A relative URL without a scheme (http: or https:) is valid, per RFC 3986: "Uniform Resource Identifier (URI): Generic Syntax", Section 4.2. If a client chokes on it, then it's the client's fault because they're not complying with the URI syntax specified in the RFC.
Your example is valid and should work. I've used that relative URL method myself on heavily trafficked sites and have had zero complaints. Also, we test our sites in Firefox, Safari, IE6, IE7 and Opera. These browsers all understand that URL format.
Roy Fielding's comment about including a body with a GET request.
Yes. In other words, any HTTP request message is allowed to contain a message body, and thus must parse messages with that in mind. Server semantics for GET, however, are restricted such that a body, if any, has no semantic meaning to the request. The requirements on parsing are separate from the requirements on method semantics.
So, yes, you can send a body with GET, and no, it is never useful to do so.
This is part of the layered design of HTTP/1.1 that will become clear again once the spec is partitioned (work in progress).
....Roy
Yes, you can send a request body with GET but it should not have any meaning. If you give it meaning by parsing it on the server and changing your response based on its contents, then you are ignoring this recommendation in the HTTP/1.1 spec, section 4.3:
...if the request method does not include defined semantics for an entity-body, then the message-body SHOULD be ignored when handling the request.
And the description of the GET method in the HTTP/1.1 spec, section 9.3:
The GET method means retrieve whatever information ([...]) is identified by the Request-URI.
which states that the request-body is not part of the identification of the resource in a GET request, only the request URI.
Update
The RFC2616 referenced as "HTTP/1.1 spec" is now obsolete. In 2014 it was replaced by RFCs 7230-7237. Quote "the message-body SHOULD be ignored when handling the request" has been deleted. It's now just "Request message framing is independent of method semantics, even if the method doesn't define any use for a message body" The 2nd quote "The GET method means retrieve whatever information ... is identified by the Request-URI" was deleted. - From a comment
From the HTTP 1.1 2014 Spec:
A payload within a GET request message has no defined semantics; sending a payload body on a GET request might cause some existing implementations to reject the request.
Best Answer
There's a very simple answer to this: Profile the performance of your web server to see what the performance penalty is for your particular situation. There are several tools out there to compare the performance of an HTTP vs HTTPS server (JMeter and Visual Studio come to mind) and they are quite easy to use.
No one can give you a meaningful answer without some information about the nature of your web site, hardware, software, and network configuration.
As others have said, there will be some level of overhead due to encryption, but it is highly dependent on:
In my experience, servers that are heavy on dynamic content tend to be impacted less by HTTPS because the time spent encrypting (SSL-overhead) is insignificant compared to content generation time.
Servers that are heavy on serving a fairly small set of static pages that can easily be cached in memory suffer from a much higher overhead (in one case, throughput was havled on an "intranet").
Edit: One point that has been brought up by several others is that SSL handshaking is the major cost of HTTPS. That is correct, which is why "typical session length" and "caching behavior of clients" are important.
Many, very short sessions means that handshaking time will overwhelm any other performance factors. Longer sessions will mean the handshaking cost will be incurred at the start of the session, but subsequent requests will have relatively low overhead.
Client caching can be done at several steps, anywhere from a large-scale proxy server down to the individual browser cache. Generally HTTPS content will not be cached in a shared cache (though a few proxy servers can exploit a man-in-the-middle type behavior to achieve this). Many browsers cache HTTPS content for the current session and often times across sessions. The impact the not-caching or less caching means clients will retrieve the same content more frequently. This results in more requests and bandwidth to service the same number of users.