# Lorentz transformation for non-inertial frame

• lovetruth
In summary, there are Lorentz transformation equations that relate non-inertial frames to inertial frames. There are also transformations that relate non-inertial frames to another non-inertial frame. By 'non-inertial frame', I mean frame of reference having absolute acceleration, jerk... or any n-th order time derivative of position.
lovetruth
Are there lorentzian transformation equations relating non-inertial frame to inertial frame. Also are there transformations relating non-inertial frame to another non-inertial frame. By 'non-inertial frame', I mean frame of reference having absolute acceleration,jerk... or any n-th order time derivative of position. Thanks for any help in advance.

JDoolin said:

I could not understand the link you have posted

lovetruth said:
I could not understand the link you have posted

Don't worry about the post. Can you see the attachment?

Lorentz transformations only relate Lorentz inertial frames. A Lorentz inertial frame is one in which the spacetime metric has the form diag(-1,1,1,1).

We can use non-inertial frames to describe physics too, but the forms of the equations in a Lorentz inertial frame and an accelerated frame are related by different transformations.

http://en.wikipedia.org/wiki/Rindler_coordinates gives the relationship between Lorentz inertial coordinates (T,X) and Rindler coordinates (t,x).

atyy said:
Lorentz transformations only relate Lorentz inertial frames. A Lorentz inertial frame is one in which the spacetime metric has the form diag(-1,1,1,1).

We can use non-inertial frames to describe physics too, but the forms of the equations in a Lorentz inertial frame and an accelerated frame are related by different transformations.

http://en.wikipedia.org/wiki/Rindler_coordinates gives the relationship between Lorentz inertial coordinates (T,X) and Rindler coordinates (t,x).

Thanks for the link. But the maths is very complicated unlike lorentz transformation.

lovetruth, Special Relativity does a full and complete job of describing acceleration, but to get there the first thing you must do is to drop the word "frame", which does not apply. A Lorentz frame is a coordinate system that fills all of space and time. While you might try to imitate this for nonzero acceleration, accelerating coordinate systems such as rotating coordinates or Rindler coordinates cannot fill all of space-time without becoming pathological. Meaning that somewhere the time coordinate becomes spacelike, and the apparent velocity exceeds the velocity of light. Paradoxes about accelerating spaceships and rotating discs are a result of disregarding this fact.

The correct approach is to describe physics as seen by a single accelerated observer. Or by a family of observers. The description will need to be local, not global. Rather than an accelerating coordinate system you introduce a tetrad, i.e. a set of basis vectors, at each point along the observer's world line. At each point there is a Lorentz transformation which locally matches the motion of the observer.

Bill_K said:
lovetruth, Special Relativity does a full and complete job of describing acceleration, but to get there the first thing you must do is to drop the word "frame", which does not apply. A Lorentz frame is a coordinate system that fills all of space and time. While you might try to imitate this for nonzero acceleration, accelerating coordinate systems such as rotating coordinates or Rindler coordinates cannot fill all of space-time without becoming pathological. Meaning that somewhere the time coordinate becomes spacelike, and the apparent velocity exceeds the velocity of light. Paradoxes about accelerating spaceships and rotating discs are a result of disregarding this fact.

The correct approach is to describe physics as seen by a single accelerated observer. Or by a family of observers. The description will need to be local, not global. Rather than an accelerating coordinate system you introduce a tetrad, i.e. a set of basis vectors, at each point along the observer's world line. At each point there is a Lorentz transformation which locally matches the motion of the observer.

In more colloquial terms, you can think of the Rindler coordinates as a rocketship. There's no problem taking that rocketship and accelerating it through the universe, but if you decide to make that rocketship several light-years long, strange things happen. The back of the rocketship must maintain a higher acceleration than the front end in order for it to keep its shape. And if you imagine people building a staircase further and further down beneath the rocketship, eventually you come to a point where the acceleration reaches infinity; go below that (an this is where it gets "pathological," maybe, and time start's to go backwards.)

A similar sort of thing happens with the spinning wheel. If you rotate a small wheel, there is no problem. But if you make that wheel bigger and bigger, so that the outside edge is moving at a significant fraction of the speed of light, it will fracture apart, quickly. Build the structure until the outside edge is going "faster" than the speed of light, then it becomes something that is mathematically impossible.

