all. You can always write any laws
> in such a way that any coordinate system can be
> used.

Yes. The usual way to do this is to specify the laws in a specific
coordinate system, and then _define_ them in any other coordinate
system to be what the originally-stated laws transform to in the
other coordinate system. This is the essence of the "comma goes to
semicolon" rule (specify the laws in an inertial frame, then replace
ordinary derivatives with covariant derivatives; implicitly one
generalizes the tensors of the inertial frame into the corresponding
tensors of the curved manifold, and the covariance group of the
tensors implicitly expands from that of the inertial frame to the
necessary general covariance of the curved manifold).

"inertial frame" = "flat manifold".

> However, there is real physical content to
> the claim that the laws of physics are invariant under
> *active* transformations.

Not really. I think what you mean is to claim that invarance related
to a _REAL_ transform (i.e. a real physical modification to the system
which leaves some property unchanged) has "real physical content". No
argument there, but that is not what is meant by an active transform.
I'll come back to this point.

As an example, let me choose rotations. To say a given law is invariant
over rotations means that the law does not refer to the orientation of
the system in any way. An active transform rotates the system relative
to the coordinates, and a passive transform rotates the coordinates
relative to to the system. These have the same content and the same
end result WHEN EXPRESSED IN TERMS OF THE COORDINATES OF THE SYSTEM.
So for laws expressed in terms of coordinates they are equivalent
(isomorphic).

[In passing I remark that this is an incentive to express the
laws in a coordinate-independent way. The law:
g^ik F_ij;k = 0
has both active and passive transforms. The law:
dF = 0
has neither, because it is not expressed in terms of
coordinates at all.]

Consider a manifold M. A coordinate system C is a 1-to-1 mapping of a
region of M onto a region of R^n. I consider only C's domain. An active
transform A is a mapping of M onto itself. The corresponding passive
transform is a mapping P of the region of R^n to itself such that
CA = PC for every point of M in the domain of C. By construction these
two mappings A and P are isomorphic within this region; each is a
mapping from this region of M onto the region of R^n.

The point is, both active and passive transforms are operations WITHIN
THE MODEL. The "real" transform I discussed above is NOT within the
model, but is a real change to the physical system (so the word
"transform" is not really appropriate, but is the best I could do).

Yes, invariances under "real" transforms like this have "real physical
content". But the distinction between active and passive transforms
is purely formal, and both are wholly within the model. Neither has
any "real physical content" at all, and they are merely different
ways of looking at the model and/or of relating the model to the world.

Getting back to rotations, you will have _great_ difficulty describing
a "real" rotation of the _ENTIRE_ system -- to what will you reference
this "rotation" to? But the active and passive transforms do not suffer
from this difficulty, as they both merely affect the relationship
between the system and the coordinates (i.e. between the "real" world
and the model coordinates). And the coordinates were arbitrary in the
first place.

A final remark: the restriction of SR to inertial frames is
a restriction which has significant physical content. But
the general covariance of GR is without such physical content,
as discussed above.

Tom Roberts tjro...@lucent.com

### z@z

Jan 16, 2000, 3:00:00 AM1/16/00
to
: = Tom Roberts
:: = Daryl McCullough
::: = Tom Roberts
:::: = Wolfgang G. in http://www.deja.com/=dnc/getdoc.xp?AN=571890451

:::: The whole LET-SR-GR-GET debate (and probably also the corresponding

:::: theories themselves) is overshadowed by the confusion between
:::: coordinate transformations (Lorentz Ether Theory) and space-time
:::: transformations (Special Relativity).

::: Not really. Apparently you have never heard of the difference between
::: "active" transforms and "passive" transforms, and attempted to make
::: up a whole new vocabulary for them. In an active transform one uses
::: a single set of coordinates and performs some modification of the
::: physical system; in a passive transform one performs the inverse
::: modification to the coordinates.

::: Both concepts are valid, and indeed are related by an isomorphism.

::: It does not matter which concept you use, as long as you remain
::: consistent throughout.

:: I would like to point out that there *is* a difference between the

:: two concepts when it comes to invariance properties. To say that the

:: laws of physics should be invariant under passive transformations is
:: really to say nothing at all. You can always write any laws in such

:: a way that any coordinate system can be used. However, there is real

:: physical content to the claim that the laws of physics are invariant
:: under *active* transformations.

