Special relativity - it confuses me.

Hi there, i've been set a few relativity questions and I -really- just don't get them, I was wondering if anyone could give me a little help to get me started!

In 2D spacetime an inertial frame

S'

moves with velocity

u

relative to an inertial frame

S

. A particle

p

moves with speed

v

with respect to

S

and

v'

with respect to

S'

, so if its position is measured at two successive instants

dx = vdt

and

dx'= v' dt'

. Suppose the two clocks agree for

p

, i.e.,

dt' = dt.

How do I show that

p

is moving with constant velocity? I've written down all the relevant equations but nothing is jumping out at me, I'm pretty sure I'm just being stupid. Differentiating

x'

with respect to t (or t') gives

v'=\gamma (v-u)

but I don't even know whether or not that's the right thing to do, let alone what to do next.

Secondly, I have to draw the following as a spacetime diagram:
A clock

C

is at rest at the spatial origin of an inertial frame

S

. A second clock

C'

is at rest at the spatial origin of an inertial frame

S'

moving with constant speed

U

relative to

S

. The clocks read

t = t' = 0

when the two spatial origins coincide. When

C'

reads

t_2'

it receives a radio signal from

C

sent out when

C

reads

t_1

. I know you're meant to post your working for any problems like these but I really don't even know where to start :frown:

I understand what's going on in the second question but I don't know how to draw the space-time diagram to represent it. Help!

Reply 1

Y__

csmith101

Hi there, i've been set a few relativity questions and I -really- just don't get them, I was wondering if anyone could give me a little help to get me started!

In 2D spacetime an inertial frame

S'

moves with velocity

u

relative to an inertial frame

S

. A particle

p

moves with speed

v

with respect to

S

and

v'

with respect to

S'

, so if its position is measured at two successive instants

dx = vdt

and

dx'= v' dt'

. Suppose the two clocks agree for

p

, i.e.,

dt' = dt.

How do I show that

p

is moving with constant velocity? I've written down all the relevant equations but nothing is jumping out at me, I'm pretty sure I'm just being stupid. Differentiating

x'

with respect to t (or t') gives

v'=\gamma (v-u)

but I don't even know whether or not that's the right thing to do, let alone what to do next.

Secondly, I have to draw the following as a spacetime diagram:
A clock

C

is at rest at the spatial origin of an inertial frame

S

. A second clock

C'

is at rest at the spatial origin of an inertial frame

S'

moving with constant speed

U

relative to

S

. The clocks read

t = t' = 0

when the two spatial origins coincide. When

C'

reads

t_2'

it receives a radio signal from

C

sent out when

C

reads

t_1

. I know you're meant to post your working for any problems like these but I really don't even know where to start :frown:

I understand what's going on in the second question but I don't know how to draw the space-time diagram to represent it. Help!

1: Use the corresponding Lorentz transformation on the 2-dimensional vector (c dt, dx).
2: Do you know how to draw spacetime diagrams? Make sure you understand how the axes of the frame S' look like in a spacetime diagram where the axes of S are perpendicular.

By the way, who set you these questions? They're from this year's course on Special Relativity for Cambridge's first year mathmos.

edit: Corrected the vector you want to transform.

Reply 2

csmith101

I am a cambridge first year mathmo :smile:

but I've been ill a lot recently so I wasn't able to attend most of Siklos' recent lectures, i've only just recently got hold of the lecture notes but the illness has left me a bit behind :frown:

Thanks a lot though, I'll try and give those hints a go. If anyone else feels like helping that'd be great too! :smile:

Reply 3

csmith101

Right, I've spent a while reading up and I think I've got the hang of it, i'm almost at the end of the sheet - but alas, I have another question!

In a laboratory frame a particle of rest mass m1 has energy E1 , and a second particle of rest mass m2 is at rest. Show that in units where c = 1, the combined energy in the centre-of-momentum frame is

\sqrt{m_1^2+m_2^2+2E_1m_2}

.

For a single particle the invariant mass is

(mc^2)^2=E^2-\|\mathbf{p}c\|^2

, but can we apply the same formula to multiple particles? I.e.

((\sum{m})c^2)^2=(\sum{E})^2-\|\sum{\mathbf{p}}c\|^2

? In which case, wouldn't we just have

\|\sum{\mathbf{p}}c\|^2=0 \rightarrow (\sum{m})c^2=(\sum{E})

so no square root in the answer for E? I assume something must be wrong somewhere...

Reply 4

Y__

csmith101

\sqrt{m_1^2+m_2^2+2E_1m_2}

.

For a single particle the invariant mass is

(mc^2)^2=E^2-\|\mathbf{p}c\|^2

, but can we apply the same formula to multiple particles? I.e.

((\sum{m})c^2)^2=(\sum{E})^2-\|\sum{\mathbf{p}}c\|^2

? In which case, wouldn't we just have

\|\sum{\mathbf{p}}c\|^2=0 \rightarrow (\sum{m})c^2=(\sum{E})

so no square root in the answer for E? I assume something must be wrong somewhere...

Well, the way I did it was averaging the particles and therefore consider the centre of momentum as a particle, i.e. let

M

be the total relativistic mass of the system,

\mathbf{P}=\sum \mathbf{p}

the total relativistic momentum, then

\mathbf{P} = \mathbf{v}_c M

where

\mathbf{v}_c

is the velocity of the c of m frame in the lab frame. Hence

\mathbf{v}_c = \mathbf{P}M^{-1} = \mathbf{P}E^{-1}

where

E

is the total energy of the system (last one works because

c=1

).
Then you can do all the crappy algebra. Don't ask me for a justification for that "consider the system as a particle" stuff, I don't know and I'm rather confused that Dr S seems to think this is obvious when he's never mentioned it in lectures (apart from today, after the sheet deadline, and then only for the non-relativistic case).