Mike_Fontenot said:
There is an easy and quick way for an accelerating observer to answer the question, "What is the current age of, and the current distance to, some particular distant person". Start with these links:

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7

https://www.physicsforums.com/showpost.php?p=2923277&postcount=1

Mike Fontenot

Again, more colloquially, Mike often discusses the "Current Age of Distant Objects." If you are riding in that accelerating rocket-ship, and looking behind you, you can calculate the age of someone far behind you, in your "Momentarily comoving reference frame". So long as they are "closer" than the rindler horizon, their "current age" will be progressing forward. But if they are further than the rindler horizon, they will be aging backwards.

It's kind of a technicality, though. Because what's really going on is that you are constantly changing inertial reference frames (As Mike stresses). If the Rindler coordinates are pathological, it is in the sense that they take something that is obviously true (that a person in an accelerating rocketship is continuously changing to new inertial reference frames) and try to make it into a steady-state "non-inertial reference frame," where in some places, time flows forward, and others, time flows backward.

Bill_K said:
lovetruth, Special Relativity does a full and complete job of describing acceleration, but to get there the first thing you must do is to drop the word "frame", which does not apply. A Lorentz frame is a coordinate system that fills all of space and time. While you might try to imitate this for nonzero acceleration, accelerating coordinate systems such as rotating coordinates or Rindler coordinates cannot fill all of space-time without becoming pathological. Meaning that somewhere the time coordinate becomes spacelike, and the apparent velocity exceeds the velocity of light. Paradoxes about accelerating spaceships and rotating discs are a result of disregarding this fact.

The correct approach is to describe physics as seen by a single accelerated observer. Or by a family of observers. The description will need to be local, not global. Rather than an accelerating coordinate system you introduce a tetrad, i.e. a set of basis vectors, at each point along the observer's world line. At each point there is a Lorentz transformation which locally matches the motion of the observer.

Thanks for the information. Change in velocity of light in accelerated frame would act as "Newton's Bucket". Can you tell more about the paradoxes or effects experienced by an accelerating observer.

JDoolin said:
In more colloquial terms, you can think of the Rindler coordinates as a rocketship. There's no problem taking that rocketship and accelerating it through the universe, but if you decide to make that rocketship several light-years long, strange things happen. The back of the rocketship must maintain a higher acceleration than the front end in order for it to keep its shape. And if you imagine people building a staircase further and further down beneath the rocketship, eventually you come to a point where the acceleration reaches infinity; go below that (an this is where it gets "pathological," maybe, and time start's to go backwards.)

A similar sort of thing happens with the spinning wheel. If you rotate a small wheel, there is no problem. But if you make that wheel bigger and bigger, so that the outside edge is moving at a significant fraction of the speed of light, it will fracture apart, quickly. Build the structure until the outside edge is going "faster" than the speed of light, then it becomes something that is mathematically impossible.

How can time flow backward? This is absolutely impossible!

JDoolin said:
In more colloquial terms, you can think of the Rindler coordinates as a rocketship. There's no problem taking that rocketship and accelerating it through the universe, but if you decide to make that rocketship several light-years long, strange things happen. The back of the rocketship must maintain a higher acceleration than the front end in order for it to keep its shape. And if you imagine people building a staircase further and further down beneath the rocketship, eventually you come to a point where the acceleration reaches infinity; go below that (an this is where it gets "pathological," maybe, and time start's to go backwards.)

A similar sort of thing happens with the spinning wheel. If you rotate a small wheel, there is no problem. But if you make that wheel bigger and bigger, so that the outside edge is moving at a significant fraction of the speed of light, it will fracture apart, quickly. Build the structure until the outside edge is going "faster" than the speed of light, then it becomes something that is mathematically impossible.

when anybody is accelerated, the momentum is transmitted from part of the object to another part through mechanical waves (i.e. compression and decompression). The whole object is not accelerated instantly, there are velocity gradients over the object. The speed of mechanical wave should be always less than speed of light. Thus, there can not be any categorical rigid body otherwise, information can be sent faster than light speed.

How can time flow backward? This is absolutely impossible!
Precisely. And that is why we cannot speak of accelerating "frames." As another example, a rotating frame cannot exist either. If you sit at the North Pole and watch the stars go by, the poor people on Alpha Centauri are going once around every 24 hours in your "frame", greatly exceeding the speed of light. An accelerating frame is simply not a useful concept, and moreover is not necessary.

lovetruth said:
How can time flow backward? This is absolutely impossible!