: Not really. I think what you mean is to claim that invariance related

: to a _REAL_ transform (i.e. a real physical modification to the system
: which leaves some property unchanged) has "real physical content". No
: argument there, but that is not what is meant by an active transform.
: I'll come back to this point.
:
: As an example, let me choose rotations. To say a given law is invariant
: over rotations means that the law does not refer to the orientation of
: the system in any way. An active transform rotates the system relative
: to the coordinates, and a passive transform rotates the coordinates
: relative to to the system. These have the same content and the same
: end result WHEN EXPRESSED IN TERMS OF THE COORDINATES OF THE SYSTEM.
: So for laws expressed in terms of coordinates they are equivalent
: (isomorphic).

: The point is, both active and passive transforms are operations WITHIN

: THE MODEL. The "real" transform I discussed above is NOT within the
: model, but is a real change to the physical system (so the word
: "transform" is not really appropriate, but is the best I could do).
:
: Yes, invariances under "real" transforms like this have "real physical
: content". But the distinction between active and passive transforms
: is purely formal, and both are wholly within the model. Neither has
: any "real physical content" at all, and they are merely different
: ways of looking at the model and/or of relating the model to the world.

If we accept this notation, we can say that both LET and SR use "REAL"
transforms. The decisive question however is, whether LET is also based
on ACTIVE transforms as SR undoubtedly is.

Lorentz presents in his well-known 1904 paper these formulae (4,5):

x' = BLx, y' = Ly, z' = Lz, t' = Lt/B - (BLx/c)(v/c)

Poincaré presents in his last pre-SR paper of June 1905:

x' = kl(x + epsilon t), y' = ly, z' = lz, t' = kl(t + epsilon x)

Taking into consideration that L = l = 1, we get

for Lorentz (a Galilean transform is implicitely assumed):

x' = gamma x, t' = t/gamma - (gamma x/c)(v/c)

for Poincaré:

x' = gamma (x + vt) t' = gamma (t + vx/c^2)

for Einstein:

x' = gamma (x - vt) t' = gamma (t - vx/c^2)

In the case of Lorentz, it is clear that only a PASSIVE transform can
be meant, because otherwise x' = gamma x would represent an expansion
and not a contraction.

In the case of Poincaré apart from the continuity with the work of
Lorentz also the fact that he uses "+" where Einstein uses "-" could
be evidence that he uses a PASSIVE transform.

If we write Einstein's version as a PASSIVE transform, we get the
following result, don't we?

x' = (x + vt) / gamma t' = (t + vx/c^2) / gamma

Whereas the math is exactly the same, there is huge physical difference
between the ACTIVE and the PASSIVE interpretation. So from a purely
mathematical point of view, a Lorentz-expansion is fully equivalent
to Lorentz-contraction, because it depends only on the way (ACTIVE or
PASSIVE) we interpret the Lorentz transform.

Wolfgang Gottfried G. (0:015.6)

### Tom Roberts

Jan 22, 2000, 3:00:00 AM1/22/00
to
"z@z" wrote:
> In the case of Poincaré apart from the continuity with the work of
> Lorentz also the fact that he uses "+" where Einstein uses "-" could
> be evidence that he uses a PASSIVE transform.

Either that, or they defined the direction of motion in the opposite
direction of each other.

> If we write Einstein's version as a PASSIVE transform, we get the
> following result, don't we?
> x' = (x + vt) / gamma t' = (t + vx/c^2) / gamma

No. You have to invert the transform, and get:

x' = gamma (x + vt) t' = gamma (t + vx/c^2)

> Whereas the math is exactly the same, there is huge physical difference
> between the ACTIVE and the PASSIVE interpretation.

Not really. They're merely different points of view, and are
_isomorphic_ to each other.

> So from a purely
> mathematical point of view, a Lorentz-expansion is fully equivalent
> to Lorentz-contraction, because it depends only on the way (ACTIVE or
> PASSIVE) we interpret the Lorentz transform.

Not true. There is never any sort of "Lorentz expansion". You made
a mistake in inverting the transform above.

