# From the direct Lorentz transformation to the inverse one.

• bernhard.rothenstein
In summary, the "inverse" Lorentz transformation can be obtained by interchanging primed and unprimed physical quantities and changing the sign of the relative velocity of the involved inertial reference frames. This is only valid if the frames are in the standard arrangement, otherwise a more general approach is needed.
bernhard.rothenstein
from the "direct" Lorentz transformation to the "inverse" one.

Consider please that we know the "direct" transformation say for the time coordinates of the same event. It is considered that we can obtain the "inverse" transfrmation by simply interchanging the corresponding unprimed physical quantities with primed ones and to change the sign of the relative velocity of the involved inertial reference frames.
Please let me know a explanation of that.
Thanks

$$x^{\prime} = \gamma (x - vt)$$

$$t^{\prime} = \gamma (t - vx/c^2)$$

Consider them as a system of two equations for two unknowns x and t, and solve them for x and t in terms of $x^{\prime}$ and $t^{\prime}$.

bernhard.rothenstein said:
It is considered that we can obtain the "inverse" transfrmation by simply interchanging the corresponding unprimed physical quantities with primed ones and to change the sign of the relative velocity of the involved inertial reference frames.
Please let me know a explanation of that.

It has nothing to do with SR. The inverse of a coordinate transformation of moving 30mph to the left is one where you're moving 30mph to the right. Full stop.

You can demonstrate this for Newtonian mechanics somewhat more easily, and is good practice for the slightly trickier SR case.

The equality $\Lambda(v)^{-1}=\Lambda(-v)$ can be derived from the Minkowski metric.

If your starting point isn't Minkowski space but instead Einstein's "postulates", and your goal is to "derive" the form of the Lorentz transformation, then the above equality should be viewed as a mathematical statement of (a weak version) of the first postulate. The first postulate isn't needed for anything else in the "derivation", but it's needed for this. Without this relationship, the gamma factor in the Lorentz transformation would have to be replaced by an arbitrary constant.

Note however that Einstein's postulates aren't well-defined enough to be taken as the starting point of a rigorous proof of anything. If you use this approach when teaching relativity, I think you should make that clear to your students. The "postulates" don't even say that spacetime is represented by $\mathbb R^4$, and they don't define what an inertial frame is. They don't explain what a "law of physics" is, and they mention "light" (which is something that can only be defined by a theory of electrodynamics) without making clear that "the speed of light" is only a reference to a particular set of curves in spacetime.

I'm not saying that it's a bad idea to use this approach. I'm just saying that it should be presented as what it really is: It's not a proof. It's just a way for us to find a mathematical model of spacetime that we can try to use in a new theory of physics. The model we end up with is of course Minkowski space. The actual postulates of SR (i.e. the statements that really define the theory) are the statements that describe the relationship between things we measure and things in Minkowski space.

Yeah, I know I'm nagging about this a lot. I do it because because it still irks me that none of the teachers who taught relativity at my university ever mentioned this, and that I didn't fully understand these things until years later. I'm thinking that if it took me a long time to figure this out, then there must be lots of other people who haven't figured it out yet (including many of those who teach relativity at universities).

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jtbell said:

$$x^{\prime} = \gamma (x - vt)$$

$$t^{\prime} = \gamma (t - vx/c^2)$$

Consider them as a system of two equations for two unknowns x and t, and solve them for x and t in terms of $x^{\prime}$ and $t^{\prime}$.

Thanks for your answer. In the spirit of it, would you aggree with the following:
"Generalizing the experience of those who derive transformation equations in special relativity theory we could postulate: If we know a "direct" transformation then we obtain the inverse one by simply interchanging the corresponding primed physical quanities with unprimed ones and changing the signj of the relative velocity of the involved inertial reference frames."
We could add: The inverse transformation could be obtained by considering the same scenario from the other involve inertial reference frame.

bernhard.rothenstein said:
Thanks for your answer. In the spirit of it, would you aggree with the following:
"Generalizing the experience of those who derive transformation equations in special relativity theory we could postulate: If we know a "direct" transformation then we obtain the inverse one by simply interchanging the corresponding primed physical quanities with unprimed ones and changing the signj of the relative velocity of the involved inertial reference frames."
We could add: The inverse transformation could be obtained by considering the same scenario from the other involve inertial reference frame.
No, this is not correct in general. If two inertial reference frames are related to each other through a rotation as well as a boost then the boost in the unprimed frame will not generally be the negative of the boost in the primed frame. The approach illustrated in jtbell's answer is the best. It works not just for SR inertial frames, but all coordinate transformations in general.

DaleSpam said:
No, this is not correct in general. If two inertial reference frames are related to each other through a rotation as well as a boost then the boost in the unprimed frame will not generally be the negative of the boost in the primed frame. The approach illustrated in jtbell's answer is the best. It works not just for SR inertial frames, but all coordinate transformations in general.
Thanks
Let me state again.
"If the inertial reference frames involved in the transformation process are in the standard arrangement then we could postulate that if we know a "direct" transformation we could obtain the inverse one by simply interchanging the corresponding primed physical quantities with unprimed and changing the sign of the relative velocity."

jtbell said:

$$x^{\prime} = \gamma (x - vt)$$

$$t^{\prime} = \gamma (t - vx/c^2)$$

Consider them as a system of two equations for two unknowns x and t, and solve them for x and t in terms of $x^{\prime}$ and $t^{\prime}$.

Thanks. I think that is a proof and not a "postulate".

Certainly. You can postulate any logically consistent idea, no matter how trivial, but the usual goal is to use as few postulates as possible. So why would you waste a postulate on a result that is so easy to derive.

## 1. What is the direct Lorentz transformation?

The direct Lorentz transformation, also known as the forward Lorentz transformation, is a mathematical equation used to describe the relationship between space and time coordinates in two different reference frames moving at constant velocities relative to each other. It was first derived by Hendrik Lorentz in the late 19th century and later expanded upon by Albert Einstein in his theory of special relativity.

## 2. How is the direct Lorentz transformation used in physics?

The direct Lorentz transformation is a fundamental tool in understanding the effects of relativity on space and time. It is used in many areas of physics, including astrophysics, particle physics, and cosmology. It allows scientists to accurately calculate and predict how measurements of space and time will differ between reference frames moving at different velocities.

## 3. What is the inverse Lorentz transformation?

The inverse Lorentz transformation is the mathematical inverse of the direct Lorentz transformation. It allows for the transformation of space and time coordinates from one reference frame to another, in the opposite direction. This is useful for calculating measurements in a stationary reference frame from measurements taken in a moving reference frame.

## 4. How is the inverse Lorentz transformation related to the direct Lorentz transformation?

The inverse Lorentz transformation is the reverse process of the direct Lorentz transformation. They are two sides of the same equation and are used together to describe the relationship between space and time coordinates in different reference frames. The direct transformation is used to calculate measurements in a moving reference frame, while the inverse transformation is used to calculate measurements in a stationary reference frame.

## 5. Are there any practical applications of the Lorentz transformations?

Yes, the Lorentz transformations have many practical applications in modern technology. They are used in the development of GPS systems, particle accelerators, and even in the correction of time dilation in satellite communications. Without the Lorentz transformations, many modern technological advancements would not be possible.

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