If at one moment, the current age of a distant object is, say, 10. And a moment later, the current age of a distant object is, say, 9. One claims, if using a "non-inertial frame" such as the Rindler coordinates, that time is "flowing backwards" in that region. Of course, time is not really flowing backwards in that region. Your rocketship can't possibly extend back that far for you to put clocks on it.

You're just in a new reference frame, where the current age of some particular distant object is actually less than the age of the same distant object in the reference frame you were in before.

Hmmm, I should also mention this region is in the opposite direction from what is usually of interest in the "twin paradox" problem. When you are accelerating toward the stay-at-home twin, you'll find the current age is not slowed down or going backwards, but greatly sped up. It is when you look in the opposite direction, toward what you are accelerating FROM, when you see the time slowed down (closer than the rindler horizon), then stop (at the rindler horizon), then flow backwards (past the rindler horizon). So the Rindler Horizon, and the Twin Paradox are sort of like two sides of the same coin, so-to-speak. The Rindler horizon is what you're accelerating away from, and the twin paradox is what you're accelerating toward.

(Edit: If you graphed the lines of simultaneity during the accelerated part of the journey in the twin paradox, the Rindler horizon would be where the consecutive lines of simultaneity intersect; I couldn't find an image of this on google-docs. Most twin-paradox space-time-diagrams have instantaneous acceleration, so the Rindler horizon is AT the Event of acceleration.)

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Bill_K said:
Precisely. And that is why we cannot speak of accelerating "frames." As another example, a rotating frame cannot exist either. If you sit at the North Pole and watch the stars go by, the poor people on Alpha Centauri are going once around every 24 hours in your "frame", greatly exceeding the speed of light. An accelerating frame is simply not a useful concept, and moreover is not necessary.

In accelerating frame(or tetrad), the apparent speed of light can vary. I think accelerating observers should age less than inertial observer due to twin-paradox. But I can see how accelerating observer will deduce that the inertial observer ages faster.

JDoolin said:
So the Rindler Horizon, and the Twin Paradox are sort of like two sides of the same coin, so-to-speak. The Rindler horizon is what you're accelerating away from, and the twin paradox is what you're accelerating toward.

Doesn't the Twin-Paradox involves a twin accelerating away & also towards the earth.

lovetruth said:
Doesn't the Twin-Paradox involves a twin accelerating away & also towards the earth.

Yes, but during the acceleration "away" you're dealing with the "current age of a nearby object" which is not nearly as interesting a question as the "current age of a distant object."

Of course the same math can be involved if you like, but it's like asking what happens on the rocketship one or two floors below you instead of 10 light-years above and below you. Nothing out-of-the-ordinary.

(Edit: The lines of simultaneity tilt on a fulcrum located at the observer. The further away the object, the more drastic the effect of the relativity of simultaneity.)

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## 1. What is the Lorentz transformation for non-inertial frames?

The Lorentz transformation is a mathematical equation used to describe the relationship between space and time in special relativity. It allows us to transform coordinates and measurements from one frame of reference to another, specifically from an inertial frame (a frame with constant velocity and no acceleration) to a non-inertial frame (a frame with acceleration).

## 2. What are the assumptions of the Lorentz transformation for non-inertial frames?

The Lorentz transformation assumes that the frames of reference are in uniform motion relative to each other and that the speed of light is constant for all observers, regardless of their relative motion. It also assumes that there are no external forces acting on the non-inertial frame.

## 3. How does the Lorentz transformation differ for inertial and non-inertial frames?

The Lorentz transformation for inertial frames involves only translations and rotations, while the transformation for non-inertial frames also includes acceleration and non-uniform motion. In non-inertial frames, the transformation is more complex and involves additional terms to account for the effects of acceleration on space and time.

## 4. Can the Lorentz transformation be used for all types of non-inertial frames?

No, the Lorentz transformation is only applicable to non-inertial frames that have constant acceleration and are undergoing uniform motion. For non-uniform or non-constant acceleration, more advanced equations, such as the Rindler transformation, must be used.

## 5. What are the practical applications of the Lorentz transformation for non-inertial frames?

The Lorentz transformation is essential for understanding and predicting the behavior of objects in non-inertial frames, such as in space travel and in systems with acceleration, such as rotating reference frames or accelerating particles in particle accelerators. It is also crucial for accurate measurements and calculations in special relativity.

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