Tom Roberts tjro...@lucent.com

### Daryl McCullough

Jan 23, 2000, 3:00:00 AM1/23/00
to
Tom says...
>
>Daryl McCullough wrote:

>> However, there is real physical content to
>> the claim that the laws of physics are invariant under
>> *active* transformations.
>

>Not really. I think what you mean is to claim that invarance related

>to a _REAL_ transform (i.e. a real physical modification to the system
>which leaves some property unchanged) has "real physical content". No
>argument there, but that is not what is meant by an active transform.
>I'll come back to this point.

I still don't agree with you here, but let me explain
why I say that there is physical content to invariance
under active transformations.

A coordinate system is basically a particular way of
describing the states of a system. Different coordinate
systems describe the same physical state in different
ways. Given any transformation T that maps one coordinate
system into another, we can define a corresponding map
m from states to states as follows:

m(s) = that state s' such that the description of s' in
coordinate system C is the same as the description
of s in coordinate system T(C).

So, as far as descriptions of states, it doesn't make any
difference whether you keep the state constant, and change
coordinates from C to T(C), or you keep the coordinate system
constant, and change state from s to m(s).

But once we start talking about dynamics, there is a big
difference between these two. To say that the laws of
dynamics are invariant under the physical change s --> m(s)
means that

d(m(s)) ds
------- = m ----
dt dt

This definitely has physical content.

### z@z

Jan 25, 2000, 3:00:00 AM1/25/00
to
: = Tom Roberts
:: = Wolfgang G. in http://www.deja.com/=dnc/getdoc.xp?AN=573350660

:: Taking into consideration that L = l = 1, we get

::
:: for Lorentz (a Galilean transform is implicitely assumed):
::
:: x' = gamma x, t' = t/gamma - (gamma x/c)(v/c)
::
:: for Poincaré:

::
:: x' = gamma (x + vt) t' = gamma (t + vx/c^2)
::
:: for Einstein:
::
:: x' = gamma (x - vt) t' = gamma (t - vx/c^2)

::
:: In the case of Lorentz, it is clear that only a PASSIVE transform can
:: be meant, because otherwise x' = gamma x would represent an expansion
:: and not a contraction.

:: In the case of Poincaré apart from the continuity with the work of

:: Lorentz also the fact that he uses "+" where Einstein uses "-" could
:: be evidence that he uses a PASSIVE transform.
:
: Either that, or they defined the direction of motion in the opposite
: direction of each other.

In any case, I just had to recognize that also Einstein uses a
PASSIVE transform.

"If Dx, Dt, etc. denote coordinate differences, the Lorentz
transformation takes in the case of a relative speed of 0.6 c
this form:

Dx' = 1.25 [Dx - 0.6 c Dt] [ 1a ]
Dt' = 1.25 [Dt - O.6/c Dx] [ 1b ]

Dx = 1.25 [Dx' + 0.6 c Dt'] [ 2a ]
Dt = 1.25 [Dt' + O.6/c Dx'] [ 2b ]

The proper lenght Dx' of the body in the moving frame S' is 1 LY.
A superficial examination could suggest that equation [2a] entails
an elongation by the factor 1.25 (resulting in 1.25 LY) for the
frame S. But correct is exactly the contrary: a contraction to

This is from a text I wrote in 1988 and translated into English in
1999. The whole argument that Einstein uses an active transform
results from my mixing up [1a] with [2a].

:: So from a purely
:: mathematical point of view, a Lorentz-expansion is fully equivalent

:: to Lorentz-contraction, because it depends only on the way (ACTIVE or
:: PASSIVE) we interpret the Lorentz transform.
:
: Not true. There is never any sort of "Lorentz expansion". You made
: a mistake in inverting the transform above.

Yes. In another respect however, EXPANSION is as present in SR as
CONTRACTION. Every moving object contracted by gamma wrt the rest
frame is expanded by the same factor (again wrt the rest frame) when
observed simultanously in the moving frame. (That this requires
constant velocities is a serious problem for SR.)

The EXPANSION can also be explained using LET. Assume a body moving
at v = sqrt(0.99)c wrt the ether. If its rest length is 10 m, then
it is contracted to 1 m. If we send from the center of the body a
signal to both edges, the sum of both signal paths is 100 m. Whereas
the path in direction of the ether wind is only around 0.25 m [1] the
signal in the opposite direction crosses an ether distance of 99.75 m
[2] before reaching the edge of the body.

