# Why is length contracted but time is dilated? [duplicate]

I've just started special relativity and have a quick question on the terminology. From a stationary frame, lorentz transformation gives both length and time of moving frame scaled by the same factor:
$$\Delta x = \gamma \Delta x'\\\Delta t = \gamma \Delta t'$$

Then why does my textbook say length is contracted but time is dilated? From the equations, both the length and time are contracted. What am I missing?

• youtu.be/-NN_m2yKAAk – BioPhysicist Dec 28 '19 at 12:13
• You really need to think about what those changes really represent, and how one measures the length of something that is moving. – BioPhysicist Dec 28 '19 at 12:19
• @Aaron Stevens thankyou for that awesome video. I'm still going through them... but both $\Delta x'$ and $\Delta t'$ are smaller than $\Delta x$ and $\Delta t$ respectively right? – AgentS Dec 28 '19 at 12:21
• Not particularly conversant in special relativity, but shouldn't gamma be in the denominator of the first equation? See hyperphysics.phy-astr.gsu.edu/hbase/Relativ/tdil.html – Bob D Dec 28 '19 at 14:40
• Please state what the delta values actually represent here. Are they for comparing two events? – BioPhysicist Dec 28 '19 at 19:29

I'm not particularly conversant in special relativity, but I think your first equation is incorrect. Assuming prime designates the moving frame, I believe gamma in the first equation should be in the denominator. Then, given $$γ>1$$

$$\Delta x=\frac{\Delta x^{'}}{γ}$$

$$\Delta t=γ\Delta t^{'}$$

Where,

$$\Delta x^{'}$$ is the length of the moving object as measured in the moving frame (proper length), and $$\Delta x$$ is the length of the moving object as measured in the stationary frame (contracted length).

$$\Delta t^{'}$$ is the elapsed time of the moving clock as measured in the moving frame (proper time), and $$\Delta t$$ is the elapsed time of the moving clock as measured in the stationary frame (dilated time).

Hope this helps.

• It's true that there is something wrong with the material the OP paraphrases from their text, but that isn't all that's wrong. Actually the way that SR treats the x and t coordinates is totally symmetric. What's asymmetric is that a ruler is not analogous to a clock. I discuss this in my answer to the question that this one duplicates. – user4552 Dec 28 '19 at 20:13
• @BenCrowell Thanks Ben. I'm a bit new to this myself. Is there anything wrong with my answer?. I would appreciate any feedback. – Bob D Dec 28 '19 at 20:21
• That was the point of my comment. The equations you give are also wrong if they're just a partial version of the Lorentz transformation -- this gets into the issue of defining the notation, which was never very clear in the OP's statement of this question. In any case, this question is a duplicate. – user4552 Dec 28 '19 at 20:30
• So I think the first equation is correct if you apply the correct Lorentz transformation for events that happen simultaneously in the primed frame. I don't think that equation is supposed to represent length contraction. – BioPhysicist Dec 28 '19 at 20:31
• So then are the equations given in the above Hyperphysics link also wrong because they appear consistent with mine? – Bob D Dec 28 '19 at 20:32

Suppose a rocket going from Earth to Alpha Centauri.

From the rocket frame, the distance between these two locations is shorter than the 4 light years that an Earth observer measures. That is length contraction.

And when the rocket arrives in Alpha Centauri, his clock will show more time passed than the clocks there (of course the later must be synchronized to the Earth). That is time dilation.

The logic is to measure the trip distance and duration, and compare with the other frame measurements.

Velocity implies equal distance, but either shorter or longer time, so to a person in a car driving on land, and seeing the road as a kilometer would be a contraction of length, but actually the road is 2 kilometers long, dilation of time is also inferred into contraction of distance such that whatever distance we see may take longer time to cover if the velocity of the moving vehicle is constant, and this greatly defines the ideology of spacetime, for example, the sun viewed from the earth seems like the sun is just outside the earth, probably around the same distance as the moon, but in reality, it is millions of miles away from the moon and earth. In essence,it all depends on the way light behaves and how our eyes can intercept and our brain can interpret the concept of space time.

Length is contracted because we're talking about the length of the object, and time is dilated because we are talking about the scale of the measuring device (length between tics).

It could just have well been "ruler dilation" and "interval contraction".

• One way to see this is wrong is to realize that you're implying that the time and distance phenomena are analogous. Actually they're not, as discussed in my answer to the question that this one duplicates. – user4552 Dec 28 '19 at 20:12
• @BenCrowell I am not talking about a physical ruler, I'm talking about "length". A moving yard stick is no longer a yard stick. – JEB Dec 29 '19 at 0:20