Understanding Negative t Coordinates in Lorentz Transformations

In summary, when performing Lorentz transformations, the sign is placed on the v, not the t. This is because the metric, s2=-t2+x2+y2+z2, must have a negative sign in order to be preserved under Lorentz transformations. If the sign were positive, different observers would measure different ds's. This can be visualized using a space-time diagram, where the red and blue coordinates represent different observers moving at the same relativistic speed. To compute the length of an object in the other observer's coordinates, you must use the red leg of the triangle, and solve using Pythagorean theorem or vector addition.
  • #1
dpa
147
0
In lorentz transformation,
why has t -sign
 
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  • #2
The sign doesn't go on the t but on the v. Depending on direction of your boost you can have either +vt or -vt.
 
  • #3
i mean for the metric,
s2=-t2+x2+y2+z2.

I beg your pardon if i did not get you. Would you mind to clarify.
 
  • #4
Oh, in the metric the sign is negative because you want a "length" which is preserved under the natural transformations of the space (Lorentz transformations in this case).

ds^2=dt^2+dx^2+dy^2+dz^2 would not be preserved under a Lorentz transformation. The version with the minus sign is. In other words, ds with the minus sign definition is something that all observers would agree on, but ds with the plus sign definition, different observers would measure different ds's.
 
  • #5
dpa said:
In lorentz transformation,
why has t -sign

Because it is the leg of a triangle that is being computed, not the hypotenuse. Google "special relativity space-time diagram" to study how to interpret the sketch below. We have red and blue guys moving in opposite directions at the same relativistic speed relative to the black coordinates. The blue guy uses the equation to compute the length of some object in the red guy's coordinates. Then, it's the red leg of the triangle that's being computed.

Given the way the coordinates are oriented for red and blue in special relativity, you can identify a right triangle--then just use Pythagorean theorem and solve for the red length. dX1'^2 +dX4^2 = dX1^2. Or, use vector addition as shown.
Minkowski_Vectors2.jpg
 
  • #6
thanks
matterwave and bobc2
 

1. Why is the t coordinate negative?

The t coordinate is often used to represent time in a coordinate system. In a Cartesian coordinate system, the x, y, and z coordinates can all be positive or negative, depending on their position relative to the origin. Similarly, the t coordinate can also be positive or negative, with the origin representing a specific point in time. The direction of the t coordinate is usually determined by the direction of the x-axis, with time increasing in the positive direction and decreasing in the negative direction.

2. Does a negative t coordinate have a specific meaning?

The meaning of a negative t coordinate can vary depending on the context in which it is used. In some cases, it may represent time before a designated starting point, while in others it may represent a time interval or duration.

3. Can the t coordinate ever be positive?

Yes, the t coordinate can be positive. This usually occurs when the designated starting point is in the past, and the t coordinate represents time after that point.

4. How is the t coordinate determined in a coordinate system?

The t coordinate is typically determined by the unit of measurement used for time in the coordinate system. For example, in a Cartesian coordinate system, the unit of measurement for time could be seconds, minutes, or hours, depending on the application.

5. Is the t coordinate always necessary in a coordinate system?

No, the t coordinate is not always necessary in a coordinate system. It is often used in physics and other scientific fields to represent time, but in other applications, it may not be relevant or useful. Additionally, some coordinate systems may use a different variable to represent time, such as the letter "t" or "tau".

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