Sorry for confusion
Wolfgang

[1] 0.5 m / (1 + sqrt(0.99) = 0.25 m
[2] 0.5 m / (1 - sqrt(0.99) = 99.75 m

### Tom Roberts

Jan 25, 2000, 3:00:00 AM1/25/00
to
"z@z" wrote:
> Yes. In another respect however, EXPANSION is as present in SR as
> CONTRACTION. Every moving object contracted by gamma wrt the rest
> frame is expanded by the same factor (again wrt the rest frame) when
> observed simultanously in the moving frame. (That this requires
> constant velocities is a serious problem for SR.)

If you intermix measurements and observations from different
coordinate systems you can obtain any sort of nonsense you like. But
in SR, when you only discuss what a given observer can observe, there
is never any expansion of a moving object (i.e. the moving object is
never measured to be larger than its proper size).

By "observer" I really mean measurements made from a single
inertial frame. In particular, the observer normally is
considered to have assistants collocated with any
measurements, and they all use clocks and rulers at rest
in the observer's inertial frame, and synchronized in it.

I have no idea what you are trying to say in your last parenthetical
remark above.

> The EXPANSION can also be explained using LET.

Yes, one can make the same mistake in LET, and obtain the same
nonsense. LET is mathematically equivalent to SR, and offers the
same opportunities to make errors (:-)).

Tom Roberts tjro...@lucent.com

### z@z

Jan 26, 2000, 3:00:00 AM1/26/00
to
: = Tom Roberts in http://www.deja.com/=dnc/getdoc.xp?AN=577450711
:: = Wolfgang G. in http://www.deja.com/=dnc/getdoc.xp?AN=577202662

:: EXPANSION is as present in SR as

:: CONTRACTION. Every moving object contracted by gamma wrt the rest
:: frame is expanded by the same factor (again wrt the rest frame) when
:: observed simultanously in the moving frame. (That this requires
:: constant velocities is a serious problem for SR.)
:
: If you intermix measurements and observations from different
: coordinate systems you can obtain any sort of nonsense you like. But
: in SR, when you only discuss what a given observer can observe, there
: is never any expansion of a moving object (i.e. the moving object is
: never measured to be larger than its proper size).

It is clear that all the following variants can directly be derived
from the SR equations:

Dx' = Dx * gamma Dt' = Dt / gamma
Dx' = Dx / gamma Dt' = Dt * gamma

To all of them correspond clearly defined physical situations. If
a fast moving dark object of length L sends a light signal from
each end at the same moving-frame time, then the distance between
the two points where the signals are emitted is L*gamma in the rest
frame. This cannot be denied in a reasonable way!

In an analoguous way, the time of a moving system goes faster by
gamma wrt an observer at rest in the rest frame. If this were not
true, then also clocks of a rest frame could not run faster wrt
the time of a particle at rest in a moving frame.

: I have no idea what you are trying to say in your last parenthetical
: remark above.

That you have no idea only shows that you do not fully understand
SR length contraction. If not even you understand such a fundamental
principle of SR, then I must conclude that (almost?) nobody on
earth really understands SR (let alone GR).

See for instance: http://www.deja.com/=dnc/getdoc.xp?AN=551801940

:: The EXPANSION can also be explained using LET. [...]
:
: Yes, one can make the same mistake in LET, and obtain the same

: nonsense. LET is mathematically equivalent to SR, and offers the
: same opportunities to make errors (:-)).

I did not make a mistake. Try again:

"The EXPANSION can also be explained using LET. Assume a body moving
at v = sqrt(0.99)c wrt the ether. If its rest length is 10 m, then
it is contracted to 1 m. If we send from the center of the body a
signal to both edges, the sum of both signal paths is 100 m. Whereas

the path in direction of the ether wind is only around 0.25 m the

signal in the opposite direction crosses an ether distance of 99.75 m

before reaching the edge of the body."

Inasfar as LET is equivalent to SR it must be able to explain that
wrt a moving observer the ether shrinks by gamma in the same way as
the moving observer shrinks wrt the ether. Because real contraction
of the ether is impossible, the only remaining possibility is an
expansion of the observer by gamma.

