# Minkowski spacetime diagrams, what does the hyperbola represent?

• knowlewj01
In summary, the conversation discusses the task of drawing a Minkowski spacetime diagram to represent two events: a farmer firing his laser gun at his cow, and the cow dying. These events are observed by two different frames of reference: S, at rest in the farmer's and cow's frame, and S', moving at 0.5c in the direction of the negative x direction. The diagram shows the two frames and the events on a lightlike region, with the angle between the axes determined by the velocity of the S' frame. The question asks for the placement of hyperbolae to determine the scale of the ct, x and ct', x' axes and if there is a frame where it appears that the cow's
knowlewj01

## Homework Statement

Draw a clearly labelled “Minkowski spacetime” diagram illustrating two events
((1) a farmer firing his laser gun at his cow, which is sitting along his positive x-direction, and
(2) the cow dying) as observed by two observers (S at rest in the farmer’s and cow’s frame,
and S’ moving at 0.5 c in the direction of the negative x direction.
Show on the diagram, qualitatively, hyperbolae which determine the scale of the ct, x and
ct’, x’ axes. Using the diagram, is there any frame S’ where it appears that the cow dying
caused the farmer to shoot his gun?

## Homework Equations

See attached file for my diagram so far

## The Attempt at a Solution

on my diagram you can see the S and S' frames, where S' is moving in the negative x direction with a velocity of 0.5c. I'm aware that the angle between the axes of the two different frames are governed by the velocity of the S' frame. The question asks for hyperbolae which determine the scale, I'm not sure where these should go. The black dot shows the farmer firing the gun and the red dot show the time when the cow dies in each frame, should the hyperbola pass through both of these points?

#### Attachments

• minkowski.jpg
10.2 KB · Views: 469
changed my mind, the gun is a laser so information about the cow's death travels at the speed of light, the dotted line shows lightlike region, the region where the event of the cow's death occurs. The cow is a certain distance from the farmer, so in the S frame the sequence of events happens as in the vertical and horizontal axes. In the S' frame the events appear as in the axes at an angle to S frame. In both frames the events are on the lightlike region. I still cannot see the parts of the diagram to draw the hyperbolae whos tangents bive the x to x' and ct to ct' transformations. Anyone know where to draw the hyperbolae?

#### Attachments

• minkowski 2.jpg
12.7 KB · Views: 464

## 1. What is a Minkowski spacetime diagram?

A Minkowski spacetime diagram is a visual representation of the mathematical concept of Minkowski spacetime, which is a four-dimensional spacetime that combines space and time into a single entity. It is used in the theory of special relativity to visualize the effects of time dilation and length contraction.

## 2. How is a Minkowski spacetime diagram constructed?

A Minkowski spacetime diagram is constructed by plotting two axes, one representing time and the other representing space, on a two-dimensional graph. The time axis is usually represented vertically, and the space axis is represented horizontally. The units of measurement for time and space are typically chosen to be the same, such as meters or seconds.

## 3. What do the diagonal lines on a Minkowski spacetime diagram represent?

The diagonal lines on a Minkowski spacetime diagram, also known as worldlines, represent the paths of objects moving at a constant velocity. The slope of the line represents the speed of the object, with steeper slopes indicating faster speeds. The worldlines of objects moving at the speed of light are represented by 45-degree angles.

## 4. What does the hyperbola in a Minkowski spacetime diagram represent?

The hyperbola in a Minkowski spacetime diagram represents the relationship between space and time for an object moving at a constant speed. It is a curved line that separates the regions where time and space are considered "positive" from the regions where they are considered "negative". The closer the hyperbola is to the origin, the slower the object is moving.

## 5. How do Minkowski spacetime diagrams help us understand special relativity?

Minkowski spacetime diagrams help us understand special relativity by providing a visual representation of the effects of time dilation and length contraction. They allow us to see how the perception of time and space changes for objects moving at different speeds, and how these changes are interconnected. They also help us understand the concept of spacetime as a unified entity, rather than separate dimensions.

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