Wolfgang

### Tom Roberts

Jan 27, 2000, 3:00:00 AM1/27/00
to
"z@z" wrote:
> Inasfar as LET is equivalent to SR it must be able to explain that
> wrt a moving observer the ether shrinks by gamma in the same way as
> the moving observer shrinks wrt the ether. Because real contraction
> of the ether is impossible, the only remaining possibility is an
> expansion of the observer by gamma.

You have only a superficial understanding when you try to explain
this in terms of "contractions" only. You forgot the possibility
that the simultaneity of the measurements is different in the
different frames, so that the different observers in different
frames do not measure the same things. Lo and behold this explains
it fully when one works out the math.

Tom Roberts tjro...@lucent.com

### Tom Roberts

Jan 28, 2000, 3:00:00 AM1/28/00
to
Lou Verdon wrote:
> ...and how different is the math from the case that clocks physically
> change Newtonian "speed" as a function of the spacetime field which is so
> well characterized in GR?

You need to be far more precise. What sort of "field" do you mean, and
how does it affect "newtonian speed"?

> Errors in the relativity of simultaneity in 4 dimensional transformations
> are well known to create foreshortening or rotation of images. Why would
> you not consider the possibility of clock error due to an external field?

No "external field" is necessary, because the effects predicted by SR
to be changes in perspective are sufficient to describe zillions of
experimental measurements. So if one added in some sort of distortions
due to some "external field" one would not obtain agreement with
experiments.

And if you think these effects due purely to the influence of some
"external field" (i.e. no perspective at all), then this field must
affect more than clock rates. And you must also explain why perspective
in 3-space is geometrical but perspective in 4-space is partly
geometrical and partly due to your "external field". Unless you also
explain the ordinary 3-d perspective as due to some "directioniferous
ether" (but so far nobody has done that).

Ordinary 3-d perspective is what causes a ruler observed from
the side to appear wider than when observed from the end. This
is isomorphic to time dilation and length contraction in SR --
they are additional manifestations of geometrical perspective
in the 4 dimensions of Minkowski spacetime.

> SR does not identify the physical cause of time dilation, but it is very
> clear that there is a physical cause.

Not true. SR identifies the "cause" of time dilation as being due to
the geometry of the measurement. In SR it is QUITE CLEAR there is no
sort of "physical cause" at all, it is merely geometry.

> Why would you consider such a potential not to exist when an average field
> of less than 1 volt per 10^11 meters of space is all that is necessary?
> Do you think that such a minute field could be measured with normal
> laboratory instruments? In fact, we cannot even approach such resolution.
> Such a field is, I believe, wrongly approximated to nil in any real
> experiment.

At Fermilab experiments have sizes on the order of 10^3 meters, so your
putative field is on the order of 10^-8 volt difference from one end
to the other. This is ENORMOUSLY too small to cause a time dilation
with gamma ~2000 as seen in pions and muons at Fermilab. But I agree
nobody would measure it (:-)).

Actually, such static fields as you propose are very difficult
to measure. If you extend a voltmeter probe from "ground"
to some point far away, the equipotential surfaces near ground
will surround the lead and the voltmeter reads 0 even if there
is a large potential difference across the spatial region.
You cannot measure a potential difference across "empty space"
with a voltmeter, you can only measure one between two objects.
You could attempt to observe it by letting a charged particle
accelerate in the potential difference, but 10^-11 V/m would
be hopelessly too small to observe that way.

> The strong force is observed in the strong field and the weak force is
> observed in the weak field, and they are both simple electrical force.

You are believing the words "strong" and "weak" far too literally. In
particular, the form of the strong force is VERY DIFFERENT from that
of any known electrical force. Speaking loosely, the strong force gets
STRONGER as you separate two strongly-charged particles; electrical
forces get weaker as you separate two electrically-charged particles.
But also, weak forces are observed inside nuclei (e.g. beta decay) as
are strong forces, so they inhabit the same region but behave very
differently; _neither_ of them behaves in a manner similar to
electrical forces (e.g. the weak force causes neutrons to decay,
but neutrons are electrically neutral).

The strong and weak forces are both short-range nuclear
forces. They were named because of their relative coupling
constants at low energies.